Exposé VIII. The finiteness theorem

1. A biduality spectral sequence1

Let us state the result we want to reach:

Proposition.

Let $A$ be a noetherian ring and let $I$ be an ideal of $A$. Set $X=\mathrm{Spec}(A)$ and $Y=V(I)$. Let $M$ be a finitely generated $A$-module of finite projective dimension. Let $F=\tilde{M}$ be the ${\mathcal{O}}_X$-Module associated with $M$.

1. There exists a spectral sequence

E₂^{p,q} = Ext^p_Y(Ext^{-q}(M, A), A) ⇒ H^{p+q}_Y(X, F).

1. There exists a spectral sequence

E₂^{p,q} = Ext^p_Y(Ext^{-q}(F, 𝒪_X), 𝒪_X) ⇒ H^{p+q}_Y(X, F).

Of course, 2) is deduced from 1) by remarking that, if $M$ and $N$ are two finitely generated $A$-modules, and if one sets $F=M$ and $G=N$, then one has isomorphisms

ℋ^•_Y(F) ≅ H̃^•_Y(X, F),
Ext^•_Y(F, G) ≅ Ẽxt^•_Y(F, G),
Ext^•_{𝒪_X}(F, G) ≅ Ẽxt^•_A(M, N).

Let $C$ be the category of $A$-modules and Ab that of abelian groups. Let $F$ be the functor

F : C → Ab    defined by    M ↦ Γ_Y(M̃).

We know from Exposé II that there is an isomorphism of $\partial$-functors

H^•_Y(X, M̃) ≅ R^• F(M).

Furthermore, let $Ex{t}_Y^{\bullet }$ denote the right derived functors in the second argument of

F ∘ Hom : C° × C → Ab.

We know from Exposé VI that there is an isomorphism of $\partial$-functors

Ext^•_Y(M, N) ≃ Ext^•_Y(M̃, Ñ).

Let us finally record the following result from Exposé VI: if $C$ is an injective $A$-module and $N$ is a finitely generated $A$-module, then the sheaf $\mathrm{Hom}(N,C)\cong \mathrm{Hom}\widetilde{N,C}$ is flasque, hence

$$R^{1}F(\mathrm{Hom}(N,C))=0.$$

It remains to prove the following result:

Lemma.

Let $A$ be a noetherian ring and let $C$ be the category of $A$-modules. Let $F:C\to Ab$ be a left exact additive functor such that, for every finitely generated $A$-module $N$ and every injective $A$-module $C$, one has $R^{1}F(\mathrm{Hom}(N,C))=0$. Let $M$ be a finitely generated $A$-module of finite projective dimension. Then there exists a spectral sequence

E₂^{p,q} = Ext^p_F(Ext^{-q}(M, A), A) ⇒ R^{p+q} F(M),

where $Ex{t}_F^p$ denotes the $p$-th right derived functor of $F\circ \mathrm{Hom}$.

We shall consider only complexes whose differential has degree +1. By the hypothesis on $M$, there exists a projective resolution of $M$ of finite length

$$u:L^{\bullet }\to M,$$

where, moreover, the $L^p$ are finitely generated modules and $L^p=0$ if $p\notin [-n,0]$. Let, on the other hand, $v:M\to I^{\bullet }$ be an injective resolution of $M$. We claim that

v ∘ u : L^• → I^•

is an injective resolution of $L^{\bullet }$. We must specify what this means.

Definition.

Let $X^{\bullet }$ be a complex of $A$-modules; by an injective resolution of $X^{\bullet }$ one means a homomorphism of complexes

$$x:X^{\bullet }\to C_X^{\bullet },$$

such that $C_X^p$ is injective for every $p\in \mathbb{Z}$, and such that $x$ induces an isomorphism on homology.

Proposition.

