Exposé II. Application to quasi-coherent sheaves on preschemes

Proposition.

Let $X$ be a prescheme, let $Z$ be a locally closed subset of the form $Z=U-V$, where $U$ and $V$ are two open subsets of $X$ such that $V\subset U$ and such that the canonical immersions $U\to X$, $V\to X$ are quasi-compact. Then for every quasi-coherent Module $F$ on $X$, the sheaves ${\mathcal{H}}_Z^i(F)$ are quasi-coherent.

By (I 2.4),¹ there is an exact sequence of relative cohomology

⋯ → ℋ^{i−1}_U(F) → ℋ^i_V(F) → ℋ^i_Z(F) → ℋ^i_U(F) → ℋ^{i+1}_V(F) → ⋯.

By (EGA III 1.4.17), in order that the ${\mathcal{H}}_Z^i(F)$ be quasi-coherent it therefore suffices that the ${\mathcal{H}}_U^i(F)$ and the ${\mathcal{H}}_V^i(F)$ be so. We may therefore assume that $Z$ is open and that the canonical immersion $j:Z\to X$ is quasi-compact.

Since $Z$ is open, we have (I 2.2)¹ a canonical isomorphism

$${\mathcal{H}}_Z^i(F)\simeq R^i{j}_{\ast }(F|Z),$$

but $j$ is separated (EGA I 5.5.1) and quasi-compact, hence (EGA III 1.4.10) the $R^i{j}_{\ast }(F|Z)={\mathcal{H}}_Z^i(F)$ are quasi-coherent, which completes the proof.

Corollary.

Let $Z$ be a closed subset of $X$ such that the canonical immersion $X-Z\to X$ is quasi-compact. Then the Modules ${\mathcal{H}}_Z^i(F)$ are quasi-coherent.

Corollary.

If $X$ is locally noetherian, then for every locally closed subset $Z$ of $X$ and every quasi-coherent Module $F$ on $X$, the ${\mathcal{H}}_Z^i(F)$ are quasi-coherent.

This follows immediately from Corollary 2 and (EGA I 6.6.4).

Corollary.

Suppose that $X$ is the spectrum of a ring $A$, and let $U$ be a quasi-compact open subset of $X$, $Y=X-U$, and $F$ a quasi-coherent Module on $X$. There is an isomorphism of cohomological functors in $F$:

$${\mathcal{H}}_Y^i(F)=(H_Y^i(X,F)).$$

In addition, one has an exact sequence functorial in $F$:

0 → H⁰_Y(X, F) → H⁰(X, F) → H⁰(U, F) → H¹_Y(X, F) → 0,

and isomorphisms functorial in $F$:

H^i_Y(X, F) ≃ H^{i−1}(U, F),    i ⩾ 2.

By Corollary 2, the ${\mathcal{H}}_Y^i(F)$ are quasi-coherent; since $X$ is affine, one therefore has $H^p(X,{\mathcal{H}}_Y^i(F))=0$ for $p>0$. The spectral sequence (I 2.3)¹ degenerates, hence

$$H_Y^i(X,F)=\Gamma ({\mathcal{H}}_Y^i(F)).$$

Equality (4.1) then follows from (EGA I 1.1.3.7); (4.2) and (4.3) follow from the cohomology exact sequence (I 2.7)¹ and from the fact that $H^i(X,F)=0$ for $i>0$, since $X$ is affine.

Under the hypotheses of 4,² $U$ is a finite union of affine open sets $X_f$; we can therefore find an ideal $I$ generated by a finite number of elements $f_{\alpha }$ and defining $Y$, say $f=(f_{\alpha })$. With the notation of (EGA III § 1),³ one has:

Proposition.

Suppose that $X$ is the spectrum of a ring $A$; let $f=(f_{\alpha })$ be a finite family of elements of $A$, $Y$ the closed subset of $X$ they define, $M$ an $A$-module, and $F$ the sheaf associated with $M$. One then has isomorphisms of $\partial$-functors in $M$:

H^i((f), M) ≃ H^i_Y(X, F).

