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VIII. Dimension

§ 1. Krull dimension of a ring

Definition 1 Let I be an ordered set. A nonempty finite totally ordered subset of I is called a chain in I. Let c be a chain in I; the smallest and greatest elements of c are called the extremi… 0CJG
Definition 2 The Krull dimension of the topological space X is, by definition, and is denoted by dim kr(X), or simply dim(X), the least upper bound in overlineR of the set of lengths of chains… 0CJH
Proposition 1 Let X be a topological space. a) If Y is a subspace of X, then dim(Y) ≤ dim(X), and dim_y(Y) ≤ dim_y(X) for every point y of Y. b) Let x be a point of X and V a neighbourhood of x… 0CJI
Corollary a) The dimension of the topological space X is the least upper bound of the dimensions of its irreducible components (II, § 4, No. 1, Def. 2). 0CJJ
Proposition 2 Let X be a topological space. Then dim(X) = sup_x ∈ X dimₓ(X). 0CJK
Corollary If (X_α)_α ∈ A is an open covering, or a locally finite closed covering, of a topological space X, one has dim(X) = sup_α ∈ A dim(X_α). 0CJL
Definition 3 Let X be a topological space. 0CJM
Remark 1 The codimension of a closed subset Y of X is therefore the greatest lower bound of the codimensions of the irreducible closed subsets of Y. One has codim(∅, X) = +∞ and, if X is no… 0CJN
Remark 2 If Y is a nonempty closed subset of X, one has codim(Y, X) ≤ dim(X). One has dim(X) = sup_Y codim(Y, X), where Y runs through the set of irreducible closed subsets of X. If Y and Y… 0CJO
Remark 3 Let Y be a closed subset of the topological space X and (X_α)_α ∈ A (resp. (Y_β)_β ∈ B) the family of irreducible components of X (resp. of Y). For each β ∈ B, let A(β) denote the… 0CJP
Remark 4 Let (Yᵢ)_i ∈ I be a finite family of closed subsets of X and Y = ⋃_i ∈ I Yᵢ; one has 0CJQ
Proposition 3 Let X be a topological space. 0CJR
Definition 4 A topological space X is said to be catenary if, for every pair (Y, Z) of irreducible closed subsets of X such that Y ⊂ Z, every saturated chain of irreducible closed subsets with… 0CJS
Proposition 4 Let X be a topological space. In order that X be catenary, it is necessary and sufficient that, for every triplet (Y, Z, T) of irreducible closed subsets of X such that Y ⊂ Z ⊂ T… 0CJT
Corollary Let X be an irreducible topological space of finite dimension. In order that X be catenary, it is necessary and sufficient that, for every pair (Y, Z) of irreducible closed subsets… 0CJU
Proposition 5 Let X be a topological space of finite dimension. Suppose that all maximal chains of irreducible closed subsets of X have the same length. Then X is catenary; for every irreducible… 0CJV
Definition 5 The Krull dimension, or simply dimension, of a (commutative) ring A is by definition the Krull dimension of the topological space Spec(A) (II, § 4, No. 3, Def. 4); it is denoted by… 0CJW
Example 1 One has dim(A) < 0 if and only if A is reduced to 0. In order that one have dim(A) leqslant 0, it is necessary and sufficient that every prime ideal of A be maximal. Integral rings… 0CJX
Example 2 Dedekind rings are noetherian integrally closed rings of dimension leqslant 1 (VII, § 2, No. 2, th. 1). More generally, by V, § 1, No. 2, cor. 2 to prop. 9, a ring is a finite prod… 0CJY
Example 3 If A is a valuation ring (VI, § 1, No. 2, def. 2), its dimension is equal to the height of the valuation (VI, § 4, No. 4, prop. 5). 0CJZ
Example 4 Let A be a ring. One has dim(A[X]) geqslant dim(A) + 1 . In fact, if p₀ ⊂ ... ⊂ pₙ is a chain of prime ideals of A, of length n, one obtains a chain p'₀ ⊂ ... ⊂ p'ₙ₊₁ of prime idea… 0CK0
Example 5 Let X be a complex analytic variety. If X is of complex dimension n at a point x of X, the local ring of germs at x of analytic functions on X is of dimension n. \ 0CK1
Example 6 Let k be a field and A a nonzero integral k-algebra. Then one has dim(A) = 0. This results from cor. 1 to prop. 1 of V, § 2, No. 1, and from the fact that dim(k) = 0. 0CK2
Example 7 If 𝔫 is a nilideal of A, Spec(A) is homeomorphic to Spec(A/𝔫) (II, § 4, No. 3, remark). Therefore one has dim(A/𝔫) = dim(A); in particular, one has dim(A) = dim(A_red) where A_red… 0CK3
Example 8 There exist noetherian rings of infinite dimension (p. 83, exercise 13). We shall see below (§ 3, No. 1, cor. 1 to prop. 2) that every noetherian local ring is of finite dimension. 0CK4
Proposition 6 Let A be a ring. a) If a is an ideal of A, one has dim(A/a) ≤ dim(A). b) If S is a multiplicative subset of A, one has dim(S⁻¹A) ≤ dim(A). c) One has dim(A) = sup dim(A/p), where p… 0CK5
Remark 1 Let ρ : A → B be a homomorphism of rings. Then dim(B) is the least upper bound of the numbers dim(B/ρ(p).B), where p runs through the set of minimal prime ideals of A: in fact, for… 0CK6
Definition 6 Let a be an ideal of a ring A. The codimension of V(a) in Spec(A) is called the height of the ideal a and is denoted by ht(a). 0CK7
Proposition 7 a) The height of a prime ideal p of A is the least upper bound of the lengths of chains of prime ideals p₀ ⊂ ... ⊂ pₙ such that pₙ = p. 0CK8
Corollary Let p be a prime ideal of A and S a multiplicative subset of A not meeting p. Then ht(p) = ht(S⁻¹p). 0CK9
Proposition 8 Let A be a ring. 0CKA
Definition 7 A ring A is said to be catenary if the topological space Spec(A) is catenary (No. 2, Definition 4). 0CKB
Remark 2 Every quotient ring of a catenary ring is catenary. In order that the ring A be catenary, it is necessary and sufficient that, for every prime ideal p of A, the ring Aₚ be catenary… 0CKC
Remark 3 By Proposition 7, b) and Proposition 4 of No. 2, the ring A is catenary if and only if, for every triple (p, q, r) of prime ideals of A such that r ⊂ q ⊂ p, one has dim(Ap/qAp) + d… 0CKD
Remark 4 Let A be a ring of finite dimension, all maximal chains of whose prime ideals have the same length. Then A is catenary, one has ht(p) + dim(A/p) = dim(A) for every prime ideal p of… 0CKE
Remark 5 We shall see in § 2, No. 4, that every finitely generated algebra over a field is a catenary ring. There exist noetherian local rings which are not catenary (p. 83, Exercise 16). 0CKF
Definition 8 Let A be a ring and M a finitely generated A-module. The Krull dimension (or simply dimension ¹) of the A-module M is the Krull dimension of the support of M (II, § 4, No. 4, Defin… 0CKG
Remark 1 Let A be a noetherian ring and M a finitely generated A-module. It is equivalent to say that dim_A(M) ≤ 0, or that the elements of Supp(M) are maximal ideals of A, or that M is of… 0CKH
Remark 2 If M is a finitely generated module over a noetherian ring A, dim_A(M) is the least upper bound of the numbers dim(A/p), where p ranges over the set Ass_A(M) of prime ideals of A a… 0CKI
Proposition 9 Let A be a ring and M a finitely generated A-module. 0CKJ
Remark 3 Under the conditions of prop. 9, c), one has codim(Supp(M), Spec(A)) = inf(codim(Supp(M'), Spec(A)), codim(Supp(M/M'), Spec(A))). This follows from the formula Supp(M) = Supp(M') ∪… 0CKK
Remark For every p ∈ Spec(A), one has z(A/p) = [V(p)]. More generally, let M be a finitely generated A-module, and let (Mᵢ)_0 ≤ i ≤ n be a composition sequence of M such that for 0 ≤ i ≤… 0CKL
Lemma 1 Let M be a finitely generated A-module and d an integer. a) In order that M be of type C_≤ d, it is necessary and sufficient that z(M) ∈ Z_≤ d; the projection z_d(M) of z(M) on Z_d… 0CKM
Proposition 10 The sequences of Z-modules and homomorphisms 0CKN
Example 1 Suppose A noetherian and integral. Then one has Z⁰ = Z.[Spec(A)]; one has C^>0 = C and z⁰(M) = rg(M).[Spec(A)]. The modules of type C^>1 are therefore the torsion modules. 0CKO
Example 2 Suppose A noetherian and integrally closed. Then Z¹ is identified with the group D(A) of divisors of A introduced in Chapter VII (§ 1, No. 3, Theorem 2, and No. 6, Theorem 3). The… 0CKP
Example 3 Modules of type C_≤ 0 are the modules of dimension ≤ 0, that is to say the modules of finite length (No. 4, Remark 1). We have long_A(M) = ε(z₀(M)) for every finite-length A-module… 0CKQ
Example 4 Suppose A is an integral domain and finite-dimensional. Put d = dim(A). Then we have C_≤ d = C, Z_d = Z.[Spec(A)] = Z⁰, z_d(M) = rg(M).[Spec(A)] = z⁰(M), and the modules of type C_… 0CKR

§ 2. Dimension of algebras

Remark 1 The condition (PM) is satisfied when there exists a finitely generated B-module, faithfully flat over A and with support equal to Spec(B). This is the case, in particular, if the A… 0CL9
Remark 2 The existence of a B-module N faithfully flat over A implies the injectivity of ρ (I, § 3, No. 5, prop. 8), and the surjectivity of the mapping “ρ : Spec(B) → Spec(A) (II, § 2, No… 0CLA
Remark 3 Suppose that ρ : A → B is a local homomorphism of local rings and that there exists a B-module N flat over A and such that N ⊗B κ(q) ≠ 0 for every prime ideal q of B. Then N is fai… 0CLB
Proposition 1 Let ρ : A → B be a homomorphism of rings satisfying the condition (PM). 0CLC
Corollary Let F be a closed subset of Spec(A). If Y is an irreducible component of the inverse image of F by the mapping “ρ : Spec(B) → Spec(A), then the closure of “ρ(Y) is an irreducible c… 0CLD
Proposition 2 Let ρ : A → B be a homomorphism of nonzero rings possessing the property (PM). One has the inequality 0CLE
Remark 4 Let ρ : A → B be a local homomorphism of noetherian local rings satisfying condition (PM). We shall see later (§ 3, no 4, prop. 7) that in this case there is equality in (1). In th… 0CLF
Corollary For every ideal a of A, one has ht(a) ≤ ht(ρ(a)B). 0CLG
Lemma 1 Let ρ : A → B be a homomorphism of rings and p a prime ideal of A. The continuous mapping ᵃ h : Spec(B ⊗_A κ(p)) → Spec(B), associated with the canonical homomorphism h : B → B ⊗_A… 0CLH
Remark 5 According to prop. 2 and Lemma 1, we therefore have, under the hypotheses of prop. 2, the inequality dim(Spec(B)) ≥ dim(Spec(A)) + inf_p ∈ Spec(A) dim(αρ⁻¹(p)) . 0CLI
Proposition 3 Let ρ : A → B be a homomorphism of rings. Put n = sup_p ∈ Spec(A) dim(B ⊗_A κ(p)). One has the inequality dim(B) + 1 ≤ (dim(A) + 1).(n + 1) . 0CLJ
Remark 1 If the rings A and B are noetherian, we shall see below (§ 3, no 4, cor. 2 to prop. 7) that one has the inequality dim(B) ≤ dim(A) + n, stronger than that of prop. 3. 0CLK
Corollary 1 Suppose that one has dim(A) < +∞ and that there exists an integer n such that dim(B ⊗_A κ(p)) ≤ n for every p ∈ Spec(A). Then one has dim(B) < +∞. 0CLL
Corollary 2 Let A be a ring and B = A[X] the ring of polynomials in one indeterminate with coefficients in A. One has : 1 + dim(A) ≤ dim(B) ≤ 1 + 2dim(A) . 0CLM
Corollary 3 If A is of finite dimension, every nonzero finitely generated A-algebra is of finite dimension. 0CLO
Lemma 2 Let ρ : A → B be a homomorphism of rings such that, for every prime ideal p of A, the κ(p)-algebra B ⊗ₓₐ κ(p) is integral (V, § 1, No. 1, def. 2). Let q and q’ be two prime ideals… 0CLP
Theorem 1 Let ρ : A → B be a homomorphism of rings making B an integral A-algebra. 0CLQ
Theorem 2 Let A be an integrally closed ring, and B a ring containing A, integral over A. Assume that B is a torsion-free A-module. For every ideal a of A, one has ht(a) = ht(aB). Let b be a… 0CLR
Remark Soient A un anneau intègre et B un anneau contenant A, entier sur A. Soit p un idéal premier de A tel que la clôture intégrale de Aₚ soit un anneau local. On peut démontrer que, po… 0CLS
Lemma 3 Let A be a k-algebra of finite type and p₀ ⊂ ... ⊂ pₘ a maximal chain of prime ideals of A. There exists an integer n ≥ m, a sequence (x₁, ..., xₙ) of elements of A, algebraically… 0CLT
Theorem 3 Let A be a k-algebra of finite type. a) For every minimal prime ideal p of A, all maximal chains of prime ideals of A with origin p have length the integer deg.trₖ κ(p). b) The rin… 0CLU
Corollary 1 Let n be a positive integer. We have 0CLV
Corollary 2 Let A be an integral k-algebra of finite type. For every prime ideal p of A, we have 0CLW
Corollary 3 Let A be a k-algebra of finite type and let f be an element of A which belongs to no minimal prime ideal of A (for example an element of A which is not a divisor of zero, cf. IV, §… 0CLX
Corollary 4 Let A be a k-algebra of finite type and let p be a prime ideal of A. 0CLY
Corollary 5 Let A be a k-algebra of finite type, p a prime ideal of A and (𝔭ᵢ)_i ∈ I the family of minimal prime ideals of A contained in p. We have: 0CLZ
Corollary 6 Let A be a finitely generated k-algebra not reduced to 0, and let n be an integer geqslant 0. The following conditions are equivalent: 0CM0
Remark 1 By cor. 5, the conditions a), b), c) of cor. 6 imply that one has dimₚ(A) = dim(A) for every prime ideal p of A. 0CM1
Proposition 4 Let A and B be two finitely generated k-algebras and ρ : A → B an algebra homomorphism. Suppose that A is integral and that the A-module B is without torsion, and denote by K the f… 0CM2
Corollary Let ρ : A → B be an injective homomorphism of finitely generated k-algebras. One has dim(A) ≤ dim(B). 0CM3
Lemma 4 Let A and B be two integral k-algebras, M an A-module without torsion, N a B-module without torsion. If the ring A ⊗ₖ B is integral, then M ⊗ₖ N is a module without torsion over A… 0CM4
Proposition 5 Let k' be an extension of k, A a k-algebra of finite type and B a k'-algebra of finite type. 0CM5
Corollary Let A be a finitely generated k-algebra, k' an extension of k, and A' the k'-algebra A \otimesk k'. 0CM6
Remark 2 Suppose that the extension k' of k is radicial. Then the canonical mapping f : Spec(A') → Spec(A) is a homeomorphism. 0CM7

