Exercise 10, § 9

Soient A un anneau, M un a-module, $x = (x_1, ..., x_n)$ a sequence of elements of A, p an integer $\leq n$. We denote by $x' = (x_1, ..., x_p)$, $x'' = (x_{p+1}, ..., x_n)$ and $M' = M/(x')$ M. a) Let k, k’ be two integers such that $k \leq k'$. Suppose that the sequence x’ is k’-sécante for M (Exercise 9). Prove that the sequence x is k-sécante for M if and only if the sequence x” is k-sécante for M’. b) Prove that if the sequence x is 1-sécante for M, the sequence x” is 1-sécante for M’. † 11) Let $A$ be a noetherian local ring, m its maximal ideal, $k = A/m$. Let M be a nonzero a-module of finite type. We call the depth of M, and denote it by prof (M), the supremum of the integers n for which there exists a sequence $(x_1, ..., x_n)$ of elements of m which is completely sécante for M. a) Prove that $\operatorname{prof}(M) = 0$ if and only if m is associated to M (X, p. 172, Exercise 28), in other words, if M contains a submodule isomorphic to k (observe that an ideal contained in a union of prime ideals is necessarily contained in one of them).

b) Prove that prof (M) is the greatest lower bound of the integers $n$ for which $\operatorname{Ext}_A^n(k, M)$ is nonzero (reason by induction on $n$). If $\operatorname{prof}(M) = n$, any sequence $(x_1, ..., x_p)$ of elements of m completely sécante for M ($p \leq n$) can be extended to a sequence $(x_1, ..., x_n)$ in m completely sécante for M.

c) If $(x_1, ..., x_k)$ is a sequence of elements of m completely sécante for M, prove that

d) Suppose that $M$ is of finite projective dimension. Prove the equality $\mathrm{dp}_A(M) + \operatorname{prof}(M) = \operatorname{prof}(A)$ (reason by induction on $\mathrm{dp}_A(M)$).

e) Prove that $\operatorname{prof}(A)$ is the supremum of the projective dimensions of the a-modules which are of finite type and of finite projective dimension.

†12) Let $A$ be a noetherian local ring, m its maximal ideal. a) Prove that if $0 < \operatorname{dh}(A) < \infty$, there exists a non-zero-divisor element in $m - m^2$ (show as in Exercise 11, a) that m would otherwise be associated to $A$, whence an exact sequence

arrive at a contradiction using Exercise 16, p. 203). b) Let $n$ be an integer. Prove that one has $\operatorname{dh}(A) = n$ if and only if there exists a sequence $(x_1, ..., x_n)$ completely sécante for $A$, generating m (reason by induction on $n$, using a) and Exercise 14, p. 203). One says that a local ring is regular if it is noetherian and of finite homological dimension.

13) Let $A$ be a regular local ring (Exercise 14) of homological dimension $n$; we denote by m its maximal ideal and $k = A/m$. a) Prove that one has $\dim_k(m/m^2) = n$. b) Prove that the $k$-algebra $\bigoplus_{r \geq 0} (m^r/m^{r+1})$ is isomorphic to an algebra of polynomials in $n$ indeterminates. c) If $n = 0$, $A$ is a field. If $n = 1$, prove that $A$ is a principal ideal domain and therefore a discrete valuation ring (prove that $A$ is integral by observing that $m \cap m^n = \bigcap m^n$, whence $\bigcap m^n = 0$). d) Prove that the ring $A[[T]]$ is regular.

14) Let $A$ be a ring, $a$ an ideal of $A$. Consider the alternating graded $(A/a)$-algebra $\operatorname{Tor}^A(A/a, A/a)$ (cf. X, p. 201, exercise 9) and the homomorphism of graded $(A/a)$-algebras $\theta : \Lambda_{A/a}(a/a^2) \to \operatorname{Tor}^A(A/a, A/a)$ deduced from the isomorphism $a/a^2 \to \operatorname{Tor}_1^A(A/a, A/a)$ (X, p. 73). Prove that if the ideal $a$ is generated by a completely transverse sequence in $A$, $\theta$ is an isomorphism.

15) Let $A$ be a noetherian ring, $a$ an ideal contained in the radical of $A$. Suppose that the $(A/a)$-module $a/a^2$ is free and that the homomorphism $\theta_2 : \Lambda_{A/a}^2(a/a^2) \to \operatorname{Tor}_2^A(A/a, A/a)$ (exercise 14) is surjective. Prove that $a$ is generated by a completely transverse sequence in $A$. (If $x$ is a minimal family of generators of $a$, show with the aid of exercise 7 (or directly) that $\operatorname{Coker}(\theta_2)$ is isomorphic to $H_1(x, A) \otimes_A A/a$.)