(We shall also write $H_J^i(M)=H_Y^i(X,F)$, if $Y$ is the closed subset of $X=\mathrm{Spec}A$ defined by an ideal $J$ of $A$.)

For $i=0$ and $i=1$, one uses the exact sequences (4.2) and (EGA III 1.4.3.2); if $i⩾2$, one uses (4.3) and (EGA III 1.4.3.1). This gives us isomorphisms functorial in $M$. One verifies that, up to a sign depending only on $i$, they are compatible with the boundary operator, whence the existence of the isomorphism of $\partial$-functors (5.1).

Now let $X$ be a prescheme, $Y$ a closed subset of $X$, $f:Y\to X$ the inclusion, and $I$ a quasi-coherent ideal defining $Y$ in $X$. Let $F$ be a sheaf on $X$.

We have seen that there exist isomorphisms of $\partial$-functors in $F$

Ext^i_{𝒪_X}(X; f_* f^{−1}(𝒪_X), F) → H^i_Y(X, F),

ℰxt^i_{𝒪_X}(f_* f^{−1}(𝒪_X), F) → ℋ^i_Y(F).

Let $n$, $m$ be integers with $m⩾n⩾0$; we denote by $i_{n,m}$ the canonical map ${\mathcal{O}}_{Y_m}={\mathcal{O}}_X/I^{m+1}\to {\mathcal{O}}_X/I^{n+1}={\mathcal{O}}_{Y_n}$, and by $j_n$ the map

$f_{\ast }{f}^{-1}({\mathcal{O}}_X)\to {\mathcal{O}}_{Y_n}$. The pairs $({\mathcal{O}}_{Y_n},i_{n,m})$ form a projective system, and the $j_n$ are compatible with the $i_{n,m}$.

Applying the functor $Ex{t}_{{\mathcal{O}}_X}^i(X;\cdot ,F)$, one deduces a morphism

ϕ′: lim→_n Ext^i_{𝒪_X}(X; 𝒪_{Y_n}, F) → Ext^i_{𝒪_X}(X; f_* f^{−1}(𝒪_X), F);

one easily shows that this is a morphism of cohomological functors in $F$. The morphism

ϕ: lim→_n Ext^i_{𝒪_X}(X; 𝒪_{Y_n}, F) → H^i_Y(X, F),

obtained by composing ${\phi }^{\prime }$ with (∗), is therefore also a morphism of cohomological functors in $F$.

One defines in the same way

ϕ̲: lim→_n ℰxt^i_{𝒪_X}(𝒪_{Y_n}, F) → ℋ^i_Y(F).

We have in view the following theorem:

Theorem.

a) Let $X$ be a locally noetherian prescheme, $Y$ a closed subset of $X$ defined by a coherent Ideal $I$, and $F$ a quasi-coherent Module. Then ϕ̲ is an isomorphism.

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

Theorem 6 will follow from 6.a) and from:

Lemma.

If the topological space underlying $X$ is noetherian and if ϕ̲ is an isomorphism, then so is $\phi$.

We first prove Lemma 7. We know that there is a spectral sequence

H^p(X, ℋ^q_Y(F)) ⇒ H^*_Y(X, F).

On the other hand, we have an inductive system of spectral sequences

H^p(X, ℰxt^q_{𝒪_X}(𝒪_{Y_n}, F)) ⇒ Ext^*_{𝒪_X}(X; 𝒪_{Y_n}, F).

It follows from the definitions of $\phi$ and ϕ̲ that these morphisms are associated with a homomorphism $\Phi$ of spectral sequences from the direct limit of (7.2n) to (7.1). If the space underlying $X$ is noetherian, then by (Godement 4.12.1)⁴

lim→_n H^p(X, ℰxt^q_{𝒪_X}(𝒪_{Y_n}, F)) ⥲ H^p(X, lim→_n ℰxt^q_{𝒪_X}(𝒪_{Y_n}, F)),

so ${\Phi }_2$ can be written as a morphism

H^p(X, lim→_n ℰxt^q_{𝒪_X}(𝒪_{Y_n}, F)) → H^p(X, ℋ^q_Y(F))

which is nothing other than the one deduced from ϕ̲.