Every left-bounded complex — i.e. such that there exists $q\in \mathbb{Z}$ with $X^p=0$ for $p<q$ — admits an injective resolution. Moreover, if $u:X^{\bullet }\to Y^{\bullet }$ is a homomorphism of complexes (both left-bounded) and if $x:X^{\bullet }\to C_X^{\bullet }$ and $y:Y^{\bullet }\to C_Y^{\bullet }$ are injective resolutions of $X^{\bullet }$ and $Y^{\bullet }$, then there exists a homomorphism of complexes

$$C_u:C_X^{\bullet }\to C_Y^{\bullet },$$

unique up to homotopy, such that the diagram

        x
X^•  ────────►  C_X^•
 │                │
 u                C_u
 │                │
 ▼      y         ▼
Y^•  ────────►  C_Y^•

is commutative up to homotopy.

The proof is left to the reader.²

Let us recall a notation introduced in Exposé V.

Notation.

Let $X^{\bullet }$ and $Y^{\bullet }$ be two complexes. We denote by ${\mathrm{Hom}}^{\bullet }(X^{\bullet },Y^{\bullet })$ the simple complex whose component of degree $n$ is

(Hom^•(X^•, Y^•))^n = ∏_{−p+q=n} Hom(X^p, Y^q),

also written ${\mathrm{Hom}}^n(X^{\bullet },Y^{\bullet })$, and whose differential is given by

∂^n : Hom^n(X^•, Y^•) → Hom^{n+1}(X^•, Y^•),
∂^n = d′ + (−1)^{n+1} d″,

where $d^{\prime }$ and $d^{\prime \prime }$ are the differentials (of degree +1) induced by those of $X^{\bullet }$ and $Y^{\bullet }$.

Let then $A^{\bullet }$ be the complex defined by $A^p=0$ if $p\ne 0$ and $A^0=A$. Let

$$a:A^{\bullet }\to C_A^{\bullet }$$

be an injective resolution of $A^{\bullet }$. Consider the double complex

Q^{p,q} = Hom(Hom(L^{-q}, A), C_A^p).

The first spectral sequence of the bicomplex $F{Q}^{\bullet \bullet }$ will yield the conclusion of Lemma 1.2.

Set

$$L^{\prime \bullet }={\mathrm{Hom}}^{\bullet }(L^{\bullet },A^{\bullet }),$$

and

$$P^{\bullet }={\mathrm{Hom}}^{\bullet }(L^{\prime \bullet },C_A^{\bullet }).$$

One sees easily that $P^{\bullet }$ is the simple complex associated with $Q^{\bullet \bullet }$. Let us compute the abutment of the spectral sequence, i.e. the homology of $F{P}^{\bullet }$. For this, using the fact that $L^{\bullet }$ is finitely generated projective in every dimension, one proves that $L^{\bullet }$ is isomorphic to ${\mathrm{Hom}}^{\bullet }(L^{\prime \bullet },A^{\bullet })$. From the homomorphism $a:A^{\bullet }\to C_A^{\bullet }$ one deduces a homomorphism

b : Hom^•(L′^•, A^•) → Hom^•(L′^•, C_A^•),

or equivalently, a homomorphism

$$c:L^{\bullet }\to P^{\bullet }.$$

This being said, it is easy to see, using the fact that $L^{\prime \bullet }$ is finitely generated projective in every dimension and left-bounded, that (1.5) is an injective resolution of $L^{\bullet }$. Applying Proposition 1.4, one concludes that $P^{\bullet }$ is homotopy-equivalent to $I^{\bullet }$, where $I^{\bullet }$ is the injective resolution of $M$ introduced earlier (1.1). One deduces that the abutment of the first spectral sequence of the double complex $F{Q}^{\bullet \bullet }$, which is $H^{\bullet }(F{P}^{\bullet })$, is isomorphic to $R^{\bullet }F(M)$.

The initial term of the first spectral sequence of the bicomplex $F{Q}^{\bullet \bullet }$ is

$$E_2^{p,q}{=}^{\prime }{H}^p{(}^{\prime \prime }{H}^q(F{Q}^{\bullet \bullet })).$$

For every $p\in \mathbb{Z}$, $C_A^p$ is injective. By the hypothesis on $F$, the functor (restricted to the category of finitely generated modules)

N ↦ F Hom(N, C_A^p)

is exact. Hence one deduces isomorphisms

″H^q(F Hom(L′^•, C_A^p)) ≅ F Hom(H^{-q}(L′^•), C_A^p).