16) Let $A$ be a local noetherian ring, $m$ its maximal ideal, $k = A/m$. Prove that the homomorphism of graded $k$-algebras $\theta : \Lambda_k(m/m^2) \to \operatorname{Tor}^A(k, k)$ is injective: if $\theta_2$ is bijective, the ring $A$ is regular (exercise 12). (If $x$ is a minimal system of generators of $m$, show that the complex $K(x, A)$ satisfies the conditions of exercise 14, p. 182; use exercises 4 and 15.)

†17) Let $A$ be a ring such that the $A$-module $A_s$ is of finite injective dimension (X, p. 202, exercise 4; a regular local ring (exercise 12) satisfies this property). a) Let $x$ be an element which is not a zero divisor and is not invertible in $A$. Prove that one has

(Let $0 \to A \to I^0 \to I^1 \to \cdots$ be a minimal injective resolution (X, p. 182, exercise 13) of $A$; show that the complex $\operatorname{Hom}_A(A/Ax, I^1) \to \operatorname{Hom}_A(A/Ax, I^2) \to \cdots$ defines a minimal injective resolution of the $(A/Ax)$-module $A/Ax$.) b) Suppose now that the ring $A$ is local noetherian. Prove the equality $\operatorname{di}(A) = \operatorname{prof}(A)$ (exercise 11; reason by induction on $\operatorname{di}(A)$). c) Let $(x_1, ..., x_n)$ be a completely transverse sequence for $A$, contained in the maximal ideal of $A$, with $n = \operatorname{di}(A)$. Prove that the ring $A$ is autoinjective (X, p. 171, exercise 26).

Soient $A$ a noetherian ring, $\mathscr{C}$ a stable set of classes of $A$-modules (X, p. 40), $x = (x_1, ..., x_n)$ a sequence of elements of $A$, $x$ the ideal generated by $x$. We denote by $\mathscr{C}_x$ the set of elements of $\mathscr{C}$ which are annihilated by $x$. Let $M$ be a finitely generated $A$-module. a) Exercise prove that for every $i \geqslant 0$, there exists an element $M_i$ of $\mathscr{C}_x$ such that one has

in $K(\mathscr{C}_x)$ (reason by induction on $n$, using Proposition 4, p. 157; in the case $n = 1$, consider the successive powers of the endomorphism $(x_1)_M$). b) Suppose that $A/x$ is of finite length. Show that there exist positive integers $m_i$ ($i \geqslant 0$) such that one has $\operatorname{long} H_i(x, M) = m_i + m_{i+1}$.

Index of notation

$(C,d)$ : p. 23.

$C_n, C^\circ$ : p. 24.

$Z(C,d), Z_n(C,d), B(C,d), B_n(C,d), H(C),$ $H_n(C,d), H^n(C,d)$ : p. 25.

$H(u), Z(u), B(u)$ : p. 26.

$C(p)$ : p. 26.

$A(\mathscr{E})$ : p. 27.

$\hat{\partial}_{u,v}, \hat{\partial}$ : p. 29.

$\operatorname{Con}(u), \operatorname{Cyl}(u)$ : p. 36.

$K(\mathscr{E}), [M]_{\mathscr{E}}, [M]$ : p. 40.

$\chi_p(M), \chi(M)$ : p. 41.

$\Omega^p_{A/k}$ : p. 43.

$\nabla$ : p. 45.

$L(M), L(f)$ : p. 50.

$I(M), I(f)$ : p. 52.

$B(A,M)$ : p. 57.

$C \otimes_A C'$ : p. 61.

$\gamma(C,C') : H(C)\otimes_A H(C') \to H(C\otimes_A C')$ : p. 62.

$\sigma(C,C')$ : p. 62.

$\operatorname{Tor}^A_n(M,N)$ : p. 67.

$\operatorname{Tor}^A(f,g)$ : p. 68.

$\psi_M(N), \psi_N(M)$ : p. 69.

$\sigma_{M,N}$ : p. 71.

$\operatorname{Homgr}_A(C,C')$ : p. 81.

$\lambda(C,C')$ : p. 82.

$\operatorname{Ext}^A(M,N), \operatorname{Ext}_A(M,N)$ : p. 86.

$\varphi_M(N), \varphi_N(M)$ : p. 88.

$\delta(M,\mathscr{E})$ : p. 90.

$\delta(\mathscr{F},N)$ : p. 92.

