Exposé VI. The functors $Ex{t}_Z^{\bullet }(X;F,G)$ and $\mathcal{E}x{t}_Z^{\bullet }(F,G)$

1. Generalities

1.1.

Let $(X,{\mathcal{O}}_X)$ be a ringed space and let $Z$ be a locally closed subset of $X$. Let $F$ and $G$ be ${\mathcal{O}}_X$-Modules; we denote by $Ex{t}_Z^i(X;F,G)$ (resp. $\mathcal{E}x{t}_Z^i(F,G)$) the $i$-th derived functor of the functor $G\mapsto {\Gamma }_Z({\mathrm{Hom}}_{{\mathcal{O}}_X}(F,G))$ (resp. $G\mapsto \Gamma Z(\mathcal{H}o{m}_{{\mathcal{O}}_X}(F,G))$, where $\Gamma Z$ denotes the sheafified sections-with-support functor and $\mathcal{H}o{m}_{{\mathcal{O}}_X}$ the sheafified Hom).

Lemma.

The sheaf $\mathcal{E}x{t}_Z^i(F,G)$ is canonically isomorphic to the sheaf associated with the presheaf $U\mapsto Ex{t}_{Z\cap U}^i(U;F|U,G|U)$.

This follows from (Tôhoku, 3.7.2) together with the fact that $\Gamma (U;\Gamma Z(\mathcal{H}o{m}_{{\mathcal{O}}_X}(F,G)))$ is canonically isomorphic to ${\Gamma }_{Z\cap U}({\mathrm{Hom}}_{{\mathcal{O}}_X|U}(F|U,G|U))$.

Theorem (Excision theorem).

Let $V$ be an open subset of $X$ containing $Z$. Then one has an isomorphism of cohomological functors

Ext^•_X(X; F, G) ≃ Ext^•_V(V; F|V, G|V).

Indeed, if $G^{\bullet }$ is an injective resolution of $G$, then $G^{\bullet }|V$ is an injective resolution of $G|V$. The theorem follows immediately.

1.4.

Let ${\mathcal{O}}_{X,Z}$ be the ${\mathcal{O}}_X$-Module defined by the following conditions ([Godement], 2.9.2): ${\mathcal{O}}_{X,Z}{|}_{X-Z}=0$ and ${\mathcal{O}}_{X,Z}{|}_Z={\mathcal{O}}_X{|}_Z$. We have seen that for every ${\mathcal{O}}_X$-Module $H$ there is a functorial isomorphism ${\Gamma }_Z(H)\simeq {\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathcal{O}}_{X,Z},H)$. From this we deduce functorial isomorphisms in $F$ and $G$:

Γ_Z(Hom_{𝒪_X}(F, G)) ≃ Hom_{𝒪_X}(𝒪_{X,Z}, Hom_{𝒪_X}(F, G)),

Γ_Z(Hom_{𝒪_X}(F, G)) ≃ Hom_{𝒪_X}(𝒪_{X,Z} ⊗_{𝒪_X} F, G),

Γ_Z(Hom_{𝒪_X}(F, G)) ≃ Hom_{𝒪_X}(F, Hom_{𝒪_X}(𝒪_{X,Z}, G)) = Hom_{𝒪_X}(F, Γ_Z(G)).

It follows in particular from (1.4.2) that there is a $\partial$-functorial isomorphism in $F$ and $G$

θ: Ext^i_{𝒪_X}(𝒪_{X,Z} ⊗_{𝒪_X} F, G) ⥲ Ext^i_Z(X; F, G).

1.5.

By definition, the functor $G\mapsto {\Gamma }_Z({\mathrm{Hom}}_{{\mathcal{O}}_X}(F,G))$ is the composite of the functor $G\mapsto {\mathrm{Hom}}_{{\mathcal{O}}_X}(F,G)$ and the functor ${\Gamma }_Z$. Since ${\Gamma }_Z$ is left exact (I 1.9), since ${\mathrm{Hom}}_{{\mathcal{O}}_X}(F,G)$ is flasque whenever $G$ is injective, and since ${\Gamma }_Z$ is exact on flasque sheaves (I 2.12), it follows from (Tôhoku, 2.4.1) that there is a spectral functor abutting to $Ex{t}_Z^{\bullet }(X;F,G)$ whose initial term is $H_Z^p(X,\mathcal{E}x{t}_{{\mathcal{O}}_X}^q(F,G))$.