By the definition of $Ex{t}_F^{\bullet }$ as the derived functor of $F\circ \mathrm{Hom}$, one deduces isomorphisms

$$E_2^{p,q}\cong Ex{t}_F^p(H^{-q}(L^{\prime \bullet }),A).$$

Now $L^{\prime \bullet }={\mathrm{Hom}}^{\bullet }(L^{\bullet },A^{\bullet })$, where $L^{\bullet }$ is a projective resolution of $M$, whence isomorphisms

$$Ex{t}^{-q}(M,A)\cong H^{-q}(L^{\prime \bullet }),$$

which gives the conclusion. QED.

2. The finiteness theorem

Theorem.³

Let $X$ be a locally noetherian prescheme, $Y$ a closed subset of $X$, and $F$ a coherent ${\mathcal{O}}_X$-Module. Suppose that $X$ is locally embeddable in a regular prescheme.⁴ Let $i\in \mathbb{Z}$. Suppose that:

a) for every $x\in U=X-Y$, one has

$$H^{i-c(x)}(F_x)=0,$$

where one has set⁵

c(x) = codim({x}̄ ∩ Y, {x}̄).

Then:

b) ${\mathcal{H}}_Y^i(F)$ is coherent.

Corollary.⁶

Under the hypotheses of the preceding theorem, condition a) is equivalent to:

c) for every $x\in U$ such that $c(x)=1$, one has $H^{i-1}(F_x)=0$.

Corollary.

Let $X$ be a locally noetherian prescheme that is locally embeddable in a regular prescheme, let $Y$ be a closed subset of $X$, let $F$ be a coherent ${\mathcal{O}}_X$-Module, and let $n$ be an integer. The following conditions are equivalent:

(i) for every $x\in U$, one has $prof{F}_x>n-c(x)$;

(ii) for every $x\in U$ such that $c(x)=1$, one has $prof{F}_x⩾n$;

(iii) for every $i\in \mathbb{Z}$, ${\mathcal{H}}_Y^i(F)$ is coherent if $i⩽n$;

(iv) $R^i{i}_{\ast }(F|U)$ is coherent for $i<n$.⁷

Suppose these results acquired when $X$ is the spectrum of a regular noetherian ring $A$ and when $F$ is the sheaf associated with an $A$-module of finite projective dimension.

Let us first remark that, if $(X_j{)}_{j\in J}$ is an open covering of $X$ by opens embeddable in a regular scheme, then each of the above conditions is equivalent to the conjunction of the analogous conditions obtained by replacing $X$ by $X_j$, $Y$ by $Y_j=Y\cap X_j$, and $F$ by $F|X_j$. Indeed, only the conditions involving $c(x)$ can present a difficulty. Let $x\in U$. If $x\in X_j$, setting

c_j(x) = codim(X_j ∩ {x}̄ ∩ Y, X_j ∩ {x}̄),

one has necessarily $c_j(x)⩾c(x)$. Let $y\in \bar{x}\cap Y$ which "gives the codimension", i.e. such that $c(x)=\dim {\mathcal{O}}_{\bar{x},y}$, and let $X_j$ be an open of the covering such that $y\in X_j$; then $x\in X_j$, hence $c_j(x)=c(x)$, which lets us conclude that a) for the $X_j$ implies a) for $X$.

At this stage, one has only a partial converse, namely that a) for $X$ implies a) for the $X_j$ such that $c(x)=c_j(x)$, which suffices for our purposes.⁸

One chooses a covering of $X$ by opens embeddable in a regular prescheme. Applying the preceding, one sees that one can suppose $X$ closed in a regular $X^{\prime }$. The reduction to $X^{\prime }$ is then immediate.

One can therefore suppose $X$ regular, and even affine by covering $X$ by affine opens. That one can suppose $F=\tilde{M}$, where $M$ is of finite projective dimension, will result from the following lemma:

Lemma.

