Exposé VII. Vanishing criteria and coherence conditions for the sheaves ℰxt^i_Y(F, G)

1. Study for $i<n$

We prove a lemma.

Lemma.

Let $X$ be a locally noetherian prescheme, $Y$ a closed subset of $X$, and $G$ a quasi-coherent ${\mathcal{O}}_X$-Module. Suppose that for every coherent ${\mathcal{O}}_X$-Module $F$ with support contained in $Y$, one has

$$\mathcal{E}x{t}^{n-1}(F,G)=0.$$

Then for every coherent ${\mathcal{O}}_X$-Module $F$ and every closed subset $Z$ of $X$ such that $Y\supset SuppF\cap Z$, one has

ℰxt^n_Z(F, G) ≅ ℋom(F, ℋ^n_Y(G)).

We first remark that

ℰxt^i_Z(F, G) = ℰxt^i_{Z ∩ Supp F}(F, G)

(trivial, cf. Exposé VI). We first carry out the proof for $Z=X$, so that $SuppF\subset Y$. The functor

$$F\mapsto \mathcal{E}x{t}^n(F,G),$$

defined on the category of coherent ${\mathcal{O}}_X$-Modules with support contained in $Y$, is left exact. By (IV 1.3), it is represented by

I = lim_{→ k} ℰxt^n(𝒪_X/𝓘^{k+1}, G),

where $\mathcal{I}$ is the ideal of definition of $Y$. Now, by (II 6), one knows that

ℋ^n_Y(G) ≅ lim_{→ k} ℰxt^n(𝒪_X/𝓘^{k+1}, G).

Whence the conclusion when $Z=X$. Still by (VI 2.3), one knows that

ℰxt^n_Z(F, G) ≅ lim_{→ k} ℰxt^n(F/𝓙^{k+1}F, G),

where $\mathcal{J}$ is the ideal of definition of $Z$. The support of $F/{\mathcal{J}}^{k+1}F$ is contained in $Y$ whenever $Z\cap SuppF\subset Y$; by what we have just proved, we therefore have

ℰxt^n_Z(F, G) ≅ lim_{→ k} ℋom(F/𝓙^{k+1}F, ℋ^n_Y(G)).

It remains to show that the natural homomorphism

lim_{→ k} ℋom(F/𝓙^{k+1}F, ℋ^n_Y(G)) → ℋom(F, ℋ^n_Y(G))

is an isomorphism when $Z\cap SuppF\subset Y$. Now $X$ can be covered by noetherian affine open sets; one is thus reduced to the case where $X$ is noetherian affine. Then $F(X)$ is a finitely generated ${\mathcal{O}}_X(X)$-Module and $Supp{\mathcal{H}}_Y^n(G)\subset Y$. Hence every homomorphism $u:F(X)\to {\mathcal{H}}_Y^n(G)(X)$ is annihilated by a power of $\mathcal{I}$, and therefore by a power of $\mathcal{J}$. QED.

Proposition.

Let $X$ be a locally noetherian prescheme, $Y$ a closed subset of $X$, $G$ a quasi-coherent ${\mathcal{O}}_X$-Module, and $n$ an integer. For any closed subsets $Z$ and $S$ of $X$ such that $Z\cap S=Y$, the following conditions are equivalent:

1. `ℋ^i_Y(G) = 0` for `i < n`;
2. there exists a coherent `𝒪_X`-Module `F`, of support `S`, such that

   ℰxt^i_Z(F, G) = 0 for i < n;
   

3. for every coherent `𝒪_X`-Module `F` with support contained in `S` (i.e.

   $SuppF\cap Z=SuppF\cap Y$), one has

   ℰxt^i_Z(F, G) = 0 for i < n;
   

4. for every coherent `𝒪_X`-Module `F`, one has

   ℰxt^i_Y(F, G) = 0 for i < n.
   

Moreover, if these conditions hold, then for every coherent ${\mathcal{O}}_X$-Module $F$ and every closed subset $Z^{\prime }$ of $X$ such that $Z^{\prime }\cap SuppF=Y\cap SuppF$, one has isomorphisms

ℰxt^n_Z(F, G) ≅ ℰxt^n_Y(F, G) ≅ ℋom(F, ℋ^n_Y(G)).

Proof. We argue by induction. The proposition is trivial for $n<0$. Suppose it has been proved for $n<q$. If one of the conditions holds for $n=q$, and for two subsets $Z$ and $S$ as stated, then by the induction hypothesis we have, for every closed subset $Z^{\prime }$ of $X$ and every coherent ${\mathcal{O}}_X$-Module $F$ such that $Z^{\prime }\cap SuppF=Y\cap SuppF$, isomorphisms

ℰxt^{q−1}_{Z'}(F, G) ≅ ℋom(F, ℋ^{q−1}_Y(G)) ≅ ℰxt^{q−1}_Y(F, G).