On the other hand, it follows from (1.4.3) that ${\Gamma }_Z({\mathrm{Hom}}_{{\mathcal{O}}_X}(F,G))$ is the composite of ${\Gamma }_Z$ and the functor $H\mapsto {\mathrm{Hom}}_{{\mathcal{O}}_X}(F,H)$.

Since the functor ${\Gamma }_Z$ takes injectives to injectives (I 1.4), it follows from (Tôhoku, 2.4.1) that there is a spectral functor abutting to $Ex{t}_Z^{\bullet }(X;F,G)$ whose initial term is $Ex{t}_{{\mathcal{O}}_X}^p(X;F,{\mathcal{H}}_Z^q(G))$.

It follows finally, from (1.4.2) and the spectral sequence for Ext, that there is a spectral functor abutting to $Ex{t}_Z^{\bullet }(X;F,G)$ whose initial term is $H^p(X;\mathcal{E}x{t}_Z^q(F,G))$. Whence the

Theorem.

There exist three spectral functors abutting to $Ex{t}_Z^{\bullet }(X;F,G)$ whose initial terms are respectively

$$H_Z^p(X,\mathcal{E}x{t}_{{\mathcal{O}}_X}^q(F,G))$$

$$H^p(X,\mathcal{E}x{t}_Z^q(F,G))$$

$$Ex{t}_{{\mathcal{O}}_X}^p(X;F,{\mathcal{H}}_Z^q(G)).$$

1.7.

Let now $Z^{\prime }$ be a closed subset of $Z$ and let $Z^{\prime \prime }=Z-Z^{\prime }$. We have an exact sequence

$$0\to {\mathcal{O}}_{X,Z^{\prime \prime }}\to {\mathcal{O}}_{X,Z}\to {\mathcal{O}}_{X,Z^{\prime }}\to 0$$

which generalizes the exact sequence of ([Godement], 2.9.3). This exact sequence splits locally; hence for every ${\mathcal{O}}_X$-Module $F$ we have a further exact sequence:

0 → F ⊗_{𝒪_X} 𝒪_{X,Z″} → F ⊗_{𝒪_X} 𝒪_{X,Z} → F ⊗_{𝒪_X} 𝒪_{X,Z′} → 0.

Let now $G$ be an ${\mathcal{O}}_X$-Module; applying the functor ${\mathrm{Hom}}_{{\mathcal{O}}_X}(\bullet ,G)$ to the exact sequence (1.7.2), one deduces from (1.4.2) and the long exact sequence for Ext the following theorem:

Theorem.

Let $Z$ be a locally closed subset of $X$, let $Z^{\prime }$ be a closed subset of $Z$, and let $Z^{\prime \prime }=Z-Z^{\prime }$. Then there is an exact sequence, functorial in $F$ and $G$:

0 → Hom_{Z′}(F, G) → Hom_Z(F, G) → Hom_{Z″}(F, G) → Ext^1_{Z′}(F, G) → ⋯
    ⋯ → Ext^i_Z(F, G) → Ext^i_{Z″}(F, G) → Ext^{i+1}_{Z′}(F, G) → ⋯

Corollary.

Let $Y$ be a closed subset of $X$ and let $U=X-Y$. Then there is an exact sequence, functorial in $F$ and $G$:

0 → Hom_Y(F, G) → Hom_{𝒪_X}(F, G) → Hom_{𝒪_X|U}(F|U, G|U) → Ext^1_Y(F, G) → ⋯
    ⋯ → Ext^i_{𝒪_X}(F, G) → Ext^i_{𝒪_X|U}(F|U, G|U) → Ext^{i+1}_Y(F, G) → ⋯

This corollary is an immediate consequence of theorem (1.3) and theorem (1.8).

2. Applications to quasi-coherent sheaves on preschemes

Proposition.

Let $X$ be a locally noetherian prescheme. For every locally closed subset $Z$ of $X$, every coherent Module $F$, and every quasi-coherent Module $G$ on $X$, the sheaves $\mathcal{E}x{t}_Z^i(F,G)$ are quasi-coherent.