Let $X$ be a regular noetherian prescheme. Let $F$ be a coherent ${\mathcal{O}}_X$-Module. The function which to each $x\in X$ assigns the projective dimension of $F_x$ is upper-bounded.

Indeed, let $x\in X$ and let $U$ be an affine open neighborhood of $x$. Let $L^{\bullet }$ be a projective resolution of the module $F(U)$, where the $L^i$ are finitely generated. By hypothesis, the ring ${\mathcal{O}}_{X,x}$ is regular, hence the projective dimension of $F_x$ is finite; let $d$ be that integer. Let

$$K=\ker (L^{-d}\to L^{-d+1}).$$

The module $K_x$ is free, because $d$ is the projective dimension of $F_x$ ([M], Ch. VI, Prop. 2.1). By (EGA 0_I 5.4.1 Errata), one deduces that the ${\mathcal{O}}_U$-Module $\tilde{K}$ is free on a neighborhood $U^{\prime }$ of $x$, with $U^{\prime }\subset U$. Choosing $f\in {\mathcal{O}}_X(U)$ such that $x\in D(f)\subset U^{\prime }$, one therefore has a projective resolution of $M_f$ (with $M=F(U)$):

$$0\to K_f\to (L^{d-1}{)}_f\to \cdots \to M_f\to 0,$$

which proves that the function under study is upper semi-continuous. Now $X$ is quasi-compact, whence the conclusion.

We henceforth suppose $X$ affine noetherian regular and we suppose that $F=\tilde{M}$, where $M$ is a finitely generated $A$-module, necessarily of finite projective dimension. We shall proceed in several steps. First, we find a condition d), equivalent to a), and prove that it is also equivalent to c). Then, using the spectral sequence of the preceding number, we prove d) ⇒ b). It then remains to prove that (iii) ⇒ (ii); indeed, (i) ⇔ (ii) ⇒ (iii) follows immediately from a) ⇔ c) ⇒ b).

Let $x\in U$; by hypothesis ${\mathcal{O}}_{X,x}$ is a regular local ring. Denoting by $D$ the dualizing functor relative to the local ring ${\mathcal{O}}_{X,x}$, it follows from (V 2.1) that

D H^{i-c(x)}(F_x) ≅ Ext^{d(x)-i}_{𝒪_{X,x}}(F_x, 𝒪_{X,x}),

where one has set

d(x) = dim 𝒪_{X,x} + c(x) = dim 𝒪_{X,x} + codim({x}̄ ∩ Y, {x}̄).

Now $X$ is noetherian and $F$ is coherent, hence

D H^{i-c(x)}(F_x) ≅ (Ext^{d(x)-i}_{𝒪_X}(F, 𝒪_X))_x.

Moreover, for a module to be zero, it is necessary and sufficient that its dual be (cf. editor's note (4) on page 54). For every $q\in \mathbb{Z}$, set

S_q = Supp Ext^q_{𝒪_X}(F, 𝒪_X),
S′_q = S_q ∩ U,    (U = X − Y),
Z_q = S̄′_q ∩ Y.

From formula (2.3), it follows that a) and c) are respectively equivalent to:

• a′) for every $q\in \mathbb{Z}$ and every $x\in S_q^{\prime }$, one has $q+i\ne d(x)$.
• c′) for every $q\in \mathbb{Z}$ and every $x\in S_q^{\prime }$, if $c(x)=1$, one has $q+i\ne d(x)$.

Here is the condition d) promised above:

• d) for every $q\in \mathbb{Z}$ and every $y\in Z_q$, one has $q+i\ne \dim {\mathcal{O}}_{X,y}$.

These conditions are equivalent:

a′) ⇒ c′) is trivial.

d) ⇒ a′). Indeed, let $q\in \mathbb{Z}$ and let $x\in S_q^{\prime }$; let $y\in \bar{x}\cap Y$ which⁹ "gives the codimension", i.e. such that

dim 𝒪_{{x}̄, y} = codim({x}̄ ∩ Y, {x}̄) = c(x).

From the fact that $X$ is regular at $y$, one deduces

dim 𝒪_{X, y} = d(x)    (cf. (2.2)).