§ 3. Dimension of noetherian rings

Proposition 1 Let A be a noetherian integral domain, x a nonzero element of A and p an element minimal in the set of prime ideals of A containing x. Then p is of height 1. 0CMQ
Proposition 2 Let A be a noetherian ring, m a positive integer and a an ideal contained in the radical of A and generated by m elements. One has 0CMR
Lemma 1 Let A be a noetherian ring, p₀ ⊂ ... ⊂ pₙ a chain of prime ideals of A of length n geqslant 1, and x an element of pₙ. There exists a chain p'₀ ⊂ ... ⊂ p'ₙ with p'₀ = p₀, p'ₙ = pₙ… 0CMS
Corollary 1 a) Every noetherian local ring is finite-dimensional. More generally, every nonzero noetherian semi-local ring (II, § 3, No. 5, def. 4) is finite-dimensional. b) Let A be a noether… 0CMT
Corollary 2 Let A be a local noetherian ring. a) Let x ∈ m_A. Then dim(A/xA) is equal to dim(A) or to dim(A) - 1. In order that one have dim(A/xA) = dim(A) - 1, it is necessary and sufficient… 0CMU
Definition 1 Let A be a noetherian local ring, M a nonzero finitely generated A-module and S a subset of the maximal ideal m_A of A. One says that S is secant for M if one has 0CMV
Remark 1 It follows from formulas (6) and (7) that S is secant for M if and only if it is secant for A/α, where α is the annihilator of M. 2) Let S and S’ be two disjoint subsets of m_A. Fo… 0CMW
Proposition 3 Let A be a noetherian local ring and M a nonzero finite a-module. For an element x of m_A to be secant for M, it is necessary and sufficient that it belong to none of the minimal e… 0CMZ
Corollary Every sequence of elements of m_A which is completely secant for M (A, X, p. 157, Definition 2) is secant for M. Let (x₁, ..., xᵣ) be a sequence of elements of m_A which is complet… 0CN0
Remark 4 It is not true in general that a sequence secant for M is completely secant for M (p. 87, Exercise 6). We shall study later the modules over a noetherian local ring for which every… 0CN1
Theorem 1 Let A be a noetherian local ring, M a nonzero finite A-module and S a subset of the maximal ideal m_A of A. 0CN2
Corollary The dimension of M is the smallest of the integers d geqslant 0 for which there exists a sequence (x₁, ..., x_d) of elements of m_A such that the A-module M / ∑ᵢ₌₁ᵈ xᵢ M is of fini… 0CN3
Lemma 2 Let A be a noetherian local ring and q an ideal of A. The following conditions are equivalent: 0CN4
Scholium The dimension of a noetherian local ring A is the smallest of the integers d ≥ 0 for which there exists an ideal of definition of A generated by d elements. 0CN5
Proposition 4 Let X be a closed subset of V, and H₁, ..., Hₘ hypersurfaces in V. Put X' = X ∩ H₁ ∩ ... ∩ Hₘ. 0CN6
Corollary 1 Let H be a nonempty hypersurface in V. The codimension of H in V is equal to 0 or 1. 0CN7
Corollary 2 Let X be a closed irreducible subset of V and H a hypersurface in V. Only three cases are possible: 0CN8
Corollary 3 If A is factorial (VII, § 3, No. 3, Prop. 2), the prime ideals of height 1 of A are the principal ideals generated by the extremal elements of A. If moreover A is local, one has di… 0CN9
Remark 1 Suppose that A is a local noetherian integral domain of dimension d; let x be a nonzero element of mA and let H = V(x). By Corollary 2 of Prop. 4, every irreducible component of H… 0CNA
Remark 2 Let x₁, ..., xₙ be elements of A, and put Hᵢ = V(xᵢ) for 1 ≤ i ≤ n. Suppose there exists a finite-type A-module M with support V = Spec(A), such that (x₁, ..., xₙ) is a completely… 0CNB
Remark 3 If the ideal α of the noetherian ring A is generated by m elements, one has ht(α) ≤ m. This follows at once from Prop. 4. 0CNC
Proposition 5 Let A be a noetherian ring and p ⊂ q a non-saturated chain of prime ideals of A. The set E of prime ideals r of A such that p ⊂ r ⊂ q is a chain is infinite. One has ⋃_r ∈ E r = q… 0CND
Proposition 6 Let A be a noetherian ring of dimension ≥ 2, and h an integer such that 0 < h < dim(A). 0CNE
Proposition 7 Let ρ : A → B be a local homomorphism of noetherian local rings, M a finitely generated A-module and N a finitely generated B-module. Put overlineB = B ⊗_A κ_A = B/ρ(m_A).B and ove… 0CNF
Corollary 1 Let ρ : A → B be a local homomorphism of noetherian local rings. 0CNG
Corollary 2 Let ρ : A → B be a homomorphism of noetherian rings. Then 0CNH
Corollary 3 Let A be a noetherian ring and n an integer geqslant 0. Then we have 0CNI
Corollary 4 Let ρ : A → B be a homomorphism of noetherian rings, and a an ideal of A. We have the inequality 0CNJ
Proposition 8 Let A be a noetherian ring, a an ideal of A, M a finitely generated A-module, hatA and hatM the separated completions of A and M respectively for the a-adic topology. 0CNK
Corollary 1 Let A be a Zariski ring (III, § 3, No. 3, Definition 2). For every A-module M of finite type, one has dim_hatA(hatM) = dim_A(M). 0CNL
Corollary 2 Let A be a noetherian ring, a an ideal of A, and hatA the separated completion of A for the a-adic topology. One has dim(hatA) leqslant dim(A), with equality when A is local and a… 0CNM
Corollary 3 Let A be a noetherian ring and n an integer geqslant 0. One has 0CNN
Remark 1 Let A be a noetherian ring and a an ideal of A. Suppose that A is separated and complete for the a-adic topology, and consider the ring R = A X₁, ..., Xₙ of restricted formal power… 0CNO
Remark 2 Let A be a local noetherian ring, identified with a subring of its completion hatA, and B a subring of hatA containing A. Suppose that B is local noetherian and that one has m_A B… 0CNP
Proposition 9 Let A be a noetherian ring, M a finitely generated A-module, a a subset of A stable under addition and multiplication, q₁, ..., qₘ prime ideals of A not containing a, and r ≥ 1 an… 0CNQ
Corollary 1 Let A be a noetherian ring, M a finitely generated A-module, p₁, ..., pₙ, q₁, ..., qₘ prime ideals of A and r an integer geqslant 1. Suppose that one has pᵢ not⊂ qⱼ for 1 leqslant… 0CNR
Corollary 2 Let A be a noetherian local ring, M a nonzero finitely generated A-module and α a subset of m_A, stable under addition and multiplication and such that long(M/α M) < +∞. There exis… 0CNS

§ 4. Hilbert-samuel series

Remark The ring A((T)) is naturally identified with the ring of fractions A[[T]]_T of the ring A[[T]] defined by the multiplicative subset formed by the powers of T. 0CO0
Lemma 1 The element 1 - T of Z((T)) is invertible. For every integer r > 0 one has 0CO1
Lemma 2 a) Let F be a nonzero element of Z((T)) such that there exists r ∈ Z, with (1-T)ʳ F ∈ Z[T, T⁻¹]; then F can be written uniquely in the form F = (1-T)⁻ᵈ.Q, where Q ∈ Z[T, T⁻¹], Q(1)… 0CO2
Theorem 1 The element Q_M of Z((T)) belongs to Z[T, T⁻¹]. 0CO3
Example 1 Suppose H₀ artinian and take M = H. Then, with the preceding notations, we have R = 0 and S = H', hence by (5), Q_H = Q_H'; since we have Q_H₀ = long(H₀), we infer by induction tha… 0CO4
Corollary a) If r = 0, then we have long_H₀(M) = c_M. 0CO5
Remark 1 The integer c_M is positive by lemma 2. One may have M ≠ 0 and c_M = 0 (cf. prop. 2). 0CO6
Remark 2 Let 0 → M' → M → M'' → 0 be an exact sequence of graded H-modules and homomorphisms of degree 0 such that M is of finite type over H and Mₙ of finite length over H₀ for each n. The… 0CO7
Remark 3 Let M(p) be the module deduced from M by shifting the grading by p (A, II, p. 165, example 3). Since we have M(p)ₙ = Mₚ₊ₙ, we have P_M(p) = T⁻ᵖP_M, Q_M(p) = T⁻ᵖQ_M and c_M(p) = c_M… 0CO8
Example 2 Suppose H₀ artinian, and let M be a free graded H-module generated by s homogeneous, linearly independent elements, of respective degrees δ₁, ..., δₛ. Then M is isomorphic to H(-δ₁… 0CO9
Example 3 Suppose again H₀ artinian; let M be a graded H-module, and suppose that there exists an exact sequence of graded H-modules and homomorphisms of degree 0 0 → Lₙ → Lₙ₋₁ → ⋯ → L₀ → M… 0COA
Remark 4 One can prove (p. 88, exerc. 4) that under the hypotheses of th. 1, the H₀-modules Torⱼᴴ(H₀, M) are of finite length, are zero for j > r, and that one has c_M = ∑ⱼ₌₀ʳ (-1)ʲ long_H₀… 0COB
Proposition 1 Let M be a graded H-module. Suppose that M is generated by M₀ and that M₀ is an H₀-module of finite length. Then one has P_M ≤ (1 - T)⁻ʳ long_H₀(M₀), c_M ≤ long_H₀(M₀). Moreover, t… 0COC
Remark 5 According to A, X, p. 160, th. 1, condition (iii) means that (X₁, ..., Xᵣ) is a completely secant sequence for the H-module M. 0COD
Proposition 2 Suppose that H₀ is a field, and let M be a finitely generated graded H-module. Let K be the field of fractions of H. Then c_M is equal to the rank of the H-module M, that is to say… 0COE
Remark 6 Under the hypotheses of Prop. 2, one therefore has c_M = 0 if and only if M is a torsion H-module, or again if and only if dim_H(M) < r (§ 1, No. 5, example 4). 0COF
Lemma 3 If F and F' are two q-good filtrations on M, there exists an integer m such that F'ₙ ⊂ Fₙ₋ₘ for every n ∈ Z. In fact, there exists n₂ such that F'ₙ ⊂ q^n-n₂M for every n, hence F'ₙ… 0COG
Lemma 4 Let F be a q-good filtration on M. If M/qM is of finite length, M/Fₙ₊₁ and Fₙ/Fₙ₊₁ are of finite length for every n ∈ Z. With the notation of (7), we have long(M/Fₙ₊₁) ≤ long(M/q^n… 0COH
Definition 1 The series H_M,F is called the Hilbert-Samuel series of the A-module M (relative to the q-good filtration F). 0COI
Proposition 3 a) If F is a q-good filtration on M, we have 0COJ
Theorem 2 Let A be a noetherian ring, q an ideal of A, M a finitely generated A-module such that M/qM is nonzero and of finite length, and F a q-good filtration on M. 0COK
Remark 1 With the notation of a), let us write R = ∑_i ∈ Z aᵢ Tⁱ, and suppose d > 0. By No. 1, the relation H_M,F = (1 - T)⁻ᵈ R may also be written (11) long_A(Fₙ/Fₙ₊₁) = ∑_i ∈ Z aᵢ binomn… 0COL
Remark 2 To say that M/qM is of finite length means that Supp(M/qM) = Supp(M) ∩ V(q) consists of maximal ideals (IV, § 2, No. 5, Proposition 7). We shall see below (No. 4, corollary to Theo… 0COM
Corollary Let A be a noetherian ring, q an ideal of A, M a finitely generated A-module such that M/qM is of finite length, and F a q-good filtration on M. a) In order that one have d_q(M) ≤… 0CON
Remark 3 Suppose q is contained in the radical of A. Then, by Nakayama's lemma, the sequence (q^nM) is stationary if and only if one has q^nM = 0 for n sufficiently large. It then follows f… 0COO
Proposition 4 Let A be a noetherian ring, x1, ..., xr elements of A, x the ideal which they generate, and M a finitely generated A-module such that M/xM is nonzero and of finite length. a) One h… 0COP
Proposition 5 Let 0 \to M' \to M \to M'' \to 0 be an exact sequence of finitely generated modules over a noetherian ring A, and q an ideal of A. a) In order that M/qM be of finite length, it is… 0COQ
Theorem 3 Let A be a noetherian local ring, q an ideal of A distinct from A, and M a finitely generated A-module such that M/qM is of finite length. Then the integer d_q(M) is the dimension… 0COR
Corollary Let A be a noetherian ring, M a finitely generated A-module and q an ideal of A such that M/qM is of finite length. Then d_q(M) is the least upper bound of the dimensions dim_Aₘ(Mₘ… 0COS
Remark 1 One also has d_q(M) = sup_m ∈ V(q) dim(Mₘ), that is, d_q(M) = dim(hatM), where hatM is the completion of M for the q-adic topology (§ 3, No. 4, prop. 8). 0COT
Remark 2 Suppose q contained in the radical of A; then dim(hatM) = dim(M) (loc. cit., cor. 1), hence d(M) = dim(M). 0COU
Lemma 5 Let A be a ring, M an A-module and (Pₙ), (Qₙ) two decreasing filtrations on M consisting of submodules. Suppose that one has Pₙ ⊃ Qₙ and long_A(Pₙ/Qₙ) < +∞ for every n ∈ Z and that… 0COV
Lemma 6 Let A be a ring, M an A-module and (Fₙ) a decreasing filtration on M consisting of submodules; suppose that there exists an integer n₁ such that F_n₁ = M. Let f be an endomorphism… 0COW
Lemma 7 Retain the notation of lemma 6. 0COX
Proposition 6 Let A be a noetherian ring, M a finitely generated A-module and q an ideal of A such that M/qM is of finite length. Let F be a q-good filtration of M, and let gr(A) = ⨁_n ≥ 0 (qⁿ/q… 0COY

