Index of Notation

References are by (N.M.K) for items in Chapter 0 and (I, N.M.K) for Chapter I.

Chapter 0

Notation	Description	Reference
$M_{[ϕ]}$	$M$ as $A$ -module via $ϕ : A \to B$ (restriction of scalars)	(0.1.0.2)
$B J$ , $J B$	Ideals in $B$ generated by image of ideal $J \subset A$	(0.1.0.3)
$r (a)$	Radical of an ideal $a$	(0.1.1.1)
$R (A)$	Radical of a ring $A$	(0.1.1.2)
$S_{f}$	Multiplicative set $f^{n} : n \geq 0$ for $f \in A$	(0.1.2.1)
$S^{- 1} A$ , $S^{- 1} M$ , $m / s$ , `iˢ`, $i_{M}^{s}$	Ring/module of fractions and canonical maps	(0.1.2.2)
$A_{f}$ , $M_{f}$ , $A_{p}$ , $M_{p}$	Specializations of fractions notation	(0.1.2.3)
$S^{- 1} u$	Localization of a homomorphism	(0.1.3.1)
$ρ_{A}^{T, S}$ , $ρ_{M}^{T, S}$ , $ρ^{T, S}$	Change-of-multiplicative-set maps	(0.1.4.1)
$S u pp (M)$	Support of an $A$ -module	(0.1.7.1)
$F ∥ U$ , `u\|U`	Restriction of sheaf and morphism	(0.3.1.5)
$F_{x}$ , $s_{x}$ , $Γ (U, F)$ , $u (s)$ , $S u pp (F)$	Stalk, germ, sections, image, support	(0.3.1.6)
$ψ_{*} (F)$	Direct image of presheaf	(0.3.4.1)
$ψ_{*} (u)$	Direct image of morphism	(0.3.4.2)
$ψ_{x}$	Stalk homomorphism for direct image	(0.3.4.4)
$G \to F$	$ψ$ -morphism notation	(0.3.5.1)
$ψ * (G)$ , $ρ_{G}$ , $v^{♭}$ , $u^{♯}$ , $σ_{F}$	Inverse image and adjunction maps	(0.3.5.3)–(0.3.5.5)
$(X, A)$ , $O_{X}$ , $O_{X, x}$ , $O_{x}$ , `1`, $e$	Ringed space, structure sheaf, stalk, unit	(0.4.1.1)
$\overset{ˇ}{F}$ , $⋀^{p} F$	Dual and exterior power	(0.4.1.4)–(0.4.1.5)
$J F$	Ideal-sheaf action on a module	(0.4.1.5)
$Ψ_{} (F)$ , $Ψ_{} (u)$	Direct image for ringed-space morphisms	(0.4.2.1)
$Ψ_{*} (C)$	Direct image of an algebra	(0.4.2.4)
$Ψ * (G)$ , $Ψ * (v)$	Inverse image for ringed-space morphisms	(0.4.3.1)
$Ψ * (C)$	Inverse image of an algebra	(0.4.3.4)
$J A$ , $J A \cdot O_{X}$	Image of ideal sheaf	(0.4.3.5)
$G \to F$	$Ψ$ -homomorphism	(0.4.4.1)
$u^{♯}$ , $v^{♭}$ , $u^{♯}$ , $v^{♭}$ , $ρ_{G}$ , $σ_{F}$	Adjunction maps for $Ψ$ -homomorphisms	(0.4.4.3)
$u_{1} \otimes u_{2}$	Tensor product of $Ψ$ -homomorphisms	(0.4.4.4)
$L^{- 1}$	Inverse of an invertible $O_{X}$ -module	(0.5.4.3)
$L^{\otimes n}$	Tensor power of an invertible sheaf	(0.5.4.4)
$Γ_{} (L)$ , $Γ_{} (L, F)$	Graded ring/module of invertible-sheaf sections	(0.5.4.6)
$M (X)$	Group of invertible $O_{X}$ -modules	(0.5.4.7)
$m_{x}$ , $κ (x)$ , $f (x)$	Maximal ideal, residue field, value of section	(0.5.5.1)
$X_{f}$	Open set where $f$ does not vanish	(0.5.5.2)
$I m (M^{'} \otimes_{A} N^{'})$	Image in tensor product	(0.6.0)
`Â`, $\hat{M}$	Completions (separated, for various topologies)	(0.7.2.3) and (0.7.3.1)
$A T_{1}, \dots, T_{r}$	Ring of restricted formal series	(0.7.5.1)
$A S^{- 1}$	Completed ring of fractions	(0.7.6.1)
$a S^{- 1}$	Completed image of an open ideal	(0.7.6.9)
$A_{f}$ , $a_{f}$	Completed ring of fractions with $S_{f}$ denominators	(0.7.6.14)–(0.7.6.15)
$A_{S}$	Inductive limit of $A_{f}$ over $f \in S$	(0.7.6.15)
$(M \otimes_{A} N)^{\land}$ , $\hat{M} \hat{\otimes}_{\hat{A}} \hat{N}$	Completed tensor product	(0.7.7.1)
$u \hat{\otimes} v$	Completed tensor product of homomorphisms	(0.7.7.3)