But $y\in \bar{x}$, hence $y\in Z_q$, whence the conclusion.

c′) ⇒ d). Let $q\in \mathbb{Z}$ and let $y\in Z_q$. Provisionally admit that there exists $x\in S_q^{\prime }$ such that

y ∈ {x}̄    and    dim 𝒪_{{x}̄, y} = 1

(one also says that $x$ follows $y$). It follows that $c(x)=1$, since $y$ "gives the codimension of $\bar{x}\cap Y$ in $\bar{x}$", because $x\notin Y$. By c′) we extract

$$q+i\ne d(x).$$

Whence the conclusion, on noting that $d(x)=\dim {\mathcal{O}}_{X,y}$ (2.6). The admitted result is expressed in the following lemma:

Lemma.

Let $X$ be a locally noetherian prescheme and let $Y$ be a closed subset of $X$. Set $U=X-Y$ and suppose that $U$ is dense in $X$. For every $y\in Y$, there exists $x\in U$ "which follows it", i.e. such that

y ∈ {x}̄    and    dim 𝒪_{{x}̄, y} = 1.

We have applied the lemma taking for $X$ the prescheme ${\bar{S}}_q^{\prime }$ and for $Y$ the part $Y\cap {\bar{S}}_q^{\prime }$.

Proof of 2.5. — There exists $x\in U$ such that $y\in \bar{x}$; let us therefore choose $x\in U$ such that $y\in \bar{x}$ and such that $\dim {\mathcal{O}}_{\bar{x},y}=r$ be minimal. We must prove that $r=1$. Since we have chosen $x$ so that every $z\in \mathrm{Spec}({\mathcal{O}}_{\bar{x},y})$, $z\ne x$, lies in $Y$, $x$ is open in $\mathrm{Spec}({\mathcal{O}}_{\bar{x},y})$. Whence the conclusion.

The second step consists in deducing b) from d).

Set $D(Z_q)=\dim {\mathcal{O}}_{X,y}|y\in Z_q$. By d), we know that, for every $q\in \mathbb{Z}$, $q+i\notin D(Z_q)$. One then applies VII.2.3, and sees that

Ext^{q+i}_Y(Ext^q(F, 𝒪_X), 𝒪_X)    is coherent.

The initial term of the spectral sequence of the preceding number is given by

E₂^{p,q} = Ext^p_Y(Ext^{-q}(F, 𝒪_X), 𝒪_X).

One deduces that $E_2^{p,q}$ is coherent for every $p\in \mathbb{Z}$ and every $q\in \mathbb{Z}$ such that $p+q=i$. Now there are only finitely many pairs $(p,q)$ with $p+q=i$, and this spectral sequence converges to $H_Y^{\bullet }(F)$, whence the conclusion.

It remains to prove that (iii) ⇒ (ii). Let us write

$$i:U\to X$$

for the canonical immersion of $U$ in $X$. Taking into account the exact homology sequence of the closed subset $Y$ (I 2.11), one sees that (iii) is equivalent to:

(iv) $R^i{i}_{\ast }(F|U)$ is coherent for $i<n$.

Indeed, one has an exact sequence

0 → ℋ^0_Y(F) → F → i_*(F|U) → ℋ^1_Y(F) → 0.

Now ${\mathcal{H}}_Y^0(F)$ is a quasi-coherent subsheaf of the coherent sheaf $F$, hence is coherent. Therefore ${\mathcal{H}}_Y^1(F)$ is coherent if and only if $i_{\ast }(F|U)$ is. Moreover, for $p>0$, the exact cohomology sequence of the closed subset $Y$ reduces to isomorphisms

$$R^p{i}_{\ast }(F|U)\stackrel{\sim }{\to }{\mathcal{H}}_Y^{p+1}(F).$$

We shall prove that (iv) ⇒ (ii). For this, recall (ii):

(ii) for every $x\in U$ such that $c(x)=1$, one has $prof{F}_x⩾n$.

We argue by induction on $n$.

If $n=0$, the two conditions are empty.