§ 5. Regular local rings

Definition 1 The noetherian local ring A is said to be regular if one has dim(A) = [𝔪_A/𝔪_A² : κ_A]. A system of coordinates of A is then any family of elements of 𝔪_A whose classes modulo 𝔪_A²… 0CPB
Example 1 Regular noetherian local rings of dimension 0 (resp. 1) are fields (resp. discrete valuation rings) (VI, § 3, No. 6, Prop. 9 and Cor. 1 of Theorem 1 below). Let A be a discrete val… 0CPC
Example 2 Let k be a field and let n be a positive integer. The ring of formal power series k[[X₁, ..., Xₙ]] is a regular noetherian local ring of dimension n (§ 3, No. 4, Cor. 3 to Prop. 8)… 0CPD
Example 3 Let A be a complete regular noetherian local ring of dimension r. The ring A X₁, ..., Xₙ of restricted formal power series is a regular noetherian local ring of dimension r + n (§… 0CPE
Example 4 Let k be a complete non-discrete valued field. The ring of formal power series in n variables which converge in a neighbourhood of 0 in kⁿ is a regular local noetherian ring of dim… 0CPF
Example 5 Let k be a field, A an integral k-algebra of finite type, and m a maximal ideal of A. The local noetherian ring Aₘ is regular if and only if one has dim(A) = [m/m² : A/m]; in fact… 0CPG
Example 6 Let X be an algebraic variety over a perfect field k. Then X is nonsingular at a point x if and only if the local ring of X at x is regular. 0CPH
Example 7 Let A be a regular local noetherian ring. We shall see later that the local noetherian ring Aₚ is regular for every prime ideal p of A. 0CPI
Proposition 1 Let A and B be local noetherian rings and ρ : A → B a local homomorphism making B a flat A-module. Suppose that one has mB = B.\rho(mA). Then dim(A) = dim(B) and B is regular if an… 0CPJ
Corollary A local noetherian ring A is regular if and only if its completion hatA is so. 0CPK
Theorem 1 Let A be a local noetherian ring. The following conditions are equivalent: (i) A is regular. (ii) The ideal m_A is generated by a secant subset for A (§ 3, No. 2, Def. 1). (iii) Th… 0CPL
Corollary 1 Every regular noetherian local ring is integrally closed, and in particular an integral domain. 0CPM
Corollary 2 Let A and B be noetherian local rings and σ a local homomorphism of B into A. Suppose that A is regular and B complete. In order that σ be bijective, it is necessary and sufficient… 0CPN
Corollary 3 Let k be a field and A a noetherian local k-algebra whose residue field is equal to k. In order that A be regular, it is necessary and sufficient that its completion hatA be isomor… 0CPO
Proposition 2 Let A be a noetherian local ring, x = (x₁, ..., xᵣ) a sequence of elements of m_A and x the ideal generated by x. The following conditions are equivalent: 0CPP
Corollary 1 Let A be a noetherian local ring, and t an element of m_A. The following conditions are equivalent: 0CPQ
Corollary 2 Let A be a regular noetherian local ring, and q an ideal of A. Then A/q is regular if and only if q is generated by a subset of a system of coordinates of A. 0CPR
Example Let k be a field, A = k[[X₁, ..., Xₙ]] and q an ideal of A, distinct from A. In order that A/q be regular, it is necessary and sufficient that one can find an integer r geqslant 0… 0CPS
Remark Let A be a regular noetherian local ring and q ⊂ m_A an ideal of A such that A/q is regular. Let (x₁, ..., xᵣ) be a sequence of elements of q whose classes modulo m_A² generate the… 0CPT
Definition 2 Let A be a ring, p a prime ideal of A, and P a polynomial in A[T]. One says that P is an Eisenstein polynomial for p if it satisfies the following conditions: a) P is monic of degr… 0CPU
Proposition 3 Let A be a ring, p a prime ideal of A and P ∈ A[T] a polynomial of Eisenstein for p. a) There does not exist a decomposition of the form P = P₁ P₂ where P₁ and P₂ are two monic pol… 0CPV
Proposition 4 a) The ring B is local and noetherian, with maximal ideal 0CPW
Corollary Let A be a regular noetherian local ring and let P ∈ A[T] be an Eisenstein polynomial for m_A. The ring B = A[T]/(P) is a regular noetherian local ring, of the same dimension as A… 0CPX
Proposition 5 Let A be an integral domain, with field of fractions K, and let L be an algebraic extension of finite degree of K. We denote by B the integral closure of A in L and by p a prime id… 0CPY
Corollary Suppose that Ap is a discrete valuation ring. Then Bq is a discrete valuation ring, t is a uniformizer of Bq, and one has 0CPZ
Example 1 Put A = \mathbf{Z} and L = \mathbf{Q}(p^{1/d}), where p is a prime number and d an integer \geqslant 2. Let B denote the integral closure of \mathbf{Z} in L. Since the polynomial T… 0CQ0
Example 2 Put A = \mathbf{Z} and L = R{pf}(\mathbf{Q}) where p is a prime number and f an integer \geqslant 1 (cf. A, V, p. 78). Thus L = \mathbf{Q}(\zeta) with \zeta = \exp(2\pi i/p^f). Let… 0CQ1
Proposition 6 Let A be a complete regular noetherian local ring, whose residue field κ_A has characteristic 0. Put n = dim(A). Then A is isomorphic to the ring of formal power series κ_A[[X₁… 0CQ2
Example 3 Let k be a field of characteristic p ≠ 0 and let n be a positive integer. The ring of formal series k[[X₁, ..., Xₙ]] is complete regular local noetherian, of dimension n, and k is… 0CQ3
Example 4 Let V be a complete discrete valuation ring and let n be a positive integer. The ring of formal series Vₙ = V[[X₁, ..., Xₙ]] is complete regular local noetherian, of dimension n +… 0CQ4
Example 5 Let us keep the preceding notation. A polynomial P in Vₙ[T] of degree d ≥ 2 is said to be special if it is of the form Tᵈ + ∑ᵢ₌₁ᵈ aᵢ Tᵈ⁻ⁱ, where a₁, ..., a_d-1 belong to m_Vₙ, a_d… 0CQ5
Theorem 2 Let A be a complete regular local noetherian ring whose residue field is of characteristic ≠ 0, and let V be a Cohen subring of A. Put n = dim(A). 0CQ6

§ 6. Dimension of graded rings

Proposition 1 a) The ideals S⁻¹H_≥ n form an exhaustive and separated filtration of the ring S⁻¹H. 0CR3
Remark 1 An element h/s of S⁻¹H with h ∈ H, s ∈ S, is invertible if and only if the component of degree 0 of h is invertible in H₀. Consequently, if the ring H₀ is local, the ring S⁻¹H is l… 0CR4
Remark 2 Suppose that H is generated by H₀ and H₁; then for every n, one has Hₙ₊₁ = H₁.Hₙ, hence H_≥ n+1 = H₁.H_≥ n and S⁻¹H_≥ n+1 = H₁.S⁻¹H_≥ n. It follows that the filtration (S⁻¹H_≥ n) o… 0CR5
Example 1 \ Let p be a graded prime ideal of C[X₀, ..., Xₙ] different from the ideal generated by the Xᵢ; let V be the algebraic subvariety of Pⁿ(C) defined by p and C the algebraic subvarie… 0CR6
Proposition 2 a) We have dim_gr(H) ≤ dim(S⁻¹H) ≤ dim(H). b) For every graded prime ideal p of H, we have ht_gr(p) ≤ ht(S⁻¹p) = ht(p). 0CR8
Lemma 1 a) If p is a prime ideal of H, pᵍʳ is a prime ideal. 0CR9
Lemma 2 Let p and q be prime ideals of H such that q ⊂ p and q ≠ p. If qᵍʳ = pᵍʳ, then q is graded, p is not, and ht(p/q) = 1. 0CRA
Proposition 3 Let p be a prime ideal of H. If p ≠ pᵍʳ, one has ht(pᵍʳ) = ht(p) - 1. 0CRB
Theorem 1 Suppose that H is noetherian. 0CRC
Remark 2 There exist non-noetherian graded rings H such that dimgr(H) < dim(H) (p. 99, Exercise 1). 0CRD
Lemma 3 Suppose that H is generated by H₀ ∪ H₁ and M generated by ⨁_i ≤ n₀ Mᵢ for a suitable integer n₀. Then the filtration (S⁻¹M_≥ n) on S⁻¹M is good for the ideal S⁻¹H_≥ 1 of S⁻¹H. 0CRE
Proposition 4 Suppose that H is noetherian and M finitely generated. Then dim_H(M) = dim_S⁻¹H(S⁻¹M). 0CRF
Proposition 5 Suppose H₀ local and artinian, H generated by H₀ ∪ H₁, H₁ a finitely generated H₀-module, M nonzero and a finitely generated H-module. Then Mₙ is an H₀-module of finite length for… 0CRG
Remark With the exception of the determination of the integer d, proposition 5 follows directly from Theorem 1 of § 4, No. 2. 0CRH
Corollary Let A be a local noetherian ring and q an ideal of definition of A. Then one has dim(A) = dim(gr_q(A)). 0CRI
Proposition 6 Suppose H₀ local and artinian, H finitely generated as an H₀-algebra and M finitely generated as an H-module. 0CRJ
Lemma 4 Let b be an element of H, homogeneous of degree > 0, and belonging to none of the minimal elements p of Supp(M) such that dim(H/p) = dim_H(M). Then dim_H(M/bM) = dim_H(M) - 1. 0CRK
Corollary 1 Suppose that H₀ is a field and that H is a finitely generated H₀-algebra. Put n = dim(H). There exist homogeneous elements a₁, ..., aₙ of H, all of the same degree > 0, such that t… 0CRL
Remark Let (h₁, ..., hᵣ) be a finite generating system of the vector space H₁ over H₀. For λ = (λ₁, ..., λᵣ) ∈ H₀ʳ, put h_λ = λ₁ h₁ + ⋯ + λᵣ hᵣ. The proofs of Prop. 6 and Corollary 1 enta… 0CRM
Corollary 2 Let A be a noetherian local ring and n ∈ N. In order that one may have dim(A) ≥ n, it is necessary and sufficient that for every integer r ≥ 0, one have [m_Aʳ/m_Aʳ⁺¹ : κ_A] ≥ binom… 0CRN
Lemma 5 Let A be a ring, r its radical, R = ⨁_i ∈ Z Rᵢ a graded A-algebra, M = ⨁_i ∈ Z Mᵢ a graded R-module. Suppose that each Mᵢ is a finitely generated A-module and that M/rM is a finite… 0CRO
Lemma 6 Let ρ : B → C be a ring homomorphism and S a multiplicative subset of B. Suppose that C is a B-algebra of finite type, and that S⁻¹C is a finite S⁻¹B-algebra. Then there exists f ∈… 0CRP
Proposition 7 Suppose that H is an H₀-algebra of finite type. Then the function p ↦ dim(H ⊗_H₀ κ(p)) is upper semicontinuous on Spec(H₀). 0CRQ
Remark 1 \ In algebraic geometry, Prop. 7 implies that the dimension of the fibres of a projective morphism of algebraic varieties is upper semi-continuous. \ 2) We shall see later that if… 0CRR
Proposition 8 a) One has dim(H) ≤ [H₊/H₊² : H₀]. b) The following conditions are equivalent: (i) dim(H) = [H₊/H₊² : H₀]; (ii) the noetherian local ring S⁻¹H is regular; (iii) H is generated as a… 0CRT
Remark 1 With the notation of c, let dᵢ be the degree of aᵢ (1 ≤ i ≤ n). Then the Poincaré series P_H = ∑_n ∈ Z [Hₙ : H₀]. Tⁿ is equal to ∏ᵢ (1 - T^dᵢ)⁻¹ (§ 4, No. 2, example 1). Consequent… 0CRU
Remark 2 Conversely, the fact that there exist integers dᵢ > 0 such that P_H = ∏ᵢ (1 - T^dᵢ)⁻¹ does not imply that H is a graded polynomial algebra; for example, if H is generated by an ele… 0CRV

§ 7. Multiplicities

Definition 1 The integer e_q(M) is called the multiplicity of the A-module M relative to the ideal q. 0CS8
Remark 1 If q’ is an ideal of A contained in the radical of A and containing q, one has e_q'(M) leqslant e_q(M) and, if the q’-adic filtration of M is q-good, one has e_q'(M) = e_q(M) (§ 4… 0CS9
Proposition 1 Let hatA and hatM be the completions of A and M for their q-adic topologies; then e_qᴬ(M) = e_qhatA(hatM). 0CSD
Proposition 2 Suppose that A is local regular (§ 5, No. 1, def. 1); one has e(A) = 1. This follows from th. 1 of § 5, No. 2. 0CSE
Remark 5 It may happen that e(A) = 1 without A being regular (p. 104, exerc. 5). In fact, a local noetherian ring A is regular if and only if hatA is integral and if one has e(A) = 1 (p. 10… 0CSF
Example By definition one has e_q,r(M) = rᵈ e_q(M) where d = dim_A(M). Consequently, if A is local regular, one has e_m_Aʳ(A) = rᵈ. For example, if A is a discrete valuation ring, one has… 0CSG
Proposition 3 Let M be a finitely generated A-module, of dimension d ≥ 0. Let Φ be the set of minimal elements p of Supp(M) such that dim(A/p) = d. Let q be an ideal of A, contained in the radic… 0CSH
Corollary Suppose A semi-local and let q be an ideal of definition of A. a) One has e_q(A) = ∑ₚ e_q(A/p), where p runs through the set of minimal prime ideals of A such that dim(A/p) = dim(A… 0CSI
Proposition 4 Let ρ : A → B be a local homomorphism of local noetherian rings, and let N be a finitely generated B-module, flat over A, and such that N ⊗A κA is a B-module of finite length. If M… 0CSJ
Corollary Suppose that B is flat over A and that ρ(mA) B = mB. Then 0CSK
Example \ Let X be a complex algebraic variety, O_X,x the local ring of X at a rational point x, Xᵃⁿ the analytic space associated with X; let x again denote the point of Xᵃⁿ corresponding… 0CSL
Proposition 5 Suppose A semi-local, and let ρ : A → B be a ring homomorphism making B a finitely generated A-module. Let N be a nonzero finitely generated B-module, and let q be an ideal of A co… 0CSM
Corollary If [B/mᵢ : A/ρ⁻¹(mᵢ)] = 1 for every i, one has e_qᴬ(N) = e_qBᴮ(N). 0CSN
Lemma 1 Let ρ : A → B be a ring homomorphism, and let p be a prime ideal of A. Consider the following two properties: (i) the canonical homomorphism tildeρ of Aₚ into Aₚ ⊗_A B is bijective… 0CSO
Lemma 2 Suppose that A is semi-local; let q be an ideal of definition of A, and let ρ : A → B be a ring homomorphism making B into a finitely generated A-module. Suppose that, for every pr… 0CSP
Proposition 6 Suppose A semi-local and reduced; let q be an ideal of definition of A; let A' be the total ring of fractions of A, and let B be a finite A-subalgebra of A'. Then B is semi-local a… 0CSQ
Example Let k be a field of characteristic ≠ 2 and take for A the local ring k[[X, Y]]/(X² + Y²) with residue field k. Take B = k[[X, T]]/(T² + 1) where T = Y/X. Distinguish two cases: if… 0CSR
Proposition 7 Suppose A local. Let s be an integer ≥ 1 and, for 1 ≤ i ≤ s, let δᵢ be an integer > 0, xᵢ an element of m_A^δᵢ, and ξᵢ its class in m_A^δᵢ/m_A^δᵢ+1. Suppose that (x₁, ..., xₛ) is a… 0CSS
Remark One can prove conversely (cf. p. 103, exerc. 4) that if A is regular and e(A/x) = δ₁ ... δₛ, then the sequence (ξ₁, ..., ξₛ) is completely secant. 0CST
Example Let A be a ring of formal series k[[X₁, ..., Xₙ]] over a field k; let F₁, ..., Fₛ be elements of A, a the ideal which they generate, and B = A/a. Let P₁, ..., Pₛ ∈ k[X₁, ..., Xₙ] b… 0CSU
Proposition 8 Let δ > 0 be an integer, x an element of q^δ, ξ its class in gr_δ(A) = q^δ/q^δ+1 and φ the multiplication by ξ in the gr(A)-module gr(M). 0CSV
Lemma 3 Let R be a noetherian graded ring with degrees ≥ 0, E a graded R-module of finite type such that Eₙ is an R₀-module of finite length for every n ∈ Z. The following conditions are e… 0CSW
Proposition 9 Let us denote by p₁, ..., pᵣ those prime ideals of the graded ring gr(A) = ⨁ₙ (qⁿ/qⁿ⁺¹) which are associated with the graded module gr(M) = ⨁ₙ (qⁿ M/qⁿ⁺¹ M) and do not contain gr₁(… 0CSX
Definition 2 Let A be a noetherian ring, q an ideal of A contained in the radical of A, and M a finitely generated A-module such that M/qM is of finite length. An element x of A is said to be s… 0CSY
Remark 1 Let δ be an integer > 0. One sometimes says that an element x of A is superficial of order δ for M relative to q if x ∈ q^δ, and if, for all n sufficiently large, the mapping qⁿ M/… 0CSZ
Theorem 1 Let A be a noetherian ring, q an ideal of A contained in the radical of A and M a finitely generated A-module such that M/qM is of finite length. Let x₁, ..., xₘ be a finite sequen… 0CT3
Corollary Suppose that A is local, with infinite residue field, and set d = dim_A(M). There exists a sequence x₁, ..., x_d of elements of q such that, on setting x = Ax₁ + ⋯ + Ax_d, one has 0CT4
Remark 5 In the situation of the preceding corollary, one has 0CT5