$\psi(S,R), \varphi(R,E)$ : p. 100.

$\psi'(S,R), \varphi'(R,E_1)$ : p. 102.

$H^n(G,M), H_n(G,M)$ : p. 111.

$a_{M,N}$ : p. 113.

$c_{M,N}, u \circ v$ : p. 114.

$\partial^m(P,\mathscr{P}), \partial^m(\mathscr{P},P), \partial^m(P,\mathscr{J}), \partial^m(\mathscr{J},M)$ : 127, 131, 132.

$c_{P,Q;M}$ : p. 128.

$c_{P;M,N}$ : p. 129.

$d_{pA}(M)$ : p. 134.

$d_h(A)$ : p. 138.

$K(u), K(u,C), K'(u,C)$ : p. 147.

$b \vee b',\ aTa'$ : p. 200, exerc. 7.

$d_iA(M)$ : p. 202, exerc. 4.

$d_{pA}(M)$ : p. 202, exerc. 7.

$td(A)$ : p. 202, exerc. 8.

$d_{a_k}(A)$ : p. 205, exerc. 25.

$d_c(G)$ : p. 206, exerc. 28.

Terminological index

Absolutely flat (ring —) : p. 169, exerc. 16. Acyclic in degree $n$ (complex —) : p. 26. $x$-adic (topology —) : p. 159. Augmented algebra : p. 196, exerc. 27. Differential graded algebra : p. 183, exerc. 18. Ascending (degree —) : p. 23. Associated (class —) with an exact sequence : p. 117. Augmentation of an algebra : p. 196, exerc. 27. Autoinjective (ring) : p. 171, exerc. 26.

Bicomplex : p. 174, exerc. 11. Bounded on the right, on the left : p. 24.

Canonical (injective resolution —) : p. 52. Canonical (free resolution —) : p. 50. Euler-Poincaré characteristic : p. 40. Cellular (complexes —) : p. 32. Chafarévitch (Golod- — theorem) : p. 192, exerc. 16. Five (five lemma) : p. 7. Universal coefficients (formula of —) : p. 78, 99. Cogenerator (module —) : p. 18. Left, right coherent (ring —) : p. 181, exerc. 11. Coherent, pseudo-coherent (module —) : p. 181, exerc. 10. Cohomology of groups : p. 111. Cohomological (dimension —) : p. 206, exerc. 28. Cohomologous : p. 194, exerc. 23. Co-induced (module —) : p. 190, exerc. 12. Commutation (isomorphism of —) : p. 71. Completely secant (family —) : p. 157. Complexes : p. 23. Koszul complexes : p. 147, 154. Composition (product of —) : p. 113. Cone of a morphism of complexes : p. 36. Conical (simplicial scheme —) : p. 177, exerc. 19. Connection : p. 45. Convergent (spectral sequence —) : p. 175, exerc. 14. Coregular (sequence —) : p. 159. Cylinder of a morphism of complexes : p. 36. Dedekind (rings of —) : p. 141, 203, exerc. 12. De Rham (complex of —) : p. 43. Descending (degree —) : p. 23. Commutative diagrams : p. 1. Differential : p. 24. Differential (algebra —) : p. 183, exerc. 18.

Exterior differential : p. 44. Cohomological dimension : p. 206, exerc. 28. Homological dimension : p. 138. Injective dimension : p. 202, exerc. 4. Flat dimension : p. 202, exerc. 7. Projective dimension : p. 134. Divisible (module —) : p. 17. Dualizing (module —) : p. 172, exerc. 30.

Injective envelope : p. 19. Euler-Poincaré (characteristic of —) : p. 40. Extensions (extension modules) : p. 81, 86. Extension of an algebra : p. 197, exerc. 1. Extension of a Lie algebra : p. 198, exerc. 4.

Faithfully flat (module —) : p. 169, exerc. 9. Differential forms : p. 44.

Goldie (theorem of —) : p. 171, exerc. 25. Golod-Chafarévitch (theorem of —) : p. 192, exerc. 16.

Hilbert (syzygy theorem of —) : p. 146. Homology (classes of —, homology module) : p. 25. Homology (exact sequence of —) : p. 30. Homology of groups : p. 111. Homological (dimension —) : p. 138. Homologism : p. 26. Homologous (cycles —) : p. 25. Homomorphisms (complexes of —) : p. 81. Homotopic to zero (complex —) : p. 34. Homotopic (morphisms —) : p. 32. Simplicially homotopic (—) : p. 177, exerc. 19. Homotopy : p. 32. Homotopism : p. 33.