If $n=1$, one supposes that $i_{\ast }(F|U)$ is coherent. Argue by contradiction and suppose there exists $x\in U$ such that $c(x)=1$ and $prof{F}_x=0$, i.e. $x\in Ass{F}_x$. Let $y\in \bar{x}\cap Y$ such that $\dim {\mathcal{O}}_{\bar{x},y}=1$. Set

A = 𝒪_{X, y}    and    X′ = Spec(A).

Carry out the base change $v:X^{\prime }\to X$, which is flat:

            v′
   U′ = X′ ×_X U  ──────►  U
        │                   │
       i′                   i
        │                   │
        ▼          v        ▼
       X′  ──────────────► X.

The morphism $i$ is separated (since it is an immersion), and of finite type (since it is an open immersion and $X$ is locally noetherian); the base change is flat, hence (EGA III 1.4.15) one has an isomorphism

$$v^{\ast }(i_{\ast }(F|U))\cong i_{\ast }^{\prime }(v^{\prime \ast }(F|U)).$$

Let us denote by $\mathfrak{x}$ (resp. $\mathfrak{y}$) the ideal of $A$ corresponding to $x$ (resp. $y$). Set $G=v^{\prime \ast }(F|U)$; then $G$ is coherent and $\mathfrak{x}\in AssG$, so there exists a monomorphism ${\mathcal{O}}_{\bar{x}}\to G$, and consequently $i_{\ast }^{\prime }({\mathcal{O}}_{\bar{x}}|U^{\prime })$ is coherent. By the choice of $y$, $\dim A/\mathfrak{x}=1$, and consequently the support of ${\mathcal{O}}_{\bar{x}}$ is reduced to $\bar{x}=x\cup y$, since $\bar{x}=\mathrm{Spec}(A/\mathfrak{x})$ as a scheme. It follows that

$$({\mathcal{O}}_{\bar{x}}|U^{\prime })(U^{\prime })=Frac(A/\mathfrak{x}),$$

the field of fractions of $A/\mathfrak{x}$, and

$$i_{\ast }^{\prime }({\mathcal{O}}_{\bar{x}}|U^{\prime })(X^{\prime })=Frac(A/\mathfrak{x}).$$

But $Frac(A/\mathfrak{x})$ is not a finitely generated $A$-module, because $\mathfrak{x}$ differs from the maximal ideal of $A$. Whence a contradiction.

Suppose $n>1$ and that the result is acquired for the $n^{\prime }<n$. By the induction hypothesis, for every $x\in U$ such that $c(x)=1$, one has $x\notin Ass{F}_x$. Let such an $x$, and let $y\in \bar{x}\cap Y$ such that $x$ follows $y$, i.e. $\dim {\mathcal{O}}_{\bar{x},y}=1$. Carry out the base change $v:\mathrm{Spec}({\mathcal{O}}_{X,y})\to X$, keeping the notation of diagram (2.7). One finds, applying (EGA III 1.4.15), isomorphisms

v^*(R^p i_*(F|U)) ≃ R^p i′_*(v′^*(F|U)),    p ∈ ℤ.

One thus reduces to the case where $X$ is the spectrum of a local ring $A$ in which $\mathfrak{x}$ is a prime ideal of dimension 1, i.e. $\dim A/\mathfrak{x}=1$. Then set $F^{\prime }={\Gamma }_Y(F)$ and $F^{\prime \prime }=F/F^{\prime }$. One sees that $F_x\simeq F_x^{\prime \prime }$ and that $\mathfrak{y}\notin Ass{F}^{\prime \prime }$. Moreover $F^{\prime }|U=0$, whence, by the exact sequence of the $R^p{i}_{\ast }$, isomorphisms

R^p i_*(F|U) ≃ R^p i_*(F″|U),    p ∈ ℤ.