IX. Complete noetherian local rings

§ 1. Witt vectors

Proposition 1 Let A be a filtered ring and (Jₙ)_n ∈ Z its filtration. Assume that one has J₀ = A and p.1_A ∈ J₁. Let m and n be integers such that m geqslant 1 and n geqslant 0, and let a₀… 0CTV
Lemma 1 If x and y are two elements of A congruent modulo Jₘ, one has x^pⁿ ≡ y^pⁿ mod Jₘ₊ₙ . By induction on n, one reduces to the case where n = 1. Let P denote the polynomial ∑ᵢ₌₀ᵖ⁻¹ Xⁱ… 0CTW
Lemma 2 Let A be a ring endowed with an endomorphism σ satisfying σ(a) ≡ aᵖ mod. p · A for every a ∈ A. Let n geqslant 1 be an integer and let a₀, ..., aₙ₋₁ be elements of A. Put uᵢ = Φᵢ(a… 0CTX
Proposition 2 Let A be a ring. 0CTY
Remark Let a = (aₙ)_n ∈ N and u = (uₙ)_n ∈ N be elements of Aᴺ such that u = Φ_A(a), and let m be an integer geqslant 0. From (10) one deduces the following assertions: 0CTZ
Example 1 We have 0CU0
Example 2 We have 0CU1
Example 3 When p ≠ 2, we have Iₙ = - Xₙ. For p = 2, we have 0CU2
Example 4 We have 0CU3
Remark Let J be the set of integers j ≥ 1. For every element j of J, define the polynomial φⱼ of Z[(Xⱼ)_j ∈ J] by the formula 0CU4
Lemma 3 Let A be a ring. There exists a surjective homomorphism of rings ρ : B → A, where B is a ring satisfying the following conditions: p is not a divisor of 0 in B, and there exists an… 0CU5
Theorem 1 a) Let A be a (commutative) ring. Endowed with the addition S_A and the multiplication P_A, W(A) is a (commutative) ring. The identity element for the addition is the sequence 0_A… 0CU6
Definition 1 Let A be a ring. The ring W(A) is called the ring of Witt vectors with coefficients in A. 0CU7
Remark Let us take up again the notations of the remark of No. 3. Let A be a ring. If a and b are elements of Aᴶ and r = (rⱼ)_j ∈ J is an element of Aᴶ, we denote by r_A(a, b) the element… 0CU8
Proposition 3 Let A be a ring. a) The mapping F_A is an endomorphism of the ring W(A). b) The mapping V_A is an endomorphism of the additive group underlying the ring W(A). c) For every a in W(A… 0CU9
Remark For the definition of mappings analogous to the mappings F and V, in the case of the universal Witt ring, see exercises 36, 37 and 38, p. 52 and following. 0CUA
Lemma 4 Let A be a ring and m geqslant 1 an integer. One has 0CUB
Proposition 4 Let a and b be in A and let x = (xₙ)_n ∈ N be an element of W(A). 0CUC
Definition 2 Let A be a ring and let n ≥ 1 be an integer. We denote by Wₙ(A) the quotient ring W(A)/Vₙ(A). 0CUD
Example 1 The homomorphism Φ₀ : W₁(A) → A is an isomorphism. 0CUE
Example 2 Let us make explicit the operations in W₂(A). We have 0CUF
Example 3 Let n ≥ 1 be an integer. If a₀, ..., aₙ₋₁, b₀, ..., bₙ₋₁ are integers such that aᵢ ≡ bᵢ bmod p for 0 ≤ i < n, one has (No. 1, prop. 1) 0CUG
Proposition 5 Let A be a ring of characteristic p (A, V, p. 2). Whatever the elements a and b of W(A), and the positive integers m, n, one has, if a = (aₙ)_n ∈ N, 0CUH
Corollary If m and n are two positive integers, one has 0CUI
Proposition 6 Let A be a ring. 0CUJ
Proposition 7 Let A be a perfect ring of characteristic p. 0CUK
Proposition 8 Let A be a field of characteristic p. The ring W(A) is a separated and complete integral local ring, with maximal ideal V₁(A) and residue field isomorphic to A. If the field A is p… 0CUL
Remark 1 Let A be a field of characteristic p. One can now prove that the ring W(A) is noetherian if and only if A is perfect (p. 43, Exercise 9). 0CUM
Remark 2 Let A be a ring of characteristic p. According to Prop. 5, one has the formulas 0CUN

§ 2. Cohen rings

Definition 1 A ring C is said to be a p-ring if the ideal pC of C is maximal, and if C is separated and complete for the pC-adic topology. 0CWA
Proposition 1 Let C be a p-ring. 0CWB
Corollary 1 If the p-ring C is an integral domain, it is a discrete valuation ring, or a field of characteristic p. 0CWC
Corollary 2 Let C be a p-ring and a an ideal of C distinct from C. The ring C/a is a p-ring. 0CWD
Example 1 For every integer n geqslant 1, the ring Z/pⁿZ is a p-ring of length n. The ring Zₚ of p-adic integers is a p-ring of infinite length. 0CWE
Example 2 Let K be a perfect field of characteristic p. By prop. 8 of § 1, No. 8, the ring W(K) of Witt vectors is a p-ring of infinite length. The mapping (aₙ)_n ∈ N ↦ a₀ induces, by passin… 0CWF
Proposition 2 Let C and C' be two p-rings and u a homomorphism of C into C'. Let v be the homomorphism of κ_C = C/pC into κ_C' = C'/pC' deduced from u by passing to quotients. 0CWG
Proposition 3 Let (Cₙ, π_n,m) be a projective system of rings relative to the set of indices N. Suppose that Cₙ is a p-ring for every n ∈ N and that the homomorphisms π_n,m are surjective. Then… 0CWH
Definition 2 Let A be a separated and complete local ring, whose residue field is of characteristic p. A Cohen subring of A means a subring C of A which is a p-ring such that A = m_A + C (i.e… 0CWI
Example Let C be a p-ring. The ring of formal series A = C[[T₁, ..., Tₙ]] is a noetherian, local, separated and complete ring, whose maximal ideal is generated by the sequence (p, T₁… 0CWJ
Theorem 1 Let A be a local, separated and complete ring, whose residue field k is of characteristic p. Let π be the canonical mapping of A onto k, and let S be a subset of A, such that π ind… 0CWK
Lemma 1 Let A' be a subring of A containing S. In order that A' contain Cₙ, it is necessary and sufficient that one have A' + m_A = A. 0CWL
Lemma 2 One has C ∩ m_A ⊂ pC. 0CWM
Remark Suppose that p1_A is not nilpotent (this takes place in particular when A is an integral ring whose field of fractions is of characteristic 0). Then C is a discrete valuation ring… 0CWN
Proposition 4 Let C and C' be two p-rings such that l(C) geqslant l(C'), and let π (resp. π') be the canonical homomorphism of C (resp. C') onto κ_C (resp. κ_C'). Let (x_λ)_λ ∈ Λ (resp. (x'_λ)_λ… 0CWO
Proposition 5 Let k be a field of characteristic p, and let n be an integer ≥ 1, or + ∞. There exists a p-ring of length n whose residue field is isomorphic to k. 0CWP
Remark 1 Let n be an integer ≥ 1 and S a p-basis of k. One can prove that the subring of Wₙ(k) generated by Wₙ(k^pⁿ) and by the elements [ξ, 0, ..., 0] (ξ ∈ S), is a p-ring of length n whos… 0CWQ
Remark 2 The reader will find in the Appendix a proof of prop. 5 which uses neither the results of § 1, nor the existence theorem for Cohen subrings (no. 2, th. 1). 0CWR
Corollary Let C be a p-ring of finite length n. There exists a p-ring C' of infinite length such that C is isomorphic to C'/pⁿC'. 0CWS
Proposition 6 Let C be a p-ring, whose residue field k be perfect. Suppose that C is of finite length n (resp. infinite). There exists a unique isomorphism u : Wₙ(k) → C (resp. u : W(k) → C) whi… 0CWT
Theorem 2 Let A be a separated complete local ring, k its residue field and π the canonical homomorphism of A onto k. It is assumed that k is a perfect field of characteristic p. 0CWU
Proposition 7 Let us retain the hypotheses and notations of th. 2. There exists a unique multiplicative subset S of A such that π induces a bijection of S onto k. In order that an element a of A… 0CWV
Remark 1 Let us retain the preceding hypotheses and notation. We have a = ∑ₙ₌₀^∞ pⁿ τₖ(aₙ^p⁻ⁿ) (a = (aₙ)_n ∈ N ∈ W(k)) by Proposition 7 of § 1, No. 8. Hence (4) u(a) = ∑ₙ₌₀^∞ pⁿ φ(aₙ^p⁻ⁿ) f… 0CWW
Remark 2 Let A be a separated and complete local ring, k its residue field and π the canonical homomorphism of A onto k. It may be shown that there exists a multiplicative subset S of A (no… 0CWX
Example 1 Let k be a perfect field of characteristic p. The multiplicative representatives of the ring W(k) are the Witt vectors τ(x) = (x, 0, 0, ...) for x ∈ k. 0CWY
Example 2 Let A be an integral, separated and complete local ring. Suppose that the residue field k of A is finite, with q = pᶠ elements, hence perfect of characteristic p. We have x^q = x f… 0CWZ
Example 3 More particularly, consider the case A = Zₚ. Then the multiplicative representatives are 0 and the (p - 1)-th roots of unity in the field of fractions Qₚ of Zₚ. 0CX0
Lemma 3 a) There exists a unique homomorphism v : B → A extending u and mapping Tᵢ to tᵢ for 1 ≤ i ≤ n. 0CX1
Lemma 4 Suppose that the noetherian local ring C is regular, and that (p₁, ..., pₘ) is a system of coordinates of C (VIII, § 5, No. 1, Def. 1). 0CX2
Theorem 3 Let A be a local, noetherian and complete ring whose residue field k is of characteristic p. Let C be a p-ring of infinite length, whose residue field is isomorphic to k (No. 3, Pr… 0CX3

§ 3. Fields of representatives

Proposition 1 Let k₀ be a field, A a k₀-algebra which is a separated and complete local ring, K a sub-k₀-extension of κ_A which possesses a separating transcendence basis (ξ_λ)_λ ∈ Λ over k₀ (A… 0CXL
Remark \ The preceding proof applies more generally to the case where one assumes only that A is a henselian local ring. The proof of uniqueness uses the hypothesis that the local ring A… 0CXM
Definition 1 Let A be a local ring. A field of representatives of A is any subfield K of A such that the canonical homomorphism of A onto κ_A induces an isomorphism of K onto κ_A (in other word… 0CXN
Theorem 1 Let A be a separated and complete local ring of characteristic p. 0CXO
Corollary 1 Let A be a separated and complete local ring, whose residue field is an algebraic extension of Q. Then there exists a unique field of representatives of A. In fact the ring A is of… 0CXP
Corollary 2 Let A be a separated and complete local ring of characteristic p ≠ 0. Assume that the residue field κ_A is perfect. Then there exists a unique field of representatives of A, namely… 0CXQ
Theorem 2 Let A be a complete noetherian local ring of dimension d containing a field. Let K be a coefficient field of A, and let m be the dimension of the vector space m_A/m_A² over the fie… 0CXR

§ 4. Integral closure of a complete local ring

Definition 1 Let A be a noetherian integral ring. One says that A is Japanese if the integral closure of A in every finite extension of its field of fractions is a finite A-algebra. 0CY1
Remark 1 It amounts to saying that A satisfies the following condition: every integral A-algebra B integral over A, contained in a finitely generated extension of the field of fractions K o… 0CY2
Example Every finitely generated integral algebra over a field is a Japanese ring (V, § 3, No. 2, th. 2). 0CY4
Proposition 1 Let A be a noetherian integral ring, K its field of fractions. Suppose that for every finite radicial extension L of K, the integral closure of A in L is a finite A-algebra. Then t… 0CY5
Corollary Suppose the field K perfect (for example of characteristic 0). Then A is Japanese if and only if its integral closure is a finite A-algebra. 0CY6
Proposition 2 Let B be a noetherian integral ring and A a noetherian subring of B, such that B is a finite A-algebra. In order that A be Japanese, it is necessary and sufficient that B be Japane… 0CY7
Theorem 1 (Tate) Let A be an integrally closed noetherian ring, a an element of A. It is assumed that the ideal aA is prime, that the ring A/aA is Japanese, and that A is complete for the aA-adic t… 0CY8
Corollary Let R be a noetherian integral domain and n an integer. If R is Japanese, the ring R[[T₁, ..., Tₙ]] is Japanese. 0CY9
Theorem 2 (Nagata) Every complete local noetherian integral domain A is Japanese. 0CYA
Corollary Let A be a semi-local noetherian ring whose completion is reduced. Then the integral closure A' of A in its total ring of fractions R is a finite A-algebra. 0CYB
Proposition 3 Let A be a semi-local noetherian integral domain, K its field of fractions. If the K-algebra K ⊗A Â is separable, the ring A is Japanese. 0CYC
Lemma 1 Let A be a semi-local noetherian ring and B a finite A-algebra. Then the ring B is semi-local and noetherian; let m₁, ..., mₙ be its maximal ideals. 0CYD
Lemma 2 Let A be a noetherian ring and M an A-module. The canonical mapping of M into the product ∏_p ∈ Ass(M) Mₚ is injective. 0CYE
Lemma 3 Let A be a noetherian ring, x an element of A, M a finitely generated A-module, and p a prime ideal of A associated with M. Suppose that the homothety x_M is injective. Let q be a… 0CYF
Lemma 4 Let A be a discrete valuation ring, B a noetherian local ring, and ρ : A → B a local and flat homomorphism. If the ring κ_A ⊗_A B is reduced, then B is reduced. 0CYG
Definition 2 A ring A is said to be a Nagata ring if it is noetherian and if, for every prime ideal p of A, the noetherian integral ring A/p is Japanese (No. 1, def. 1). 0CYH
Example 1 Every finitely generated algebra over a field is a Nagata ring (No. 1, example). 0CYI
Example 2 Every complete noetherian local ring is a Nagata ring (No. 2, th. 2). 0CYJ
Example 3 The ring Z is a Nagata ring (No. 1, example and corollary to prop. 1). 0CYK
Example 4 One can prove (exercise 30) that every finitely generated algebra over a Nagata ring is a Nagata ring. 0CYL
Proposition 4 Let A be a Nagata ring. a) Every finite A-algebra is a Nagata ring. b) For every multiplicative subset S of A, the ring S⁻¹A is a Nagata ring. a) Let B be a finite A-algebra, ρ : A… 0CYM
Theorem 3 (Zariski-Nagata) Let A be a semi-local noetherian ring. The following conditions are equivalent: (i) A is a Nagata ring; (ii) for every prime ideal p of A, the κ(p)-algebra κ(p) ⊗_A hatA is separab… 0CYN
Corollary 1 The completion of a local reduced Nagata ring is reduced. It is enough, in fact, to put R = A in Th. 3, (iii). 0CYO
Corollary 2 (Chevalley) Let A be a reduced algebra of finite type over a field, and p a prime ideal of A. The completion of the local ring Aₚ is reduced. Since A is reduced, the local ring Aₚ is reduced… 0CYP
Corollary 3 Let k be a field of characteristic 0, and A a local noetherian k-algebra. In order that A be a Nagata ring, it is necessary and sufficient that, for every prime ideal p of A, the r… 0CYQ

App. 0. Exercises

Proposition 1 a) The ring A is local, with maximal ideal m = lim overrightarrowm_α. For every α ∈ I, the homomorphism φ_α is local and flat, and one has φ_α(m_α) A = m. 0CZN
Lemma 1 Let P ∈ A[X] be a monic polynomial whose image overlineP in κ_A[X] is irreducible. Then the A-algebra B = A[X]/(P) is local and finite over A, with maximal ideal m_A B, the canonic… 0CZO
Definition 1 Let A be a local ring. An A-algebra B is said to be an elementary enlargement of A if B is isomorphic to the A-algebra A[X], or else if there exists a monic polynomial P in A[X], w… 0CZP
Definition 2 Let A be a local ring. An A-algebra B is said to be an enlargement of A if there exists a well-ordered set Λ having a greatest element ω, and an increasing family (B_λ)_λ ∈ Λ of su… 0CZQ
Example 1 Every field extension is an inflation. Let indeed K be an extension of a field k. Let us furnish K with a well-ordering for which 0 is the greatest element, and for λ ∈ K, let K_λ… 0CZR
Example 2 Let A be a local ring, and I a set of indices. Denote by A[(Xᵢ)_i ∈ I] the local ring of the polynomial ring A[(Xᵢ)_i ∈ I] at the prime ideal m_A A[(Xᵢ)_i ∈ I]. The A-algebra A[(Xᵢ… 0CZS
Remark With the notation of Def. 2, the ring B_μ is an inflation of B_λ when λ ≤ μ. 0CZT
Proposition 2 Let A be a local ring and B an inflation of A. 0CZU
Corollary Suppose that A is noetherian. a) One has dim(A) = dim(B). b) Suppose that A is regular, and let (x₁, ..., xₙ) be a system of coordinates of A. Then B is regular and the sequence (x… 0CZV
Proposition 3 Let A, B, C be three local rings and u : A → B, v : B → C two inflations. Then v ∘ u is an inflation. Let (B_λ)_λ ∈ Λ and (C_μ)_μ ∈ M be families of sub-A-algebras of B and of sub-… 0CZW
Theorem 1 Let f : A → A' be a surjective local homomorphism of local rings and let B' be an inflation of A'. There exists an inflation B of A and an isomorphism of A-algebras of B ⊗_A A' ont… 0CZX
Corollary Let A be a local ring and K an extension of its residue field κ_A. Then there exist a local ring B and an inflation A → B such that the κ_A-algebra κ_B is isomorphic to K. 0CZY
Proposition 4 Let p be a prime number, k a field of characteristic p, and let n be an integer ≥ 1, or + ∞. Then there exists a p-ring (S 2, No. 1, Definition 1) of length n whose residue field i… 0CZZ