Hence:

• (i) ⇒ (iv), by taking $Z^{\prime }=Y$ in (1.1);

• (iv) ⇒ (iii), by taking $Z^{\prime }=Z$ in (1.1);

• (iii) ⇒ (ii), by taking $F={\mathcal{O}}_S$;

• (ii) ⇒ (i), by taking $Z^{\prime }=Z$ in (1.1); this gives $\mathcal{H}om(F,{\mathcal{H}}_Y^{q-1}(G))=0$. One then remarks that

  Supp ℋ^{q−1}_Y(G) ⊂ Y = Z ∩ S ⊂ S = Supp F,
  

  and one applies the following lemma:

Lemma.

Let $X$ be a prescheme, let $P$ be a coherent ${\mathcal{O}}_X$-Module, and let $H$ be a quasi-coherent ${\mathcal{O}}_X$-Module such that

ℋom(P, H) = 0 and Supp P ⊃ Supp H.

Then $H=0$.

It suffices to prove the lemma when $X$ is affine, since the affine open sets form a base of the topology of $X$ and the hypotheses are preserved by restriction to an open set.

Now in that case one is reduced to a problem on $A$-modules, where $X=\mathrm{Spec}(A)$. One applies the formula (valid under the sole hypothesis that $M$ is of finite type)

Ass Hom_A(P, H) = Supp P ∩ Ass H.

One knows that Ass H ⊂ Supp H ⊂ Supp P and that $Ass{\mathrm{Hom}}_A(P,H)=\varnothing$; hence $AssH=\varnothing$, so $H=0$.

To complete the proof of the proposition, it remains to observe that (iv) allows us to apply 1.1.

Corollary.

Let $G$ be a coherent Cohen-Macaulay ${\mathcal{O}}_X$-Module, and let $n\in \mathbb{Z}$. The conditions of 1.2 are equivalent to:

1. codim(Y ∩ Supp G, Supp G) ⩾ n.
   

Recall first that an ${\mathcal{O}}_X$-module is said to be Cohen-Macaulay if, for every $x\in X$, the stalk $G_x$ is a Cohen-Macaulay ${\mathcal{O}}_{X,x}$-module, i.e. one has for every $x\in S=SuppG$:

prof G_x = dim G_x = dim 𝒪_{S,x}.

By Proposition III 3.3, condition (i) of 1.2 is equivalent to

prof_Y G = inf_{x ∈ Y} prof G_x ⩾ n,

and therefore also to

prof_Y G = inf_{x ∈ Y ∩ S} prof G_x ⩾ n,

since the depth of a zero module is infinite. Now, by definition,

codim(Y ∩ S, S) = inf_{x ∈ S ∩ Y} dim 𝒪_{S,x},

whence the conclusion, by applying formula (1.2).

We shall now prove a result that lets us deduce the coherence conditions we have in view from certain vanishing criteria.

Lemma.

Let $X$ be a locally noetherian prescheme. Let $T^{\bullet }$ be an exact contravariant $\partial$-functor, defined on the category of coherent ${\mathcal{O}}_X$-Modules, with values in the category of ${\mathcal{O}}_X$-Modules. Let $Y$ be a closed subset of $X$. Let $i\in \mathbb{Z}$. Suppose that, for every coherent ${\mathcal{O}}_X$-Module with support contained in $Y$, $T^{i}F$ and $T^{i-1}F$ are coherent. Let $F$ be a coherent ${\mathcal{O}}_X$-Module. For $T^{i}F$ to be coherent, it is necessary and sufficient that $T^i{F}^{\prime \prime }$ be coherent, where we have set

$$F^{\prime \prime }=F/{\Gamma }_Y(F).$$

Indeed, $F^{\prime }={\Gamma }_Y(F)$ is coherent because $X$ is locally noetherian; the cohomology exact sequence of $T^{\bullet }$ then gives

T^{i−1} F' → T^i F'' → T^i F → T^i F',

where the outer terms are coherent, whence the conclusion.

Lemma.

If $F$ and $G$ are coherent, and if $SuppF\subset Y$, then $\mathcal{E}x{t}_Y^i(F,G)$ is coherent.

Indeed, $\mathcal{E}x{t}_Y^i(F,G)$ is isomorphic to $\mathcal{E}x{t}^i(F,G)$; this is valid, moreover, on any ringed space $X$: if $Z$ is a closed subset containing $Y\cap SuppF$, then $\mathcal{E}x{t}_Z^i(F,G)$ is isomorphic to $\mathcal{E}x{t}_Y^i(F,G)$ (cf. Exposé VI).

Proposition.

Suppose $F$ and $G$ are coherent, and set $S=SuppF$, $S^{\prime }=S\cap (X-Y)$. Suppose that, for every $x\in Y\cap S^{\prime }$, one has $prof{G}_x⩾n$. Then $\mathcal{E}x{t}_Y^i(F,G)$ is coherent for $i<n$.

