Index of notation

The reference numbers indicate the chapter, section and sub-section (or exercise) in that order.

1, (E a set), U . V, UV (U, V additive subgroups), $a^0$ (a an ideal): Preliminary conventions of Chapter I $E : F$ : I.2.10 $A[S^{-1}], a/s$ (A a ring, S a subset of A, $a \in A$, s a product of elements of S): II.2.1 $i_A^S$: II.2.1 $S^{-1}A, A_p$ (S a multiplicative subset, p a prime ideal): II.2.1 $M[S^{-1}], m/s, i_M^S$ (M an A-module, S a subset of A, $m \in M$, s a product of elements of S): II.2.2 $S^{-1}M, M_p$ (M an A-module, S a multiplicative subset of A, p a prime ideal of A): II.2.2 $S^{-1}u, u_p$ (u an A-module homomorphism): II.2.2 $r(a)$ (a an ideal): II.2.6 $V(M), V(f)$ (M a subset of the ring A, $f \in A$): II.4.3 $\operatorname{Spec}(A)$: II.4.3 $X_f$ ($f \in A, X = \operatorname{Spec}(A)$): II.4.3 $\mathfrak{g}(Y)$ (Y a subset of $\operatorname{Spec}(A)$): II.4.3 $ah$ (h a ring homomorphism): II.4.3 $\operatorname{Supp}(M)$ (M an A-module): II.4.4 $A, M_f, u_f$ (A a ring, M an A-module, u an A-homomorphism, $f \in A$): II.5.1 $\operatorname{rg}_p(P)$ (P a projective module): II.5.3 $\operatorname{rg}(P)$ (P a projective module): II.5.3 $P(A), \operatorname{cl}(M)$ (A a ring, M a projective A-module of rank 1): II.5.4 $\mathfrak{C}, \mathfrak{C}(A)$: II.5.7 $\det(u), \chi_u$ (u an endomorphism of a projective module of rank n): II.5. Ex. 9 $A^{(d)}, M^{(d,k)}, M^{(d)}$ (A a graded ring, M a graded A-module): III.1.3 $A_{(p)}, M_{(p)}$ (A a graded ring, p a graded prime ideal of A, M a graded A-module): III.1.4

$gr_n(G), gr(G)$ (G a filtered group): III.2.3 $gr(h)$ (h a homomorphism compatible with the filtrations): III.2.4 $\mathbf{Z}_n$ (n an integer > 1): III.2.12 $\hat{\mathbf{Z}}$: III.2.13 $A\{X_1, \ldots, X_p\}$ (A a linearly topologized ring): III.4.2 $f(b_1, \ldots, b_p)$ (f a restricted formal power series): III.4.2 $f \circ g, M_f, M_f(\mathbf{X}), J_f, J_f(\mathbf{X}), \mathbf{X}, \mathbf{1}$, ($\mathbf{f}, \mathbf{g}$ systems of formal power series, $\mathbf{g}$ without constant term): III.4.4 $f(x)$ (f a system of formal power series, x a system of topological nilpotent elements): III.4.5 $m \times n$ (m an ideal): III.4.5 $Ass_A(M), Ass(M)$ (M an A-module): IV. 1.1 $Ass_f(M)$: IV.1. Ex. 17 $A^G$ (A an algebra, G a group operating on A): V.1.9 $G^Z(p'), G^Z, A^Z(p'), A^Z$ (G a group operating on a ring A', p' a prime ideal of A'): V.2.2 $G^T(p'), G^T, A^T(p'), A^T$ (A group operating on a ring A', p' a prime ideal of A): v.2.2 $K^Z(p'), K^Z, K^T(p'), K^T$ (K the field of fractions of an integrally closed domain A, p' a prime ideal of the integral closure of A in a quasi-Galois extension of K): V.2.3 $Y^p$ (where p = $(p_1, \ldots, p_m)$, the $p_i$ being integers $\geqslant 0$): V.3.1 $m(A), \kappa(A), U(A)$ (A a local ring): VI $R, a$: VI.2.1 $+\infty$: VI.3.1 $\Gamma_A, v_A$: VI.3.2 $a(M)$ (M a major set): VI.3.5 $h(G)$ (G a totally ordered group): VI.4.4 $\mathscr{T}_v$ (v a valuation): VI.5.2 $e(v'/v), e(A'/A), e(L/K)$: VI.8.1 $f(v'/v), f(A'/A), f(L/K)$: VI.8.1 $\varepsilon(G, H)$ (G a totally ordered group, H a subgroup of G of finite index): VI.8.4 $\varepsilon(v'/v)$ (v a valuation, v' an extension of v): VI.8.4 $\operatorname{mod}(x), \operatorname{mod}_K(x)$ (K a non-discrete locally compact field, x $\in$ K): VI.9.1 $r(G)$ (rational rank of a commutative group): VI.10.2 $d(K'/K), s(v'/v), r(v'/v)$ (v a valuation on K, v' an extension of v to a transcendental extension K' of K): VI.10.3 $I(A), D(A)$ (A an integral domain): VII.1.1 $a \prec b, \operatorname{div}(a), \operatorname{div}(x)$ (a, b fractional ideals, x an element of the field of fractions): VII.1.1 $\tilde{a}$ (a a fractional ideal): VII.1.1 $d_1 \leq d_2$ (d_1, d_2 divisors): VII.1.1

$b : a$ (a, $b$ fractional ideals): VII.1.1 $J(A)$ (A an integral domain): VII.1.2 $P(A)$ (A a Krull domain): VII.I.3 $p^{(n)}$ ($p$ a divisorial prime ideal): VII.1.4 $v_p$ ($p$ a prime ideal of height 1 in a Krull domain): VII.1.10 $F(A), C(A)$ (A a Krull domain): VII.1.10 $e(\mathfrak{P}/p)$ ($p \in P(A), \mathfrak{P} \in P(B), A \subset B, \mathfrak{P} \cap A = p$) VII.1.10 $i$ (homomorphism from $D(A)$ to $D(B)$, or of $C(A)$ to $C(B)$): VII.1.10 $\bar{i}$ (homomorphism from $C(A)$ to $C(B)$): VII.1.10 $A, A_0, \Delta(K)$ (rings of restricted adèles): VII.2.4 $\mathfrak{P}^*, \mathfrak{P}^*(A)$ (A an integral domain): VII.3.2 $M^*$ (dual lattice of a lattice $M$): VII.4.2 $l_p(T), \chi(T)$ (T a torsion A-module, $p$ a prime ideal of height 1): VII.4.5 $F(A), T(A), cl(M)$: VII.4.5 $\chi(M, M')$ (M, $M'$ lattices): VII.4.6 $c(d)$ (da divisor): VII.4.7 $c(M), r(M), \gamma(M)$ (M a lattice): VII.4.7 $\mathfrak{P}|p, e_{\mathfrak{P}/p}, f_{\mathfrak{P}/p}, f(\mathfrak{P}/p)$ (A $\subset$ B Krull domains such that B is a finite A-algebra $P \in P(A), \mathfrak{P} \in P(B), \mathfrak{P} \cap A = p$): VII.4.8 $N_{B/A}, N, i_{B/A}$: VII.4.8