X. Depth, regularity, duality

§ 1. Depth

Proposition 1 Let A be a ring, J an ideal of A and 0 → M' → M → M'' → 0 an exact sequence of A-modules. Put 0D06
Remark 5 Suppose that one has p = p' and that the injection u : M' → M which occurs in the exact sequence of prop. 1 belongs to J Hom_A(M', M). One then has p'' = p - 1. In fact, the assump… 0D07
Proposition 2 Let A be a ring, J an ideal of A, M an A-module and N an A-module annihilated by a power of J. One has Ext_Aⁱ(N, M) = 0 for every integer i < prof_A(J; M). 0D08
Corollary 1 Let m be an integer 0 and let J' be an ideal of A which contains J^m. One has profA(J ; M) \leq profA(J' ; M). 0D09
Corollary 2 Suppose the ideal J to be of finite type, and let J' be an ideal of A such that V(J) \supset V(J'). 0D0A
Proposition 3 Let A be a ring, C a complex of A-modules bounded on the left and p an integer. Suppose that for every pair of integers (m, n) with m \geq n \geq p, the depth of the A-module Cm re… 0D0B
Corollary 1 Let A be a ring, J an ideal of A, C a complex of A-modules bounded on the left and p an integer. Suppose that one has JHm(C) = 0 and profA(J ; Cm) m - p for m \geq p. Then one has… 0D0C
Corollary 2 Let A be a local ring, C a complex of A-modules bounded to the left, p an integer. Suppose that for m ≥ p, Hₘ(C) is of finite length and Cₘ of depth > m - p. Then Hₙ(C) = 0 for n ≥… 0D0D
Theorem 1 Let A be a ring, J an ideal of A, x = (xᵢ)_i ∈ I a generating family of J, M an A-module. The depth of M relative to J is the greatest lower bound (in N ∪ +∞) of the integers n suc… 0D0E
Corollary 1 Suppose the ideal J to be of finite type and JM ≠ M. Then prof_A(J; M) is finite and ≤ Card(I) ; in order that it be equal to Card(I), it is necessary and sufficient that the famil… 0D0F
Corollary 2 Let A be a local ring, J an ideal of finite type of A distinct from A, M a nonzero A-module of finite type. Put r = [J/m_AJ : κ_A]. One has prof_A(J; M) ≤ r ; there is equality if… 0D0G
Proposition 4 Let ρ : A → B be a ring homomorphism, J an ideal of A and N a B-module. One has the equality prof_A(J; N) = prof_B(JB; N). 0D0H
Corollary Let A be a local ring, a an ideal of A distinct from A and M an A-module annihilated by a. One has prof_A(M) = prof_A/a(M). 0D0I
Lemma 1 If the A-module B is flat, the homomorphism vᵖ is bijective for every integer p. 0D0J
Proposition 5 Let A be a ring, J an ideal of finite type of A and M an A-module. Let Ω denote the set of maximal ideals of A belonging to Supp(M) and containing J. Then 0D0K
Proposition 6 Let A be a ring, J an ideal of finite type of A and M an A-module. Let B be a ring and ρ : A → B a ring homomorphism making B into a flat A-module. 0D0L
Corollary Let A be a noetherian ring, J an ideal of A, M a finitely generated A-module, widehatA and widehatM the separated completions of A and M for the J-adic topology. Then one has prof_… 0D0M
Proposition 7 Let A be a ring, J an ideal of A, M an A-module, and (x₁, …, xᵣ) an M-regular sequence of elements of J. Then prof_A(J; M) = r + prof_A(J; M/(x₁M + … + xᵣM)) and in particular prof… 0D0N
Theorem 2 Let A be a noetherian ring, J an ideal of A and M a finitely generated A-module. 0D0O
Corollary 1 For every M-regular sequence (x₁, …, xᵣ) of elements of J, the following properties are equivalent: 0D0P
Corollary 2 Let A be a noetherian local ring, M a nonzero finitely generated A-module. One has prof_A(M) leqslant dim_A(M) < +∞ . 0D0Q
Remark 1 Let r be an integer. By Proposition 2 of No. 1 and II, § 4, No. 4, Corollary 2 of Proposition 17, the inequality prof_F(M) geqslant r is equivalent to the following property: for e… 0D0R
Remark 2 Suppose that the A-module M is finitely generated. By Remark 2 of No. 1 and Theorem 2 of No. 4, one has the following equivalences 0D0S
Proposition 8 Let A be a noetherian ring, F a closed subset of Spec(A), and M a finitely generated A-module. One has 0D0T
Remark 3 If q is a point of Supp(M), we therefore have prof_A(q;M) = inf_p ⊇ q prof_Aₚ(Mₚ). In particular, one has the inequality prof_A(q;M) leqslant prof_A_q(M_q); there is equality when… 0D0U
Proposition 9 Let A be a noetherian ring, M and N two A-modules of finite type, and F the support of N. Then prof_F(M) is the greatest lower bound (in N ∪ +∞) of the set of integers n such that… 0D0V
Remark 4 Let A be a noetherian ring, N a finitely generated A-module. The greatest lower bound in N ∪ +∞ of the set of integers n such that Ext_Aⁿ(N, A) is nonzero is sometimes called the g… 0D0W
Lemma 2 Let ρ : A → B be a local homomorphism of noetherian local rings, N a finitely generated B-module and y an element of m_B. The following two conditions are equivalent: 0D0X
Proposition 10 Let ρ : A → B be a local homomorphism of noetherian local rings, N a finitely generated B-module and y = (y₁, …, yₛ) a sequence of elements of m_B. Let 𝔶 denote the ideal of B gene… 0D0Y
Proposition 11 Let ρ : A → B be a local homomorphism of noetherian local rings, M a finitely generated A-module and N a finitely generated B-module; suppose that the A-module N is flat. 0D0Z
Remark Recall that, under the preceding assumptions, 0D10
Corollary Let ρ : A → B be a local homomorphism of noetherian local rings making B a flat A-module. One has 0D11
Proposition 12 Let A be a noetherian local ring, M a nonzero finitely generated A-module and J an ideal of A distinct from A. One has the sequence of inequalities 0D12
Remark 1 Consider the chain of inequalities of Prop. 12. 0D13
Lemma 3 Let A be a noetherian ring, p ⊂ p1 ⊂ ... ⊂ p{r−1} ⊂ q a saturated chain of length r of prime ideals of A, M a finitely generated Λ-module and n an integer. If the A-module Ext^n{Ap… 0D14
Proposition 13 Let A be a noetherian ring, M a finitely generated A-module, p and q two prime ideals of Supp(M) with p ⊂ q. One has 0D15
Corollary 1 One has the inequality 0D16
Corollary 2 Let A be a local noetherian ring and M a finitely generated A-module. One has the inequality 0D17
Remark 2 One has sup_p ∈ Ass(M) dim(A/p) = dim(M) (VIII, § 1, No. 4, remark 2), and one recovers the inequality prof(M) ≤ dim(M) for M ≠ 0 (No. 4, cor. 2 of th. 2). 0D18
Proposition 14 Let A be a noetherian ring. The following conditions are equivalent: 0D19
Lemma 4 Let A be a noetherian ring, F a closed subset of Spec(A), U the complementary open set, and u : M → N a homomorphism of finitely generated A-modules. 0D1A
Remark 1 Let A be a noetherian ring, F a closed subset of Spec(A), U the complementary open set. In order that prof_F(A) geqslant 1, it is necessary and sufficient that Ass(A) ⊂ U (remark 2… 0D1B
Theorem 3 (Hartshorne) Let A be a noetherian ring, F a closed subset of Spec(A) and U the complementary open set. Suppose that prof_F(A) geqslant 2. Then, for every connected component Y of Spec(A), the… 0D1C
Corollary 1 The mapping which associates to each connected component of U its closure in Spec(A) is a bijection of the set of connected components of U onto the set of connected components of… 0D1D
Corollary 2 For every noetherian local ring B of depth ≥ 2, the topological space Spec(B) - m_B is connected. 0D1E
Corollary 3 Under the hypotheses of Theorem 3, suppose that Spec(Aₚ) is irreducible (resp. that Aₚ is an integral domain) for every p ∈ U; then Spec(Aₚ) is irreducible (resp. Aₚ is an integral… 0D1F
Corollary 4 Let A be a noetherian ring whose spectrum is connected. Suppose there exists an integer d ≥ 1 such that one has prof(Aₚ) ≥ 2 for every prime ideal p of A of height > d. 0D1G
Remark 2 When A is a Macaulay ring, one can apply corollary with d = 1 (§ 2, No. 5). 0D1H
Example (Complete intersection formed by four coordinate planes of an affine space of dimension 4) Let k be a field. Let S denote the polynomial ring k[T₁, T₂, T₃, T₄]. Recall (VIII, § 2, No. 4, Theorem 3) that every maximal chain of prime ideals of S is of length 4. Let m denot… 0D1I
Proposition 15 Let A be a noetherian ring satisfying the following two conditions: (i) for every minimal prime ideal p of A, the ring Aₚ is reduced; (ii) for every prime ideal p of A of height ≥… 0D1J
Proposition 16 Let A be an integrally closed noetherian ring, J an ideal of A of height geqslant 2, and M a reflexive A-module of finite type. One has prof_A(J; M) geqslant 2. 0D1K
Corollary A noetherian local integrally closed ring of dimension geqslant 2 has depth geqslant 2. 0D1L
Theorem 4 (Serre) Let A be a noetherian ring satisfying the following conditions: (i) for every minimal prime ideal 𝔭 of A, the ring A_𝔭 is reduced; (ii) for every prime ideal 𝔭 of A of height 1, th… 0D1M
Remark 1 A convenient form of th. 4 is the following: let A be a noetherian ring, such that for every prime ideal p of A the ring Aₚ is integrally closed or of depth geqslant 2; then A is n… 0D1N
Corollary 1 Let A be a noetherian ring, F a closed subset of Spec(A), U the complementary open set. Suppose that prof_F(A) is geqslant 2 (resp. geqslant 1) and that, for every p ∈ U, the ring… 0D1O
Corollary 2 Let ρ : A → B be a homomorphism of noetherian rings making B into a flat A-module. 0D1P
Remark 2 ht(p) = 1 and ht(q) leqslant 1. Then Aₚ is a discrete valuation ring; let π be a uniformizing element of Aₚ. Since B_q is faithfully flat over Aₚ, the element π 1_B_q of B_q is not… 0D1Q
Remark 3 ht(p) = 1 and ht(q) ≥ 2. Then, by the relation dim(B_q) = dim(Aₚ) + dim(κ(p) ⊗_A B_q) (VIII, § 3, No. 4, Cor. 1 of Prop. 7), the ring κ(p) ⊗_A B_q is of dimension ≥ 1. It is reduce… 0D1R
Remark 4 ht(p) ≥ 2. Since Aₚ is of depth ≥ 2 (Cor. of Prop. 16), the same is true of B_q by loc. cit. 0D1S
Corollary 3 Let ρ : A → B be a homomorphism of noetherian rings. Assume that B is a flat A-module, that A is normal, and that κ(p) ⊗_A B is normal for every p ∈ Spec(A). Then B is normal. 0D1T