One shows, as in (1.6.3), that the Modules $\mathcal{E}x{t}_Z^i(F,G)$ are the abutment of a spectral sequence with initial term $\mathcal{E}x{t}_{{\mathcal{O}}_X}^p(F,{\mathcal{H}}_Z^q(G))$. By (II, cor. 3) the ${\mathcal{H}}_Z^q(G)$ are quasi-coherent, and so are the $\mathcal{E}x{t}_{{\mathcal{O}}_X}^p(F,{\mathcal{H}}_Z^q(G))$, since $F$ is coherent. The proposition follows immediately.

2.2.

Let now $Y$ be a closed subprescheme of $X$ and let $\mathcal{I}$ be a defining ideal of $Y$. Let $m$ and $n$ be integers with $m⩾n⩾0$; we denote by $i_{n,m}$ the canonical map ${\mathcal{O}}_{Y_m}={\mathcal{O}}_X/{\mathcal{I}}^{m+1}\to {\mathcal{O}}_X/{\mathcal{I}}^{n+1}={\mathcal{O}}_{Y_n}$ and by $j_n$ the map ${\mathcal{O}}_{X,Y}\to {\mathcal{O}}_{Y_n}$. The system $({\mathcal{O}}_{Y_n},i_{n,m})$ forms a projective system, and the maps $j_n$ are compatible with the $i_{n,m}$.

Applying the functor $Ex{t}_{{\mathcal{O}}_X}^i(F\otimes \bullet ,G)$, one deduces a morphism

φ′: lim_{→ n} Ext^i_{𝒪_X}(X; F ⊗ 𝒪_{Y_n}, G) → Ext^i_{𝒪_X}(X; F ⊗ 𝒪_{X,Y}, G);

this is a morphism of cohomological functors in $G$. The morphism

φ: lim_{→ n} Ext^i_{𝒪_X}(X; F ⊗ 𝒪_{Y_n}, G) → Ext^i_Y(X; F, G)

obtained as the composite of ${\varphi }^{\prime }$ with $\theta$ (cf. 1.4) is therefore likewise a morphism of cohomological functors in $G$.

One defines in the same way

φ̲: lim_{→ n} ℰxt^i_{𝒪_X}(F ⊗ 𝒪_{Y_n}, G) → ℰxt^i_Y(F, G).

Theorem.

Let $X$ be a locally noetherian prescheme, let $Y$ be a closed subset of $X$ defined by a coherent ideal $\mathcal{I}$, let $F$ be a coherent Module and let $G$ be a quasi-coherent Module. Then:

a) φ̲ is an isomorphism.

b) If $X$ is noetherian, $\varphi$ is an isomorphism.

The proof of b) being almost word for word that of (II 6 b)), thanks to the spectral sequence 1.6.2, we shall not reproduce it.

For the proof of a), one may, by (2.1), assume $X$ affine with ring $A$, $F$ (resp. $G$) defined by an $A$-module $M$ (resp. $N$), and $\mathcal{I}$ by an ideal $I$. It suffices to prove that the homomorphism

lim_{→ n} Ext^i_A(M/I^n M, N) → Ext^i_Y(X, F, G)

deduced from φ̲ is an isomorphism.

Indeed, for $i=0$, one can canonically identify both sides of (2.3.1) with the submodule of ${\mathrm{Hom}}_A(M,N)$ consisting of those elements of ${\mathrm{Hom}}_A(M,N)$ annihilated by some power of $I$. One then sees that the homomorphism (2.3.1) is none other than the identity map.

The functor $N\mapsto {\lim }_{\to n}Ex{t}_A^{\bullet }(M/I^{n}M,N)$ is a universal $\partial$-functor. We shall show that the same holds for the functor $N\mapsto Ex{t}_Y^{\bullet }(M,N)$. Indeed, if $N$ is an injective module, by (9 and 11), ${\mathcal{H}}_Y^q(N)=0$ for $q\ne 0$; and by (IV.2.2), ${\mathcal{H}}_Y^0(N)$ is injective.

It follows then that $Ex{t}_{{\mathcal{O}}_X}^p(X;M,{\mathcal{H}}_Y^q(N))=0$ for $p+q\ne 0$; hence, by (1.6.3), $Ex{t}_Y^i(M,N)=0$ for $i\ne 0$ and $N$ injective. This completes the proof.

Bibliography

Same references as those listed at the end of Exp. I, cited respectively [T\hat{o}hoku] and [Godement].