Induced (module —) : p. 190, exerc. 12. Injective (modules —) : p. 15. Injective (dimension —) : p. 202, exerc. 4. Iterated (connecting homomorphisms —) : p. 127, 131, 132.

Koszul (complex of —) : p. 147, 154. Künneth (formula of —) : p. 79.

Connecting (homomorphism of —) : p. 29. Connecting (homomorphism of —) of extension modules : p. 90, 92. Connecting (homomorphism of —) of torsion products : p. 72. Length of a complex : p. 24. Length (of a presentation) : p. 180, exerc. 6.

Minimal (injective resolution —) : p. 182, exerc. 13. Minimal (projective resolution —) : p. 54. Morphism of complexes : p. 25.

Zero on the right, on the left : p. 24.

Flat (absolutely —) : p. 169, exerc. 16. Flat (faithfully —) : p. 169, exerc. 9. Flat (dimension —) : p. 202, exerc. 7. Flat (modules —) : p. 8. Poincaré (Euler- — characteristic) : p. 40. Poincaré (polynomial of —) : p. 41. Precomplex : p. 174, exerc. 13. Presentation, finite presentation : p. 10. Presentation of length $n$ : p. 180, exerc. 6. Composition product : p. 113, Torsion product : p. 61, 67. Projective (dimension —) : p. 134. Prüferian (ring —) : p. 203, exerc. 11. Pseudo-coherent (module —) : p. 181, exerc. 10.

Regular (local ring —) : p. 208, exerc. 12. Regular (sequence —) : p. 158. Regular (spectral sequence —) : p. 175, exerc. 14. Relatively projective, injective : p. 170, exerc. 19. Resolutions : p. 48. Canonical injective resolution : p. 52. Canonical free resolution : p. 50. Graded resolution : p. 56. Minimal projective resolution : p. 54. Projective, injective resolution of a complex : p. 183, exerc. 17.

Simplicial scheme : p. 177, exerc. 19. Split (complex —) : p. 35. Splitting : p. 35. Secant (completely —) : p. 157. Snake (diagram of the —) : p. 3. Simplex : p. 178, exerc. 19. Simplicial (scheme —) : p. 177, exerc. 19. Simplicially homotopic : p. 177, exerc. 19. Singular (homology —) : p. 31. Spectral (sequence —) : p. 175, exerc. 14. Standard (resolution —) : p. 57. Stationary (spectral sequence —) : p. 175, exerc. 14. Exact homology sequence : p. 30. Spectral sequence : p. 175, exerc. 14. Syzygies (theorem of —) : p. 146.

Tensor product of two complexes : p. 61. Torsion (product of —) : p. 61, 67. Tor-dimension : p. 202, exerc. 8. $x$-adic topology : p. 159. ($p$-th) Translated : p. 27. Transfer : p. 191, exerc. 14. Trivial (cohomologically —) : p. 192, exerc. 18.

Universal (coefficients —) : p. 78, 99.

Table of contents

CHAPTER X. — HOMOLOGICAL ALGEBRA ............................................. X.1

§ 1. Complements of linear algebra ............................................. X.1 1. Commutative diagrams .................................................. X.1 2. The snake diagram ...................................................... X.3 3. Flat modules ............................................................... X.8 4. Modules of finite presentation ....................................... X.10 5. Homomorphisms of a module of finite presentation ............ X.12 6. Structure of flat modules ............................................. X.12 7. Injective modules ....................................................... X.19 8. Injective cogenerators ............................................... X.20 9. Injective envelopes ................................................... X.21 10. Structure of injective modules .................................... X.22

§ 2. Complexes of A-modules .................................................. X.23 1. Complexes of A-modules ............................................. X.23 2. Operations on complexes ............................................. X.27 3. The connecting homomorphism and the exact homology sequence. X.29 4. Homotopies ............................................................. X.32 5. Split complexes ....................................................... X.34 6. Cone and cylinder of a morphism of complexes ................ X.36 7. The cone of an injective morphism; new definition of the connecting homomorphism ............................................. X.39 8. Euler-Poincaré characteristics .................................... X.40 9. Complexes of right modules, complexes of multimodules .... X.43 10. Example: de Rham complex ......................................... X.43

§ 3. Resolutions ............................................................... X.46 1. Extension of morphisms of complexes ............................ X.46 2. Resolutions ............................................................ X.48 3. The canonical free resolution ...................................... X.50 4. The canonical injective resolution ............................... X.52 5. Resolutions of finite type ......................................... X.53 6. Minimal projective resolutions .................................... X.54 7. Graded resolutions .................................................. X.56 8. The standard resolution ........................................... X.57 9. Resolutions and Grothendieck groups ............................ X.58