§ 2. Macaulay modules and rings

Definition 1 Let A be a noetherian ring and M a finitely generated A-module. One says that M is Macaulay or is a Macaulay module if, for every maximal ideal m ∈ Supp(M), one has prof_Aₘ(Mₘ) = d… 0D2C
Example 1 Every A-module of finite length is Macaulay. 0D2D
Example 2 Let M' be a direct factor submodule of a finitely generated Macaulay A-module M. Then M' is Macaulay; in fact, for every maximal ideal m of A, the Aₘ-module M'ₘ is a direct factor… 0D2E
Example 3 Let M be a finitely generated Macaulay A-module and (x₁, …, xₙ) an M-regular sequence of elements of A. Then the A-module overlineM = M/(x₁M + ⋯ + xₙM) is Macaulay. Let indeed m be… 0D2F
Example 4 Let M be a finitely generated A-module, and let a be an ideal of A such that aM = 0. In order that the A-module M be Macaulay, it is necessary and sufficient that it be Macaulay as… 0D2G
Proposition 1 Let A be a noetherian ring, M a finitely generated A-module, and p and q prime ideals of Supp(M) such that p ⊂ q. Suppose that dim_A_q(M_q) = prof_A_q(M_q). Then one has dim_Aₚ(Mₚ)… 0D2H
Corollary Let A be a noetherian ring and M a finitely generated A-module. The following conditions are equivalent: (i) the A-module M is Macaulay; (ii) one has prof_Aₚ(Mₚ) = dim_Aₚ(Mₚ) for e… 0D2I
Remark Let S be a multiplicative subset of A and M a finitely generated Macaulay A-module. Then S⁻¹M is a Macaulay S⁻¹A-module. In fact, let q ∈ Spec(S⁻¹A); let i_A^S : A → S⁻¹A denote th… 0D2J
Proposition 2 Let A be a noetherian ring and M a finitely generated Macaulay A-module. a) The A-module M has no embedded associated prime ideals.\footnote{Recall (cf. IV, § 2, No. 3, remark) tha… 0D2K
Remark 1 Under the hypotheses of Prop. 2, it may happen that two irreducible components X₁ and X₂ of Supp(M) have an intersection Y reduced to a point and that one has dim X₁ ≠ dim X₂ and d… 0D2L
Corollary Let A be a noetherian local ring and M a nonzero finitely generated Macaulay A-module. 0D2M
Remark 2 Let M be a finitely generated Macaulay A-module, and p an element of Supp(M). In view of Theorem 2 of § 1, No. 4, one has prof_A(p; M) < +∞, and there exists an M-regular sequence… 0D2N
Proposition 3 Let A be a noetherian local ring, M a nonzero finitely generated A-module, and d the dimension of M. The following conditions are equivalent: 0D2O
Lemma 1 — Let A be a noetherian local ring, M a finitely generated and pure A-module, and x an element of mA. The following conditions are equivalent: 0D2P
Proposition 4 Let A be a noetherian local ring, M a nonzero finitely generated A-module, and (x₁, ..., xr) a sequence of elements of mA secant for M. The following conditions are equivalent: 0D2Q
Theorem 1 Let A be a noetherian local ring, M a nonzero finitely generated A-module, d the dimension of M, x = (x₁, ..., xd) a sequence of elements of mA secant for M, and J the ideal genera… 0D2R
Lemma 2 Let overlineG be the image of G in A/Ann(M). One has ht(overlineG) = codim(Supp(M/SM), Supp(M)) . When moreover SM ≠ M, one has ht(overlineG) ≤ Card(S) . 0D2S
Definition 2 Let A be a noetherian ring, M a finitely generated A-module, and S a finite subset of A. One says that S is strongly secant for M if one has Card(S) ≤ codim(Supp(M/SM), Supp(M)) . 0D2T
Remark 1 Every finite subset S of A such that SM = M is strongly secant for M. When SM ≠ M, it follows from lemma 2 that, for S to be strongly secant for M, it is necessary and sufficient t… 0D2U
Remark 2 If the ring A is local and the nonzero module M, every subset S of m_A strongly secant for M is secant for M. In fact, since the A-module M/SM is nonzero, one has 0D2V
Proposition 5 Let A be a noetherian ring, M a finitely generated A-module, and S a finite subset of A. The following conditions are equivalent: 0D2W
Corollary Let A be a noetherian ring and M a finitely generated A-module. Every M-regular sequence is strongly secant for M. 0D2X
Proposition 6 Let A be a noetherian ring, M a finitely generated Macaulay A-module, and S a finite subset of A strongly secant for M. Then the A-module M/SM is Macaulay. 0D2Y
Definition 3 One says that a ring A is Macaulay, or is a Macaulay ring, if it is noetherian and the A-module A is Macaulay. 0D2Z
Example 1 Every artinian ring is a Macaulay ring (No. 1, Example 1). 0D30
Example 2 A Macaulay ring has no embedded associated prime ideals (No. 2, Prop. 2). Conversely, let A be a noetherian ring of dimension leqslant 1 which has no embedded associated prime idea… 0D31
Example 3 A normal noetherian ring of dimension leqslant 2 is a Macaulay ring (§ 1, No. 10, text preceding Theorem 4). Conversely, let A be a Macaulay ring whose local ring Aₚ at every prime… 0D32
Example 4 If A is a Macaulay ring, the same is true of S⁻¹A for every multiplicative subset S of A (No. 1, Remark). Conversely, if the ring Aₘ is a Macaulay ring for every maximal ideal m of… 0D33
Example 5 Let A be a noetherian ring and J an ideal of A. In order that A/J be a Macaulay ring, it is necessary and sufficient that it be a Macaulay A-module (No. 1, Example 4). 0D34
Example 6 Let A be a noetherian local ring and J an ideal of A generated by an A-regular sequence. In order that A/J be a Macaulay ring, it is necessary and sufficient that A be one (Example… 0D35
Example 7 For a noetherian local ring A to be a Macaulay ring, it is necessary and sufficient that it possess an ideal of definition generated by an A-regular sequence: this follows from Pro… 0D36
Proposition 7 Let A be a noetherian ring. The following conditions are equivalent: 0D37
Remark Let ρ : A → B be a homomorphism of rings, and let p ∈ Spec(A). Let us denote by overlineB the ring κ(p) ⊗_A B. It may be identified with S⁻¹B/p(S⁻¹B), where S is the multiplicative… 0D38
Proposition 8 Let ρ : A → B be a homomorphism of Noetherian rings and let N be a B-module which is a finitely generated A-module. In order that the A-module N be Macaulay, it is necessary and su… 0D39
Corollary 1 Let ρ : A → B be a homomorphism of Noetherian rings. If B is a finite A-algebra and a Macaulay A-module, it is a Macaulay ring. If moreover ρ is injective, one has ht(aB) = ht(a) f… 0D3A
Corollary 2 Let A be an integrally closed noetherian ring and B a ring containing A. Suppose that B is a torsion-free A-module, of finite type. If B is a Macaulay ring, the A-module B is Macau… 0D3B
Corollary 3 Let A be an integrally closed ring, K its field of fractions, L a finite K-algebra such that [L : K] 1_A is invertible in A, and B a sub-A-algebra of L, finite over A. 0D3C
Example Corollary 3 applies in particular in the following two situations: 0D3D
Proposition 9 Let ρ : A → B be a homomorphism of noetherian rings, M a finitely generated A-module and N a finitely generated B-module, flat over A. Denote by ᵃρ : Spec(B) longrightarrow Spec(A)… 0D3E
Corollary 1 Let ρ : Λ → B be a local homomorphism of noetherian local rings, M a nonzero finitely generated A-module and N a nonzero finitely generated B-module which is a flat A-module. In or… 0D3F
Corollary 2 Let ρ : A → B be a homomorphism of noetherian rings making B into a faithfully flat A-module and let M be a finitely generated A-module. In order that the B-module M ⊗A B be a Maca… 0D3G
Corollary 3 Let A be a noetherian ring, B a finite flat Λ-algebra, M a finitely generated Macaulay A-module. Then the B-module M ⊗A B is a Macaulay module. 0D3H
Corollary 4 Let A be a noetherian ring, J an ideal of A and M a finitely generated A-module. Denote by Ā and Ĝ the separated completions of A and M for the J-adic topology, and by S the multip… 0D3I
Proposition 10 Let ρ : A → B be a homomorphism of noetherian rings making B into a flat A-module. The following conditions are equivalent: 0D3J
Corollary 1 Every finite flat algebra over a Macaulay ring is a Macaulay ring. 0D3K
Corollary 2 Let A be a Macaulay ring and n a positive integer; then A[X₁, …, Xₙ] and A[[X₁, …, Xₙ]] are Macaulay rings. 0D3L
Corollary 3 Every algebra of finite type over a noetherian Macaulay ring is catenary. 0D3M

§ 3. Depth and homological dimension

Example Let J be an ideal of A generated by an A-regular sequence x = (x₁, …, xᵣ). One has dpA(A/J) ≤ r. In fact, this is clear if A is zero; in the contrary case, the Koszul complex K_bul… 0D3Y
Proposition 1 Let A be a ring, M an A-module and n an integer ≥ 0. The following conditions are equivalent: 0D3Z
Proposition 2 Let A be a ring, M and N A-modules, i an integer and S a multiplicative subset of A. One has a canonical isomorphism of S⁻¹A-modules 0D40
Corollary Let A be a ring, M and N A-modules, i an integer. 0D41
Proposition 3 Let A be a noetherian ring, M a finitely generated A-module, and N an A-module. 0D42
Corollary 1 For every multiplicative subset S of A , one has dp_S⁻¹A(S⁻¹M) ≤ dp_A(M) , di_S⁻¹A(S⁻¹N) ≤ di_A(N) , dh(S⁻¹A) ≤ dh(A) . 0D43
Corollary 2 If one has dp{Am}(Mm) < +\infty for every maximal ideal m of Supp(M) , one has dpA(M) < +\infty . 0D44
Remark Let A be a regular noetherian ring of infinite dimension (VIII, § 5, Exerc. 6 c)). We shall see below (No. 7, Th. 2 and § 4, No. 1, Prop. 1) that one has di{Am}(Am) = dh(Am) < +\in… 0D45
Proposition 4 Let A be a noetherian local ring, M a finitely generated A-module, and n an integer ≥ 0. The following conditions are equivalent : 0D46
Corollary 1 Let A be a noetherian local ring and n an integer ≥ 0. The following conditions are equivalent : 0D47
Corollary 2 One has dh(A) = dp_A(κ_A). 0D48
Remark 1 Let A be a local ring. The A-module Tor₁ᴬ(κ_A, κ_A) is isomorphic to 𝔪_A / 𝔪_A² (A, X, p. 72, Example). Hence when A is noetherian, in order that Tor₁ᴬ(κ_A, κ_A) be zero, it is nec… 0D49
Remark 2 Let A be a noetherian local ring, M a finitely generated A-module of finite projective dimension n, N a nonzero finitely generated A-module. The A-module Ext_Aⁿ(M, N) is nonzero: l… 0D4A
Remark 3 Let A be a noetherian ring, M a finitely generated A-module of finite projective dimension, N a finitely generated A-module whose support is equal to Spec(A). According to the prec… 0D4B
Proposition 5 Let A be a noetherian ring, M a finitely generated A-module and n an integer geqslant 0. The following conditions are equivalent : 0D4C
Remark 4 Let A be a noetherian ring and n an integer geqslant 0. If m and m' are two distinct maximal ideals of A, the A-modules Ext_Aⁿ(A/m, A/m') and Torₙᴬ(A/m, A/m') are annihilated by m… 0D4D
Proposition 6 Let A be a noetherian ring, N an A-module, n an integer geqslant 0. The following conditions are equivalent: 0D4E
Remark 5 Let N be a finitely generated A-module; the condition Ext_Aⁿ(A/𝔪, N) = 0 for every maximal ideal 𝔪 of A does not necessarily imply di_A(N) < n. If, for example, the ring A is local… 0D4F
Proposition 7 Let A be a ring, x a cancellable element of A, M an A-module such that the homothety x_M is injective, N an A-module annihilated by x, n an integer. The canonical homomorphisms of… 0D4G
Corollary Let A be a noetherian local ring, M a finitely generated A-module and J an ideal of A generated by a sequence (x₁, …, xᵣ) of elements of m_A which is both A-regular and M-regular… 0D4H
Theorem 1 (Auslander-Buchsbaum) Let A be a noetherian local ring and M a finitely generated A-module of finite projective dimension. Then one has the equality 0D4I
Remark In view of cor. 2 of prop. 4 (No. 3), theorem 1 applied to the A-module κ_A implies that one is in one or the other of the following cases: 0D4J
Corollary 1 Retain the hypotheses of theorem 1. 0D4K
Corollary 2 Retain the hypotheses of theorem 1 and suppose moreover that A is a Macaulay ring. Then dp_A(M) is the sum of the two positive integers dim(A) - dim_A(M) and dim_A(M) - prof(M). 0D4L
Corollary 3 Let A be a noetherian ring, M a finitely generated A-module of finite projective dimension, i an integer ≥ 0, N a finitely generated A-module, and F the support of the A-module Ext… 0D4M
Corollary 4 Let A be a noetherian Macaulay ring and M a finitely generated A-module of finite projective dimension. 0D4N
Proposition 8 Let A be a noetherian ring, M a finitely generated A-module. Then dim_A(M) ≤ di_A(M). 0D4O
Proposition 9 Let A be a noetherian local ring and M a nonzero finitely generated A-module of finite injective dimension. Then di_A(M) = prof(A). 0D4P
Definition 1 A ring A is said to be a Gorenstein ring if it is noetherian and if the Aₘ-module Aₘ is of finite injective dimension for every maximal ideal m of A. 0D4Q
Proposition 10 Let A be a Gorenstein ring; then A is a Macaulay ring, and satisfies di_A(A) = dim(A). 0D4R
Example 1 For every multiplicative subset S of a Gorenstein ring A, the ring of fractions S⁻¹A is a Gorenstein ring: in fact, let q be a maximal ideal of S⁻¹A; it is of the form S⁻¹p, where… 0D4S
Example 2 Let A be a Gorenstein ring and let J be an ideal of A, generated by an A-regular sequence x. The quotient ring A/J is a Gorenstein ring: in fact, for every maximal ideal m of A con… 0D4T
Example 3 Let A be a noetherian local ring, J an ideal of A generated by an A-regular sequence of elements of m_A. If A/J is a Gorenstein ring, then the same is true of A (No. 4, Cor. of Pro… 0D4U
Example 4 Let A be a regular local noetherian ring; then A is a Gorenstein ring. In fact, let x be a system of coordinates of A (VIII, § 5, No. 1, Def. 1). The sequence x is A-regular (loc… 0D4V
Example 5 Every quotient ring of a principal ideal domain is a Gorenstein ring (Example 2). In particular, every monogenic algebra over a field is a Gorenstein ring. 0D4W
Lemma 1 Let A be an artinian local ring. The following conditions are equivalent: 0D4X
Lemma 2 Let A be a local noetherian ring such that di_A(A) = +∞. Then Ext_Aⁱ(κ_A, A) ≠ 0 for every integer i geqslant dim(A). 0D4Y
Theorem 2 (Bass) Let A be a noetherian local ring; put d = dim(A). Let x = (x₁, …, x_d) be a maximal secant sequence of elements of m_A, and x the ideal which it generates. The following conditions… 0D4Z
Corollary Let A be a noetherian local ring of dimension d. The A-module Extᵈ_A(κ_A, A) is not zero. 0D50
Proposition 11 Let A be a noetherian ring. The following conditions are equivalent: 0D51
Proposition 12 Let ρ : A → B be a local homomorphism of noetherian local rings, making B into a flat A-module. The following conditions are equivalent: 0D52
Corollary 1 Let ρ : A → B be a homomorphism of noetherian rings making B into a flat A-module. The following conditions are equivalent: 0D53
Corollary 2 Let A be a noetherian ring, J an ideal of A, hatA the separated completion of A for the J-adic topology. Consider the following conditions: 0D54
Corollary 3 Let A be a Gorenstein ring and (Tᵢ)_i ∈ I a finite family of indeterminates. The rings A[(Tᵢ)_i ∈ I] and A[[Tᵢ]_i ∈ I] are Gorenstein rings. 0D55

§ 4. Regular rings

Proposition 1 Let A be a regular noetherian local ring and n its dimension. Then dh(A) = n and, for every integer i geqslant 0, 0D5M
Proposition 2 A regular noetherian local ring is factorial. 0D5N
Proposition 3 Let A be a regular noetherian local ring and M a nonzero finitely generated A-module. Its projective dimension is finite, and one has 0D5O
Corollary 1 One has dp_A(M) geqslant dim(A) - dim(M); for equality to hold, it is necessary and sufficient that M be a Macaulay module. 0D5P
Corollary 2 For the A-module M to be free, it is necessary and sufficient that it be a Macaulay module and of dimension dim(A), or again that it be of depth geqslant dim(A). 0D5Q
Corollary 3 Every finitely generated reflexive module over a regular noetherian local ring of dimension 2 is free. 0D5R
Corollary 4 Let ρ : A → B be a local homomorphism of noetherian local rings. Suppose that A is regular and that ρ makes B into a finitely generated A-module. Then dp_A(B) ≥ dim(A) - dim(B). Fo… 0D5S
Remark Corollary 4 makes it possible to characterize Macaulay local rings in several important cases. Let A be a noetherian local ring; it is a Macaulay ring if and only if the same is tr… 0D5T
Theorem 1 (Serre) A noetherian local ring is regular if and only if its homological dimension is finite. 0D5U
Definition 1 One says that a ring A is regular if it is noetherian and if the local ring Aₘ is regular for every maximal ideal m of A. 0D5V
Proposition 4 Let A be a noetherian ring. The following conditions are equivalent: (i) A is regular; (ii) every finitely generated A-module has projective dimension < +∞; (iii) for every maximal… 0D5W
Example 1 If the ring A is regular, the ring of fractions S⁻¹A is regular for every multiplicative subset S of A: this follows for example from characterization (iii) above. 0D5X
Example 2 In order that a ring be regular, it is necessary and sufficient that it be isomorphic to the product of a finite family of integral regular rings; this follows in fact from the fac… 0D5Y
Example 3 The integral regular rings of dimension ≤ 1 are the Dedekind rings (VIII, § 5, No. 1, example 1 and VII, § 2, No. 2, theorem 1). 0D5Z
Corollary 1 Let A be a noetherian ring. The following conditions are equivalent: 0D60
Corollary 2 A regular ring is normal, Gorenstein and Macaulay. 0D61
Corollary 3 Let A be a noetherian ring, J an ideal of A and hatA the separated completion of A for the J-adic topology. 0D62
Corollary 4 Let A be a regular ring and P a finitely generated projective A-module. The symmetric algebra S_A(P) is a regular ring. 0D63
Corollary 5 Let A be a regular ring, and (Tᵢ)_i ∈ I a finite family of indeterminates. The polynomial ring A[(Tᵢ)_i ∈ I] and the ring of formal series A[[Tᵢ]_i ∈ I] are regular. 0D64
Proposition 5 Let ρ : A → B be a homomorphism of noetherian rings and N a B-module. Suppose that a) the ring A is regular, b) N is a finitely generated A-module, c) the B-module N is Cohen-Macau… 0D65
Corollary Let B be an integral noetherian ring and let A be a regular subring of B. Suppose that the A-module B is finitely generated. In order that the ring B be a Macaulay ring, it is nece… 0D66
Example Let B be a nonzero integral algebra of finite type over a field K. By VIII, § 2, No. 4, Cor. 1 of Th. 3, there exists a subalgebra A of B isomorphic to a polynomial algebra over K… 0D67
Proposition 6 a) Every ring of fractions of a presentable ring is presentable. Every algebra of finite type over a presentable ring is a presentable ring. 0D68
Proposition 7 Let A be a presentable ring. 0D69
Proposition 8 Let ρ : A → B be a homomorphism of noetherian rings making B a faithfully flat A-module. 0D6A
Corollary Let B be a regular integral ring and let A be a noetherian subring of B such that B is a finitely generated A-module. The following conditions are equivalent: 0D6B
Lemma 1 One has 0D6C
Proposition 9 Let ρ : A → B be a local homomorphism of noetherian local rings. The following conditions are equivalent: 0D6D
Example Let X, Y be two complex analytic varieties, locally of finite dimension, f a morphism of X into Y, and x a point of X. Consider the local homomorphism ρ : O_Y, f(x) → O_X, x associ… 0D6E
Corollary Let ρ : A → B be a homomorphism of noetherian rings making B a flat A-module. If A is regular and if κ(ρ⁻¹(𝔫)) ⊗_A B is regular for every maximal ideal 𝔫 of B, the ring B is regula… 0D6F