Chapter I

Notation	Description	Reference
$Spec (A)$ , $j_{x}$ , $m_{x}$ , $κ (x)$ , $f (x)$ , $M_{x}$ , $r (E)$ , $V (E)$ , $V (f)$ , $D (f)$	Affine-scheme notation	(I, 1.1.1)
$j (Y)$	Annihilator ideal of a subset of $Spec (A)$	(I, 1.1.3)
$^{a} ϕ$	Map of spectra associated with a ring homomorphism	(I, 1.2.1)
$S_{f}$ , $S_{f}^{'}$	Multiplicative set and its saturation	(I, 1.3.1)
$ρ_{g, f}$	Localization map for $D (g) \subset D (f)$	(I, 1.3.3)
`Ã`, $\tilde{M}$ , $O_{X}$ , $θ_{f}$	Structure sheaf, sheaf associated with a module, canonical map	(I, 1.3.4)
`ũ`	Sheaf morphism associated with module morphism	(I, 1.3.5)
$\tilde{ϕ}$	Sheaf morphism associated with ring homomorphism	(I, 1.6.1)
$A (X)$	Ring of an affine scheme $X$	(I, 1.7.1)
$O_{X / Y}$	Local ring of $X$ along closed irreducible $Y$	(I, 2.1.6)
$Hom (X, Y)$	Morphisms of preschemes	(I, 2.2.1)
$Hom_{S} (X, Y)$ , `1_X`	$S$ -morphisms and identity	(I, 2.5.2)
$Γ (X / S)$	$S$ -sections of $X$	(I, 2.5.5)
$X ⨆ Y$	Sum of preschemes	(I, 3.1.1)
$X \times_{S} Y$ , $X \times Y$ , $(g, h)_{S}$ , $u \times_{S} v$ , $u \times v$	Product and morphism of products	(I, 3.2.1)
$X \otimes_{A} B$ , $(g, h)_{A}$ , $u \otimes_{A} v$	Base change for $A$ -preschemes	(I, 3.2.1)
$X_{(S^{'})}$	Base change of an $S$ -prescheme to $S^{'}$	(I, 3.3.6)
$f_{(S^{'})}$	Base change of a morphism	(I, 3.3.7)
$Γ_{f}$	Graph morphism	(I, 3.3.14)
$X (T)$	$T$ -valued points of $X$	(I, 3.4.1)
$P \times_{R} Q$	Fiber product of sets	(I, 3.4.2)
$X (T)_{S}$	$T$ -valued points over $S$	(I, 3.4.3)
$X (B)$ , $X (B)_{A}$	$B$ -valued points of an $A$ -prescheme	(I, 3.4.4)
$X \otimes_{y} B$ , `X ⊗_𝒪_y B`	Base change to algebra $B$ over $κ (y)$	(I, 3.6.3)
$Z \lor Y$	Union of subpreschemes (in (I, 4.1.10))	(I, 4.1.10)
$f^{- 1} (Y)$	Inverse image of a subprescheme	(I, 4.4.1)
$N$	Nilradical sheaf	(I, 5.1.1)
$X_{re d}$	Reduced prescheme associated with $X$	(I, 5.1.3)
$f_{re d}$	Morphism of reduced preschemes	(I, 5.1.5)
$Δ_{X / S}$ , $Δ_{X}$ , $Δ$	Diagonal morphism	(I, 5.3.1)
$r g_{se p} (X)$	Separable rank of a finite $K$ -scheme	(I, 6.4.5)
$n (X)$	Geometric number of points of a finite $K$ -scheme	(I, 6.4.8)
$R (X / S, Y)$ , $R a t_{S} (X, Y)$	Set of rational $S$ -maps $X ⇢ Y$	(I, 7.1.2)
$R (X)$	Ring of rational functions on $X$	(I, 7.1.3)
$K (X)$ , $K_{X}$	Sheaf of rational functions	(I, 7.3.1)
$L (A)$	Local rings of an integral ring $A$	(I, 8.1.2)
$δ (f)$	Domain of definition of a rational map	(I, 8.2.1)
$F \otimes_{O} G$ , $F ⊠_{S} G$	Tensor product on possibly distinct preschemes	(I, 9.1.2)
$\tilde{G}$ (extension)	Extension of a quasi-coherent sheaf	(I, 9.4.1)
`Ȳ`	Closure of a subprescheme	(I, 9.5.11)
$Sp f (A)$ , $O_{X}$ ( $X = Sp f (A)$ )	Formal spectrum and structure sheaf	(I, 10.1.2)
$D (f)$	Open set in $Sp f (A)$ corresponding to $f$	(I, 10.1.4)
$^{a} ϕ$ , $\tilde{ϕ}$ (formal)	Morphism associated with continuous ring hom.	(I, 10.2.1)
$J A$ (formal)	Ideal sheaf for ideals of definition	(I, 10.3.1)
$X \hat{\times}_{S} Y$ , $X \hat{\times} Y$	Fiber product of formal preschemes	(I, 10.7.3)
$F_{/ X^{'}}$ , $u_{/ X^{'}}$	Formal completion of sheaf/morphism along closed subset	(I, 10.8.4)
$X_{/ X^{'}}$ , $\hat{X}$	Formal completion of a prescheme	(I, 10.8.5)
$\hat{f}$	Extension of a morphism to completions	(I, 10.9.1)
$M^{\land}$ , $u^{\land}$	Completion of a module/homomorphism (adic)	(I, 10.10.1)
$X \cap Y$ , `X_Y`	Intersection of formal subpreschemes / restriction	(I, 10.15.1)

Éléments de Géométrie Algébrique — English

Index of Notation

Chapter 0

Chapter I