§ 4. Torsion products ...................................................... X.61 1. Tensor product of two complexes ................................. X.61 2. Tensor products and homotopy ................................... X.64 3. Tensor product by a right bounded flat complex .............. X.66 4. Definition and first properties of the torsion product ..... X.67 5. The connecting homomorphisms and exact sequences ........ X.71 6. Flat modules and torsion products ............................. X.74 7. Künneth formula .................................................... X.76 8. Bounded flat complexes over a Noetherian ring .............. X.79 9. Generalization to complexes of multimodules ................ X.80

§ 5. Extension modules .................................................... X.81 1. Complexes of homomorphisms ................................... X.81 2. Complexes of homomorphisms and homotopies ................ X.83 3. Definition and first properties of extension modules ...... X.86 4. The connecting homomorphisms and exact sequences ........ X.89 5. Projective modules, injective modules and extension modules. X.93 6. Universal coefficient formula ................................... X.94 7. Generalization to complexes of multimodules; the canonical isomorphisms ............................................. X.98

§ 6. Use of non-canonical resolutions ................................. X.100 1. Calculation of the modules Tor^A(P, M) and Ext_A(M, N) ... X.100 2. Calculation of the mappings Tor^A(g, f) and Ext_A(f, h) ... X.103 3. Calculation of the connecting homomorphisms ................. X.104 4. Finiteness of extension and torsion modules ................ X.107 5. The homomorphisms Tor^B(P, N) ⊗_A Q → Tor^B(P, N ⊗_A Q) and Ext_B(M, N) ⊗_A Q → Ext_B(M, N ⊗_A Q) .................. X.108 6. The homomorphisms Tor^B(P, N ⊗_A Q) → Tor^A(P ⊗_B N, Q) and Ext_A(Q, Hom_B(N, M)) → Ext_B(N ⊗_A Q, M) ............... X.109 7. The homomorphisms B ⊗_A Tor^A(E, F) → Tor^B(E ⊗_A B, B ⊗_A F) and B ⊗_A Ext_A(E, F) → Ext_B(B ⊗_A E, B ⊗_A F) ... X.110 8. Application: homology and cohomology of groups .......... X.111

§ 7. Composition product .............................................. X.113 1. The homomorphism Ext_A(N, P) ⊗ Ext_A(M, N) → Ext_A(M, P) X.113 2. The seven calculations of the composition product ........ X.115 3. The class associated with an exact sequence ................ X.116 4. Properties of the class associated with an exact sequence . X.118 5. Relation between exact sequences and elements of Ext_A(M, N) ... X.121 6. Composition product and connecting homomorphisms of extension modules ............................................. X.125 7. The homomorphism Ext_A(P, Q) ⊗ Tor^A(P, M) → Tor^A(Q, M) X.128

8. Composition products and connecting homomorphisms of torsion products ... X.130 9. Calculation of composition products by shifting of resolutions ... X.132 § 8. Homological dimension .................................. X.134 1. Projective dimension of a module .................... X.134 2. The homomorphism TorⁿA(P, M) → HomA(ExtⁿA(M, A), P).... X.137 3. Homological dimension of a ring ..................... X.138 4. Rings of homological dimension 0 .................... X.140 5. Rings of homological dimension 1 .................... X.140 6. Homological dimension of polynomial rings ........... X.141 7. Homological dimension of graded modules ............. X.143 § 9. Koszul complexes ...................................... X.147 1. The complexes K(u), K.(u, C), K⁻(u, C) .............. X.147 2. Functoriality ....................................... X.150 3. Example 1: the complex S(L) ⊗A Λ(L) ................ X.151 4. Example 2: the case of a free module ................ X.153 5. Example 3: the case L = A ........................... X.156 6. Completely secant families .......................... X.157 7. A criterion for completely secant sequences ........ X.159 8. Proof of Theorem 1: first part ...................... X.161 9. Proof of Theorem 1: second part ..................... X.163 10. Class of extensions associated with a regular sequence ... X.165 Exercises of § 1 .............................................. X.168 Exercises of § 2 .............................................. X.173 Exercises of § 3 .............................................. X.178 Exercises of § 4 .............................................. X.184 Exercises of § 5 .............................................. X.187 Exercises of § 6 .............................................. X.189 Exercises of § 7 .............................................. X.197 Exercises of § 8 .............................................. X.202 Exercises of § 9 .............................................. X.206 Index of notations ............................................. X.210 Terminological index .......................................... X.211