If ϕ̲ is an isomorphism, then so is ${\Phi }_2$, and consequently so is $\phi$ by (EGA 0_III

11.1.5); Lemma 7 is therefore proved.

We shall now prove 6.a); this is a local question on $X$. By Corollary 4 and (EGA I 1.3.9 and 1.3.12) we may assume that $X$ is the spectrum of a ring $A$. It therefore suffices to show that, under the hypotheses of Theorem 6.a), the canonical homomorphism

lim→_n Ext^i_A(A/I^n, M) → H^i_Y(X, M)

is an isomorphism.

Let $f_{\alpha }$ be a finite number of elements of $A$ generating $I$, with $f=(f_{\alpha })$; the sequence of ideals $(f^n)$ is then decreasing and cofinal with the sequence of $I^n$, so that (7.3) is equivalent to a morphism of $\partial$-functors in $M$:

lim→_n Ext^i_A(A/(f^n), M) → H^i_Y(X, M).

On the other hand, one has canonical isomorphisms

lim→_n Hom_A(A/(f^n), M) ≃ lim→_n { m ∈ M | (f^n)m = 0 } ≃ H⁰((f), M).

Since $\lim {\to }_{n}Ex{t}_A^i(A/(f^n),M)$ is a universal $\partial$-functor in $M$, there is a unique morphism of $\partial$-functors in $M$:

lim→_n Ext^i_A(A/(f^n), M) → H^i((f), M),

which coincides in degree zero with (7.5).

Since the composite of (7.3) and (5.1) is a morphism of $\partial$-functors in $M$ that coincides with (7.6) in degree 0, it coincides with (7.6) in every degree. Theorem 6.a) is therefore an immediate consequence of:

Lemma.

Let $A$ be a noetherian ring, $I$ an ideal generated by a finite system $f=(f_{\alpha })$ of elements, and $M$ an $A$-module. Then the homomorphisms (7.6) are isomorphisms.

Lemma.

Let $A$ be a ring, $f=(f_{\alpha })$ a finite system of elements of $A$, $I$ the ideal generated by $f$, and $i>0$ an integer. The following conditions are equivalent:

a) The homomorphism (7.6) is an isomorphism for every $M$.

b) $H^i((f),M)=0$ for $M$ injective.

c) The projective system $(H^i(f^n,A))=H_{i,n}$ is essentially zero, that is: for every $n$, there exists $n^{\prime }>n$ such that $H_{i,n^{\prime }}\to H_{i,n}$ is zero.

a) entails b) trivially.

b) entails a): indeed, b) implies that $M\mapsto H^i((f),M)$ is a universal cohomological functor, so (7.6) is then a morphism of universal cohomological functors. It is an isomorphism in degree zero, hence in every degree.

c) entails b): indeed, if $M$ is injective, one has for every $n$

H^i(f^n, M) = Hom(H^i(f^n, A), M) = Hom(H_{i,n}, M),

so c) implies that for every $i$ the inductive system $(H^i(f^n,M){)}_{n\in \mathbb{Z}}$ is essentially zero, whence b).

b) entails c). Indeed, let $n>0$, and let $j$ be a monomorphism of $H_{i,n}$ into an injective module $M$. Let $n^{\prime }⩾n$, and let $j_{n^{\prime }}\in \mathrm{Hom}(H_{i,n^{\prime }},M)$ be the composite of $j$ with the transition homomorphism $t_{n^{\prime },n}:H_{i,n^{\prime }}\to H_{i,n}$. The $j_{n^{\prime }}$ define an element of $H^i((f),M)$, which is zero by hypothesis. There therefore exists $n_0$ such that $j_{n^{\prime }}=0$ for $n^{\prime }>n_0$. But since $j$ is a monomorphism, $j_{n^{\prime }}=0$ entails $t_{n^{\prime },n}=0$, whence the proposition.