§ 5. Complete intersections

Definition 1 Let A be a ring, J an ideal of A. One says that the ideal J is completely secant at the point p of V(J) if the ideal Jₚ of Aₚ is generated by a completely secant sequence for Aₚ. O… 0D6T
Proposition 1 Let A be a ring, J an ideal of A, generated by a finite sequence x = (x₁, …, xᵣ) of elements of A. The following conditions are equivalent: 0D6U
Remark 1 When the ring A is noetherian, one may replace in conditions (ii) and (ii') "completely secant" by "regular" (A, X, p. 160, cor. 1). 0D6V
Remark 2 Let A be a ring, J an ideal of finite type of A and p a prime ideal of A containing J. By cor. 2 of th. 1 of § 1, No. 3, one has 0D6W
Theorem 1 Let A be a noetherian ring and J an ideal of A. The following conditions are equivalent: 0D6X
Remark 1 Suppose the ideal J completely secant; let (x1, ..., xr) be a sequence of elements of J such that for every maximal ideal m ∈ V(J) the canonical images of the xi in J/mJ form a bas… 0D6Y
Corollary 1 Let hatA be the separated completion of A for the J-adic topology and hatJ = JhatA the separated completion of J. In order that the ideal hatJ of hatA be completely secant, it is n… 0D6Z
Corollary 2 Let ρ : A → B be a homomorphism of noetherian rings making B into a flat A-module, and J an ideal of A. 0D70
Corollary 3 Let A be a noetherian ring and J a completely secant ideal of A. If A is a Macaulay ring (resp. a Gorenstein ring), the same is true of A/J. 0D71
Remark 2 One says that a noetherian ring A is a complete intersection ring if, for every maximal ideal m of A, the complete local ring widehatAₘ is isomorphic to the quotient of a complete… 0D72
Proposition 2 Let A be a noetherian ring. The following conditions are equivalent: 0D73
Proposition 3 Let A be a noetherian ring and J an ideal of A such that A/J is regular. The following conditions are equivalent: 0D74
Proposition 4 Let A be a regular ring, J an ideal of A, and A₀ a subring of A such that the canonical homomorphism A₀ → A/J is bijective. 0D75
Theorem 2 Let A be a regular ring, graded of type N. The ring A₀ consisting of the elements of degree 0 in A is regular; there exists a finitely generated projective A₀-module P graded in de… 0D76
Corollary 1 Let B be a regular ring, graded in positive degrees. Suppose that every finitely generated projective B₀-module is free. a) The ring B₀ is an integral domain and regular, and the B… 0D77
Corollary 2 Let k be a field, B a finitely generated graded polynomial k-algebra, and A a graded subalgebra of B. The following conditions are equivalent: (i) B is a free graded A-module; (ii)… 0D78
Proposition 5 Let ρ : A → B be a local homomorphism of noetherian local rings, N a finitely generated B-module, x = (x1, ..., xr) a sequence of elements of mB, and u : A[T1, ..., Tr] → B the uni… 0D79
Corollary Let k be a field, Λ a noetherian local k-algebra, x = (x1, ..., xr) a sequence of elements of mA, and M a finitely generated Λ-module. Let Ā and Ĝ be the completions of A and M for… 0D7A
Proposition 6 Let ρ : A → B be a homomorphism of noetherian rings, and J an ideal of B. The following conditions are equivalent: 0D7B
Remark 1 Suppose that the equivalent conditions of Prop. 6 are satisfied. Since B/J is flat over A, for every A-algebra A', the canonical sequence of A'-modules 0 → A' ⊗_A J → A' ⊗_A B → A'… 0D7C

§ 6. Extension of scalars in regular algebras

Example 1 To say that an extension L of a field K is a K-algebra essentially of finite type means that it is an extension of finite type in the sense of A, V, p. 11, Def. 2. The K-algebra L… 0D7O
Example 2 In order that a local k-algebra be essentially of finite type, it is necessary and sufficient that it be isomorphic to a k-algebra of the form Aₚ, where A is a k-algebra of finite… 0D7P
Proposition 1 If the ring k is noetherian, every k-algebra essentially of finite type is a noetherian ring. 0D7Q
Proposition 2 a) Every quotient algebra of a k-algebra essentially of finite type is essentially of finite type. 0D7R
Corollary Let A be a k-algebra essentially of finite type and B a noetherian k-algebra. Then the ring A ⊗ₖ B is noetherian. 0D7S
Proposition 3 Let k be a ring, A a k-algebra and B an A-algebra. If A is essentially of finite type over k and B essentially of finite type over A, then B is essentially of finite type over k. 0D7T
Corollaire The tensor product of two k-algebras essentially of finite type is a k-algebra essentially of finite type. 0D7U
Proposition 4 Let k be a field, A a k-algebra essentially of finite type and B a k-algebra. 0D7V
Corollary 1 Let k be a field, K an extension of k and A a k-algebra essentially of finite type. For A_(K) to be a Macaulay ring (resp. a Gorenstein ring), it is necessary and sufficient that t… 0D7W
Corollary 2 Let k be a noetherian ring, A and B k-algebras. Suppose that A is flat and essentially of finite type over k. If A and B are Macaulay rings (resp. Gorenstein rings), then A ⊗ₖ B is… 0D7X
Lemma 1 Let A be a noetherian ring, the union of an increasing filtered family (A_α)_α ∈ I of noetherian subrings. 0D7Y
Lemma 2 Let k be a field, and let K and L be extensions of k. Suppose that K is of finite type and that one of the extensions K or L is separable. Then the ring K ⊗ₖ L is regular. 0D7Z
Proposition 5 Let k be a field, A a k-algebra and K an extension of k. Suppose that A is essentially of finite type or that the extension K is of finite type. 0D80
Definition 1 Let k be a field and A a k-algebra. One says that A is absolutely regular (resp. absolutely normal)¹ if the ring A_(k') is regular (resp. normal) for every radicial extension k' of… 0D81
Example 1 If k is perfect, every regular (resp. normal) k-algebra is absolutely regular (resp. absolutely normal). 0D82
Example 2 Let A be an artinian k-algebra. If A is normal, it is reduced, hence isomorphic to a finite product of extensions of k (A, VIII, § 8, No. 1, prop. 2). The following conditions are… 0D83
Example 3 Let A be a regular local k-algebra. If the extension κ_A of k is separable, the algebra A is absolutely regular. Let indeed k' be a finite-degree extension of k. The A-module A_(k'… 0D84
Proposition 6 Let k be a field and A a noetherian k-algebra. 0D85
Lemma 3 Let k be a field and K a finite-type extension of k. There exist an extension L of K, of finite degree, and a subextension k' of L over k, radicial and of finite degree over k, suc… 0D86
Proposition 7 Let k be a field, A and B two k-algebras one of which is essentially of finite type. Suppose that A is absolutely regular (resp. absolutely normal). If the ring B is regular (resp… 0D87
Corollary 1 Let k be a field. The tensor product of two absolutely regular (resp. absolutely normal) k-algebras, one of which is essentially of finite type, is an absolutely regular (resp. abs… 0D88
Corollary 2 Let k be a field, A an absolutely regular (resp. absolutely normal) k-algebra, and K an extension of k. Suppose that A is essentially of finite type or that the extension K of k is… 0D89
Corollary 3 Let k be a field, A a k-algebra, and K an extension of k. Suppose that the k-algebra A is essentially of finite type or that the extension K of k is of finite type. In order that t… 0D8A
Corollary 4 Let k be a field, A a k-algebra essentially of finite type, and K an extension of k which is a perfect field. In order that A be absolutely regular (resp. absolutely normal), it is… 0D8B
Proposition 8 Let k be a field and A a k-algebra essentially of finite type. Let I denote the kernel of the homomorphism of k-algebras μ : A ⊗ₖ A → A such that μ(x ⊗ y) = xy for x and y in A. Th… 0D8C
Theorem 1 Let k be a field and A a local k-algebra essentially of finite type. Put n = \dim(A) + \deg.\mathrm{tr}k(\kappaA). 0D8D
Lemma 4 Let k be a field, K an algebraic extension of k , B a k-algebra whose spectrum is connected. For every canonical factor C of B{(K)} , one has \dim(C) = \dim(B) . 0D8E
Lemma 5 Let k be an algebraically closed field, A a finitely generated k-algebra whose spectrum is connected, n an integer and (x1, \ldots , xn) a finite sequence of elements of A . Suppos… 0D8F
Example When A is an extension of finite type of k, theorem 1 yields again Cor. 1 of A, V, p. 128, taking account of Example 2 of No. 4. 0D8G
Corollary 1 Let k be a field and A a k-algebra essentially of finite type. The set of elements 𝔭 of Spec(A) such that the k-algebra A_𝔭 is absolutely regular is open in Spec(A). 0D8H
Corollary 2 Let k be a field and A a k-algebra essentially of finite type. In order that A should be absolutely regular, it is necessary and sufficient that the A-module Ωₖ(A) be projective an… 0D8I
Remark Suppose the k-algebra A absolutely regular. Then the total ring of fractions F of A is identified with the product of the A_q, where q runs through the set of minimal prime ideals… 0D8J
Corollary 3 Let k be a field of characteristic 0 and A a k-algebra essentially of finite type. For A to be regular, it is necessary and sufficient that the A-module Ωₖ(A) be projective. 0D8K

§ 7. Smooth algebras

Proposition 1 If φ admits a lifting, the mapping (δ, tildeφ) ↦ δ + tildeφ defines a simply transitive operation of the group of k-derivations of A into N on the set of liftings of φ. 0D8S
Example Let B be a k-algebra, N a B-module. Endow the k-module B ⊕ N with the structure of a k-algebra defined by (b, x)(b', x') = (bb', bx' + b'x) (cf. A, III, p. 127), so that N is a squ… 0D8T
Proposition 2 Endow the k-module A/J ⊕ J/J² with the structure of a k-algebra defined in the above example. The mappings v ↦ hᵥ and h ↦ ψₕ induce bijections between the following sets : 0D8U
Definition 1 One says that A is formally smooth over k, or is a formally smooth k-algebra, if it satisfies the following condition: whatever the k-algebra C and the square-zero ideal N of C may… 0D8V
Remark 1 Let k be a ring, A a k-algebra and J an ideal of A. If the k-algebra A/J is formally smooth (for the discrete topology), the identity mapping of A/J admits a lifting to A/J²; conse… 0D8W
Remark 2 Let k be a ring, A a linearly topologized formally smooth k-algebra, M an A-module whose annihilator is open in A. Then every derivation δ of k into M extends to a derivation of A… 0D8X
Proposition 3 Let k be a ring. 0D8Y
Proposition 4 Let k be a ring, A and B k-algebras, J an ideal of A, K an ideal of B. 0D8Z
Corollary Let k be a field, K a separable extension of k and A a linearly topologized K-algebra. If A is formally smooth over K, it is formally smooth over k. 0D90
Remark 1 Let k be a field. Every étale k-algebra (A, V, p. 28, def. 1) is formally smooth (loc. cit., p. 34, th. 4, d) and No. 2, prop. 3, b)). 0D91
Remark 2 We shall see below (cor. 2 of th. 2 of No. 5) that a field extension which is formally smooth is absolutely regular, hence separable (§ 6, No. 4, example 2). 0D92
Proposition 5 Let A be a formally smooth linearly topologized k-algebra. Every continuous homomorphism of k-algebras φ : A → C/Cₘ admits a continuous lifting to C. 0D93
Example Let C be a k-algebra and N a nilpotent two-sided ideal of C. The proposition applies to the algebra C endowed with the N-adic filtration. If A is a formally smooth linearly topolog… 0D94
Theorem 2 Let k be a ring, A a k-algebra, and J an ideal of A such that the k-algebra A/J is formally smooth. Endow A with the J-adic topology. The following conditions are equivalent: 0D95
Corollary 1 Let k be a field and A a noetherian local k-algebra such that the extension κA of k is separable. The following conditions are equivalent: 0D96
Corollary 2 Let k be a field, A a noetherian k-algebra and J an ideal of A contained in the radical of A. Suppose that the k-algebra A is formally smooth for the J-adic topology. Then it is ab… 0D97
Corollary 3 Let k be a ring and A a formally smooth k-algebra. 0D98
Lemma 1 Suppose that the A-algebra B admits a generating family t = (tᵢ)_i ∈ I, with defining relations f_λ ∈ A[T] (λ ∈ A). The B-linear homomorphism ψ : (B ⊗_A Ωₖ(A)) ⊕ B⁽ᴵ⁾ longrightarro… 0D99
Lemma 2 Suppose the ring A local and noetherian. Then Aₙ is local and noetherian. The following conditions are equivalent: 0D9A
Proposition 6 Let k be a field of characteristic p ≠ 0, k' a purely inseparable extension of k, of finite degree and of height leqslant 1, and A a regular local k-algebra. Then A_(k') is a local… 0D9B
Proposition 7 Let k₀ be a ring, k a k₀-algebra, A a k-algebra, m a maximal ideal of A. Suppose that k and A/m are formally smooth over k₀. In order that A be formally smooth over k for the m-adi… 0D9C
Remark 1 When A is noetherian, condition (i) means that the local ring A_𝔪 is regular (VIII, § 5, No. 2, Theorem 1). 0D9D
Proposition 8 Let k be a field and A a noetherian local k-algebra. The following conditions are equivalent: 0D9E
Remark 2 Let k be a field and A a k-algebra such that the ring A_(k') is regular for every purely inseparable extension k' of k, of finite degree and of height ≤ 1; then A is absolutely reg… 0D9F
Proposition 9 Let A be a ring, M a finitely generated A-module, N a projective A-module, and u : M → N an A-linear mapping. 0D9G
Lemma 3 Let K be an ideal of A containing J. Suppose that there exists an integer m such that J ∩ Kᵐ is contained in JK (this condition is satisfied if A is noetherian). If A/J is formally… 0D9H
Lemma 4 Suppose that A is formally smooth over k for the J-adic topology. In order that A/J be formally smooth over k, it is necessary and sufficient that the canonical mapping bard have a… 0D9I
Theorem 3 Let k be a ring, A a formally smooth k-algebra, and J a finitely generated ideal of A; put B = A/J. 0D9J
Corollary 1 Let k₀ be a ring, k a noetherian k₀-algebra formally smooth, and B a local k-algebra essentially of finite type. If the k₀-algebra B is formally smooth for the 𝔪_B-adic topology, i… 0D9K
Corollary 2 Let k₀ be a ring, k a noetherian k₀-algebra formally smooth, and B a k-algebra essentially of finite type. The set U of prime ideals 𝔭 of B such that the k₀-algebra B_𝔭 is formally… 0D9L
Remark 1 Corollaries 1 and 2 apply in particular when k₀ is a field and one is in one of the following two cases: a) B is an algebra essentially of finite type over a separable extension of… 0D9M
Remark 2 Under the hypotheses of cor. 3, the A-module Ω(A) is projective (No. 5, cor. 3 of th. 2), hence free; every derivation of A into κ_A therefore lifts to a derivation of A. The condi… 0D9N
Corollary 4 (Zariski) Let k be a field and A a k-algebra essentially of finite type or a complete noetherian local ring. The set of prime ideals 𝔭 of A such that the local ring A_𝔭 is regular is open in… 0D9O
Lemma 5 Let ρ : A → B be a local homomorphism of noetherian local rings. Suppose that B is essentially of finite type over A. For the A-algebra B to be formally smooth, it is necessary and… 0D9P
Theorem 4 Let A be a noetherian ring and B an A-algebra essentially of finite type. The following conditions are equivalent: 0D9Q
Definition 2 Let A be a noetherian ring. An A-algebra B is said to be smooth if it is essentially of finite type and if it satisfies the equivalent conditions of theorem 4. 0D9R
Proposition 10 Let A be a noetherian ring. 0D9S
Example 1 The smooth algebras over a field k are the k-algebras essentially of finite type and absolutely regular. 0D9T
Example 2 Let A be a noetherian ring, T = (Tᵢ)_i ∈ I a finite family of indeterminates. The A-algebra A[T] is smooth. More generally, let F₁, …, Fₘ be elements of A[T], and let B be the A-al… 0D9U