Indeed, 1.6 allows us to apply 1.5 to $T^{\bullet }(F)=\mathcal{E}x{t}_Y^{\bullet }(F,G)$. Setting $F^{\prime \prime }=F/{\Gamma }_Y(F)$, one sees that $Supp{F}^{\prime \prime }=S^{\prime }$. Now, by III 3.3, the hypothesis on the depth of $G$ ensures the vanishing of ${\mathcal{H}}_{Y\cap S^{\prime }}^i(G)$ for $i<n$; by 1.2, one deduces the vanishing of $T^i{F}^{\prime \prime }$ for $i<n$, whence the conclusion by 1.5.

2. Study for $i>n$

Let $X$ be a locally noetherian regular prescheme, that is, one all of whose local rings are regular. Let $Y$ be a closed subset of $X$. Let $F$ and $G$ be two coherent ${\mathcal{O}}_X$-Modules. Set $S=SuppF$, $S^{\prime }=S\cap (X-Y)$. Set

m = sup_{x ∈ Y ∩ S} dim 𝒪_{X,x},
n = sup_{x ∈ Y ∩ S'} dim 𝒪_{X,x};

one has $n⩽m$.

Proposition.

In the situation just described, one has:

1. $\mathcal{E}x{t}_Y^i(F,G)=0$ for $i>m$,
2. $\mathcal{E}x{t}_Y^i(F,G)$ is coherent for $i>n$.

Note first that $\mathcal{E}x{t}_Y^i(F,G)$ is coherent for every $i$ when $SuppF\subset Y$. Moreover, setting $F^{\prime \prime }=F/{\Gamma }_Y(F)$ as above, one sees that $Supp{F}^{\prime \prime }=S^{\prime }$, so that (2) follows from (1) and from 1.3.

To prove (1), one first remarks that

ℰxt^i_Y(F, G) ≅ lim_{→ k} ℰxt^i(F/𝓙^k F, G),

where $\mathcal{J}$ is the ideal of definition of $Y$. On the other hand, it follows from Theorem 4.2.2 of (A. Grothendieck, "Sur quelques points d'algèbre homologique", Tôhoku Mathematical Journal 9 (1957), pp. 119–221) that the Ext sheaves commute with the formation of stalks, at least when $X$ is a locally noetherian prescheme and the first argument is coherent; since the same is true of direct limits, one finds isomorphisms

(ℰxt^i_Y(F, G))_x ≅ lim_{→ k} Ext^i_{𝒪_{X,x}}((F/𝓙^k F)_x, G_x)

for every $x\in X$. Since $Supp\mathcal{E}x{t}_Y^i(F,G)\subset S\cap Y$, to conclude it suffices to remark that $x\in Y\cap S$ entails $\dim {\mathcal{O}}_{X,x}⩽m$, hence

Ext^i_{𝒪_{X,x}}((F/𝓙^k F)_x, G_x) = 0 for i > m,

since the global cohomological dimension of a regular local ring is equal to its dimension.¹

Let $X$ be a locally noetherian prescheme; for every subset $P$ of $X$, set

D(P) = { dim 𝒪_{X,p} | p ∈ P }.

Lemma.

If $P$ is the underlying space of a connected subprescheme of $X$, then $D(P)$ is an interval.

Indeed, let $\alpha$ and $\beta$ belong to $D(P)$, corresponding to points $p$ and $q$ of $P$. We show that there exists a sequence of points of $P$, $(p=p_1,\cdots ,p_n=q)$, such that for $1⩽i<n$ one has $|\dim {\mathcal{O}}_{X,p_i}-\dim {\mathcal{O}}_{X,p_{i+1}}|=1$; it will follow that $D(P)$ contains the interval $[\alpha ,\beta ]$. For this, one remarks that $p$ and $q$ can be joined by a chain of irreducible components of $P$ such that two successive components meet. One is reduced to the case where $p$ is the generic point of an irreducible component $Q$ of $P$, and where $q\in Q$, and so $q\supset p$ as ideals of ${\mathcal{O}}_q$, where the assertion is trivial from the definition of dimension.

Proposition.

Let $X$ be a locally noetherian regular prescheme, $Y$ a closed subset of $X$, and $F$ a coherent ${\mathcal{O}}_X$-Module. Let $P=Y\cap SuppF\cap (X-Y)$. Let $n\in \mathbb{Z}$, and suppose that $n\notin D(P)$. Then $\mathcal{E}x{t}_Y^n(F,{\mathcal{O}}_X)$ is coherent.

The conclusion is local and the hypotheses are preserved by restriction to an open set. Now $P$ is closed and so locally noetherian, hence locally connected; we may therefore assume $X$ affine and noetherian, and $P$ connected. Set D(P) = [a, b[, which is legitimate by the preceding lemma. If $n>b$, we conclude by 2.1; if $n<a$, then $n<\dim {\mathcal{O}}_{X,x}=prof{\mathcal{O}}_{X,x}$ for every $x\in P$, and we conclude by 1.7.

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