Corollary.

Suppose that the space underlying $X=\mathrm{Spec}(A)$ is noetherian. In order that the preceding conditions be satisfied for every finite family of elements of $A$ and every $i>0$ (or equivalently: for $i=1$), it is necessary and sufficient that for every injective $A$-module $M$, the sheaf $F$ associated with $M$ be flasque.

It is necessary: indeed, let $f=(f_{\alpha })$ be a finite system of elements of $A$, let $Y$ be the closed subset defined by $f$, and $U=X-Y$; one then has the exact sequence

H⁰(X, F) → H⁰(U, F) → H¹((f), M) → 0,

and, thanks to 9.b), $H^0(X,F)\to H^0(U,F)$ is surjective.

It is sufficient by virtue of (5.1) and of the fact that for every closed subset $Y$ of $X$ and every flasque sheaf $F$ on $X$, $H_Y^i(X,F)=0$ for $i>0$.

Lemma.

Under the hypotheses of Lemma 9, for every noetherian $A$-module $N$ and every $i>0$, the projective system $(H_{i,n}(N){)}_{n\in \mathbb{Z}}$, where $H_{i,n}(N)=H^i(f^n,N)$, is essentially zero.

Proof by induction on the number $m$ of elements of $f$.

If $m=1$, $f$ reduces to a single element, say $f$; then $H_{i,n}(N)$ is zero for $i>1$, and $H_{1,n}(N)$ is canonically isomorphic to the annihilator $N_{(n)}$ of $f^n$ in $N$, the transition homomorphism $N_{(n^{\prime })}\to N_{(n)}$, $n^{\prime }⩾n$, being multiplication by $f^{n^{\prime }-n}$. The $N_{(n)}$ form an increasing sequence of submodules of $N$, and since $N$ is noetherian there exists $n_0$ such that $N_{(n)}=N_{(n_0)}$ for $n⩾n_0$. Thus all the $N_{(n)}$ are annihilated by $f^{n_0}$, and the transition homomorphisms $N_{(n^{\prime })}\to N_{(n)}$ are all zero for $n^{\prime }⩾n+n_0$. The lemma is therefore proved for $m=1$.

We now assume that $m>1$ and that the lemma is proved for integers $m^{\prime }<m$; let then $g=(f_1,\cdots ,f_{m-1})$ and $h=f_m$.

For every $n>0$, one has (EGA III 1.1.4.1) an exact sequence

0 → H⁰(h^n, H^i(g^n, N)) → H^i(f^n, N) → H¹(h^n, H^{i−1}(g^n, N)) → 0,

and, as $n$ varies, a projective system of exact sequences. It follows from the induction hypotheses that for $i>0$ the $H^i(g^n,N)$ form an essentially zero projective system, and hence so do the $H^0(h^n,H^i(g^n,N))$, which one identifies with quotients of $H^i(g^n,N)$. For the right-hand terms one factors the transition morphisms from $n^{\prime }$ to $n$ through

H¹(h^{n′}, H^{i−1}(g^{n′}, N)) → H¹(h^{n′}, H^{i−1}(g^n, N)) → H¹(h^n, H^{i−1}(g^n, N)).

Since $H^{i-1}(g^n,N)$ is a noetherian module, it follows from the case $m=1$ that there exists, for given $n$,

an $n^{\prime }>n$ such that the second arrow is zero. We see therefore that in this projective system of exact sequences, the outermost projective systems are essentially zero, and hence so is the middle projective system.

We have thus proved Lemma 11, hence Lemma 8, and consequently Theorem 6.

Remark.

One can also obtain Theorem 6 by establishing the condition of Corollary 10 with the help of the structure theorems for injective modules over a noetherian ring (Matlis, Gabriel).