§ 8. Duality of modules of finite length

Proposition 1 Let A be a noetherian ring. For every prime ideal 𝔭 of A, let e_𝔭 : A/𝔭 → I(𝔭) be an injective envelope of A/𝔭 (A, X, § 1, No. 9). 0DAA
Remark 1 Let I be an indecomposable injective A-module, and let 𝔭 be the unique element of Ass(I); by Prop. 1, I contains a submodule isomorphic to A/𝔭 of which it is the injective envelope… 0DAB
Theorem 1 Let A be a noetherian ring. For every injective A-module I, there exists one and only one family of cardinals (a_𝔭)_𝔭 ∈ Spec(A) such that I is isomorphic to ⨁_𝔭 I(𝔭)^(a_𝔭). 0DAC
Remark 2 Let A be a noetherian ring, M an A-module, e : M → I an injective envelope of M. The set Ass(M) is equal to Ass(I): in fact, the inclusion Ass(M) ⊂ Ass(I) is obvious. On the other… 0DAD
Example Let A be a principal ideal domain, and let K be its field of fractions. The injective A-modules are the divisible A-modules (A, X, p. 17, Cor. 2). The A-module K is an injective en… 0DAE
Lemma 1 Let A be a ring, a an ideal of A, and I an A-module. For every integer n geqslant 0, let Iₙ denote the a-submodule of I consisting of the elements annihilated by aⁿ. 0DAF
Proposition 2 Let A be a noetherian ring, α an ideal of A, E an A-module and M a sub-A-module of E annihilated by α. The following conditions are equivalent: (i) E is an injective envelope of M… 0DAG
Lemma 2 Let A be a ring, α an ideal of finite type of A, and M an A-module each element of which is annihilated by a power of α. Let widehatA denote the separated completion of A for the α… 0DAH
Proposition 3 Let A be a noetherian ring, p a prime ideal of A, and e : A/p → I an injective envelope of the A-module A/p. For every integer n ≥ 0, let Iₙ denote the submodule of I consisting of… 0DAI
Remark It follows from the preceding proof that the annihilator of Iₙ in widehatAₚ (resp. in Aₚ) is 𝔭ⁿ widehatAₚ (resp. 𝔭ⁿ Aₚ). Hence the annihilator of the A-module Iₙ is the inverse ima… 0DAJ
Corollary Let J be an injective A-module such that AssA(J) = {p}. 0DAK
Definition We say that an A-module I is a Matlis A-module if it is injective, if mA is its unique associated prime ideal, and if the κA-vector space HomA(κA, I) is of dimension 1. 0DAL
Lemma 3 Let I be a Matlis A-module. Then: 0DAM
Lemma 4 Let I be a Matlis A-module and M a A-module. The following conditions are equivalent : 0DAN
Lemma 5 Every artinian widehatA-module is artinian as a A-module. 0DAO
Theorem 2 Let M be an A-module. 0DAP
Remark Suppose the ring A artinian. One has long_A(I) = long_A(D_A(A)) = long(A) (Theorem 2, c)). Let M be a finitely generated A-module; it admits an injective envelope isomorphic to Iˢ… 0DAQ
Proposition 4 For the a-module M to be of finite length, it is necessary and sufficient that D(M) be of finite length. One then has z₀(D(M)) = z₀(M), long_A(M) = long_A D(M), Ann_A(M) = Ann_A(D(… 0DAR
Example Let A be a principal ideal domain, K its field of fractions. The A-module K/A is a Matlis module: in fact the canonical mapping from K/A into ∏_m ∈ Ω K/Aₘ induces an isomorphism fr… 0DAS
Example 1 Let J be a Matlis A-module. Put T(M) = Hom_A(M, J) for every A-module M of finite length and T(f) = Hom_A(f, 1_J) for every A-linear mapping f between A-modules of finite length. T… 0DAT
Example 2 Let C be an injective complex of A-modules and d an integer such that Hⁱ(Homgr_A(κ_A, C)) is zero for i ≠ d and is of length 1 for i = d. For every A-module M of finite length, one… 0DAU
Example 3 Let Ω be an A-module and d an integer geqslant 0 such that Ext_Aⁱ(κ_A, Ω) is zero for i ≠ d and has length 1 for i = d. Set T(M) = Ext_Aᵈ(M, Ω) for every A-module M of finite lengt… 0DAV
Example 4 If A is a Gorenstein ring, for example a regular ring, one can apply Example 3 by taking Ω = A and d = dim(A) (§ 3, No. 7, Prop. 11). 0DAW
Theorem 3 a) The A-module I is a Matlis module. For every integer m geqslant 0, Iₘ is the a-submodule of I consisting of the elements annihilated by m_Aᵐ. 0DAX
Example 5 Let us take up again the hypotheses and notations of Example 1. Then T(A/𝔪_Aⁿ) = Hom_A(A/𝔪_Aⁿ, J) is identified with the submodule Jₙ of J formed by the elements annihilated by 𝔪_A… 0DAY
Example 6 Let us take up again the hypotheses and notations of Example 3. One obtains that I = varinjlim Ext_Aᵈ(A/𝔪_Aⁿ, Ω) is a Matlis A-module. For every A-module of finite length M, there… 0DAZ
Proposition 5 Let ρ : A → B be a local homomorphism of noetherian local rings, such that the residual extension κ_A → κ_B induced by ρ is of finite degree. Let I_A be a Matlis A-module. 0DB0
Corollary a) If the A-algebra B is finite, the B-module I_B = Hom_A(B, I_A) is a Matlis B-module. 0DB1
Proposition 6 Suppose the A-module J injective; for every A-module N, put D(N) = Hom_A(N, J). 0DB2
Remark 1 For every homomorphism f : N → N' of A-modules, let us denote by D(f) : D(N') → D(N) the homomorphism Hom(f, 1_J). Let u : M → M' and v : P → P' be homomorphisms of A-modules. Choo… 0DB3
Remark 2 Let 0DB4

§ 9. Dualizing modules

Definition 1 Let A be a noetherian ring. One says that a A-module Ω is dualizing if it is finitely generated and if, for every maximal ideal m of A, the A/m-vector space Ext_Aⁱ(A/m, Ω) is zero… 0DBF
Example 1 If the ring A is local and artinian, the dualizing A-modules are the finitely generated injective A-modules Ω such that Hom_A(κ_A, Ω) is of dimension 1 (S 3, No. 3, prop. 6), that… 0DBG
Example 2 For a noetherian ring A to be Gorenstein, it is necessary and sufficient that the A-module A be dualizing (S 3, No. 7, prop. 11). In particular, the A-module A is dualizing when A… 0DBH
Remark 1 Let A be a local noetherian ring and Ω a finitely generated A-module. The residue field κ_widehatA identifies with κ_A, and the widehatA-module widehatΩ with widehatA ⊗_A Ω (III, S… 0DBI
Remark 2 Let Ω be a dualizing A-module; for every projective A-module L of rank 1, the A-module Ω ⊗_A L is dualizing (A, X, p. 108, prop. 7, b)). We shall see below (No. 4, prop. 6) that ev… 0DBJ
Proposition 1 Let A be a noetherian ring and Ω a dualizing A-module. 0DBK
Proposition 2 Let A be a noetherian ring, Ω a dualizing A-module. For every prime ideal p of A, the Aₚ-module Ωₚ is dualizing. 0DBL
Lemma 1 Let B be a local noetherian integral ring of dimension 1, and M a finitely generated torsion-free B-module. Suppose that one has long_B(M/xM) = long_B(B/xB) for every nonzero eleme… 0DBM
Corollary 1 For every multiplicative subset S of A, the S⁻¹A-module S⁻¹Ω is dualizing. 0DBN
Corollary 2 The support of Ω is equal to Spec(A). 0DBO
Corollary 3 Let M be a finitely generated A-module, and let i be an integer. The A-module Ext_Aⁱ(M, Ω) is finitely generated, and its support is of codimension geqslant i in Spec(A). 0DBP
Proposition 3 Let A be a local noetherian ring, Ω a dualizing A-module and M a finitely generated A-module. 0DBQ
Corollary If M is Macaulay, then Ext_Aⁱ(M, Ω) = 0 for i ≠ c; the A-module Ext_Aᶜ(M, Ω) is Macaulay, and its support is equal to that of M. 0DBR
Proposition 4 Let A be a noetherian ring, J an ideal of A generated by an A-regular sequence x, and Ω a finitely generated A-module. 0DBS
Corollary 1 Let A be an integral noetherian ring. Every dualizing A-module is torsion-free and of rank 1. 0DBT
Corollary 2 Let A be a local Macaulay ring, \Omega a finitely generated A-module, and x a maximal secant sequence of elements of mA, generating an ideal J. The following conditions are equival… 0DBU
Proposition 5 Let ρ : A → B be a homomorphism of noetherian rings, making B into a flat A-module. Assume that for every maximal ideal n of B, the ring κ(ρ⁻¹(n)) ⊗_A B is a Gorenstein ring. Let Ω… 0DBV
Corollary Let A be a noetherian ring admitting a dualizing module Ω; let B be a polynomial algebra over A in a finite number of indeterminates. The B-module Ω_(B) is dualizing. 0DBW
Proposition 6 Let A be a noetherian local ring and Ω a dualizing A-module. Let B be a finite A-algebra; suppose that the A-module B is Macaulay. The B-module Ext_Aⁱ(B, Ω) is zero for i ≠ dim(A)… 0DBX
Corollary 1 Let A be a noetherian ring, Ω a dualizing A-module, and B a finite A-algebra; suppose that the A-module B is Macaulay. The B-module Ext_A(B, Ω) is dualizing. 0DBY
Remark Let us retain the assumptions of cor. 1 and suppose further that the canonical homomorphism ρ : A → B is injective. Then one has dim(Aₘ) = dim(Bₘ) for every maximal ideal m of A (V… 0DBZ
Corollary 2 If a noetherian ring A has a dualizing module, every finitely generated A-algebra which is a Macaulay ring has a dualizing module. 0DC0
Corollary 3 Every presentable Macaulay ring (in particular, every complete local Macaulay ring) has a dualizing module. 0DC1
Lemma 2 Let A be a noetherian ring, M and N finitely generated A-modules, u : M → N a homomorphism. Let x be an element of the radical of A, such that the homothety x_N is injective. If th… 0DC2
Proposition 7 Let A be a noetherian ring and Ω a dualizing A-module. 0DC3
Corollary 1 In order that A be a Gorenstein ring, it is necessary and sufficient that the A-module Ω be projective of rank 1. 0DC4
Corollary 2 Suppose that the ring A is presentable. The set of prime ideals p of A such that Aₚ is a Gorenstein ring is open in Spec(A). 0DC5
Theorem 1 Let M be a finitely generated A-module. Then α_M is a homologism: we have Hᵢ(D(D(M))) = 0 for i ≠ 0 and the homomorphism α_M is bijective. 0DC6
Corollary If the A-module M is Macaulay, the homomorphism β_M is bijective. 0DC7
Proposition 8 The A-module V/Ω is a Matlis module. 0DC8
Proposition 9 Let B be the integral closure of A, and c = A : B. Suppose that B is a finitely generated A-module. One has the inequality [B/c] ≤ 2[B/A] in Z₀(A). For there to be equality, it is… 0DC9

§ 10. Local cohomology, grothendieck duality

Remark 1 Let X be the topological space Spec(A), O_X the structural sheaf of rings and widetildeM the O_X-module associated with M. The graded A-module H_A(M) is identified with the module… 0DCM
Example 1 If M is injective, the A-module H_Aⁱ(M) is zero for i > 0 and injective for i = 0 (§ 8, No. 2, lemma 1, c)). 0DCN
Example 2 If H_A⁰(M) = M (for example if M is artinian), H_Aⁱ(M) is zero for i > 0. Let in fact (I, e) be an injective envelope of M. The submodule H_A⁰(I) of I is injective (example 1) and… 0DCO
Example 3 Let Ω be a dualizing A-module. For i ≠ dim(A), one has Ext_Aⁱ(A/α, Ω) = 0 for every ideal of definition α of A (§ 8, No. 5, example 3), whence H_Aⁱ(Ω) = 0; for i = dim(A), the A-mo… 0DCP
Example 4 Let A be a noetherian local integral ring; let K denote its field of fractions, and suppose that A ≠ K. It is an injective A-module (A, X, p. 18, example 1), so that the module H_A… 0DCQ
Example 5 Let us retain the hypotheses of the preceding example and suppose moreover that dim(A) = 1. Let N be a torsion A-module; since every nonzero ideal of A distinct from A is an ideal… 0DCR
Proposition 1 Let A be a noetherian local ring and M a finitely generated A-module. 0DCS
Corollary Let M be a nonzero Macaulay A-module of finite type. The A-module H_Aⁱ(M) is zero for i ≠ dim(M) and nonzero for i = dim(M). 0DCT
Remark 2 For every ideal of definition α of A, the A-module Ext_A(A/α, M) is annihilated by α, and A/α identifies with widehatA/α widehatA; consequently, the graded A-module Ext_A(A/α, M) i… 0DCU
Remark 1 Let f : M → N be a homomorphism of a-modules. Using A, X, p. 103, prop. 2, one proves that the following diagrams are commutative: 0DCV
Remark 2 Let (E) 0 → M → N → P → 0 be an exact sequence of a-modules. Using A, X, p. 104, prop. 3 and p. 106, prop. 4, one proves that the following diagrams are commutative: 0DCW
Remark 3 Consider the isomorphism τᵈ(M) : M ⊗_A H_Aᵈ(A) longrightarrow H_Aᵈ(M) For x ∈ M, let us denote by fₓ the mapping a ↦ ax of A into M. For every u ∈ H_Aᵈ(A), one has τᵈ(M)(x ⊗ u) = H… 0DCX
Proposition 2 Let A be a local Macaulay ring, of dimension d, and let Ω be a dualizing A-module. The homomorphism ω : H_Aᵈ(A) → D(Ω) is bijective. 0DCY
Corollary The homomorphism D(ω) : widehatΩ → D(Hᵈ_A(A)) is an isomorphism. 0DCZ
Theorem 1 (Grothendieck Duality) Let A be a local Macaulay ring, of dimension d, and let Ω be a dualizing A-module. 0DD0
Corollary Let A be a local Macaulay ring, and M a nonzero finitely generated A-module, of dimension e. The A-module H_Aᵉ(M) is nonzero. 0DD1
Remark 1 When the A-module M is finitely generated, the widehatA-module Ext_Aᵈ⁻ⁱ(M, widehatΩ) is identified with widehatA ⊗_A Ext_Aᵈ⁻ⁱ(M, Ω) (A, X, p. 108, Proposition 7, c)), and δⁱ(M) can… 0DD2
Remark 2 Let u : M → M' be a homomorphism of A-modules. By Remark 1 of No. 2 and that of § 8, No. 7, the following diagrams are commutative: 0DD3
Remark 3 Let 0DD4
Example Let A be an integral noetherian local ring of dimension 1; let K denote its field of fractions. Let Ω be a dualizing A-module, and let M be a finitely generated A-module. The A-mod… 0DD5