Since $n>1$, one deduces that neither $\mathfrak{x}$ nor $\mathfrak{y}$ belongs to $Ass{F}^{\prime \prime }$. Now $\mathfrak{x},\mathfrak{y}$ are the only prime ideals of $A$ containing $\mathfrak{x}$; it follows (III 2.1) that there exists an element $g\in \mathfrak{x}$ which is $M$-regular, where one has set $F=\tilde{M}$, $M=F(X)$. Whence an exact sequence

0 → M ──g·→ M → N → 0,

in which $g\cdot$ denotes multiplication by $g$ in $M$. By the exact homology sequence, one sees that

R^p i_*(Ñ|U)    is coherent for p < n − 1,

hence, by the induction hypothesis, $prof(\tilde{N}{)}_x⩾n-1$, and therefore $prof{F}_x⩾n$. QED.

3. Applications

One deduces from these results a coherence condition for the higher direct images of a coherent sheaf under a morphism that is not proper.

Theorem.

Let $f:X\to Y$ be a morphism of preschemes. Suppose that $Y$ is locally noetherian and that $f$ is proper. Suppose that $X$ is locally embeddable in a regular prescheme. Let $n\in \mathbb{Z}$. Let $U$ be an open of $X$ and let $F$ be a coherent ${\mathcal{O}}_U$-Module. Suppose that, for every $x\in U$ such that $codim(\bar{x}\cap (X-U),\bar{x})=1$, one has $prof{F}_x⩾n$. Then the ${\mathcal{O}}_Y$-Modules $R^p(f\circ g{)}_{\ast }(F)$ are coherent for $p<n$, where $g$ is the canonical immersion of $U$ in $X$.

Indeed, there exists a Leray spectral sequence whose abutment is $R^{\bullet }(f\circ g{)}_{\ast }(F)$ and whose initial term is given by

E₂^{p,q} = R^p f_*(R^q g_*(F)).

Moreover, there exists a coherent ${\mathcal{O}}_X$-Module $G$ such that $G|U\simeq F$ (EGA I 9.4.3). It then follows from the preceding paragraph that condition (iv) of page 74 is satisfied, i.e. that $R^q{g}_{\ast }(G|U)$ is coherent for $q<n$. One then applies the finiteness theorem of EGA III 3.2.1 to $f$ and to the sheaves $R^q{g}_{\ast }(F)$, and finds that $E_2^{p,q}$ is coherent for $q<n$, whence the conclusion.

Proposition.

Let $X$ be a locally noetherian prescheme that is locally embeddable in a regular prescheme. Let $U$ be an open of $X$ and let $i:U\to X$ be the canonical immersion. Let $n\in \mathbb{Z}$. Finally, let $F$ be a coherent and Cohen-Macaulay ${\mathcal{O}}_U$-Module (on $U$!). The following conditions are equivalent:

(a) $R^p{i}_{\ast }(F)$ is coherent for $p<n$;

(b) for every irreducible component $S^{\prime }$ of the closure $\bar{S}$ of the support $S$ of $F$, one has

codim(S′ ∩ (X − U), S′) > n.

Let $G$ be a coherent ${\mathcal{O}}_X$-Module such that $G|U\simeq F$ (EGA I 9.4.3). Applying Corollary 2.3 to $G$, one finds that condition (a) is equivalent to:

(c) for every $x\in \bar{S}$, one has $prof{F}_x>n-c(x)$, with

c(x) = codim({x}̄ ∩ Y, {x}̄).

(a) ⇒ (b). Indeed, let $S^{\prime }$ be an irreducible component of $\bar{S}$ and let $s$ be its generic point. Since $F$ is Cohen-Macaulay, one has $prof{F}_s=\dim {\mathcal{O}}_{\bar{S},s}=0$. Moreover, $\bar{s}=S^{\prime }$, whence the conclusion.

(b) ⇒ (a). Let $x\in \bar{S}$ and let $S^{\prime }$ be an irreducible component of $\bar{S}$ such that $x\in S^{\prime }$. Let $s$ be the generic point of $S^{\prime }$. Since $F$ is Cohen-Macaulay, one knows that

prof F_x = dim 𝒪_{{s}̄, x}.

If $c(x)=+\infty$, there is nothing to prove. Otherwise, there exists $y\in Y\cap \bar{x}$ such that

c(x) = dim 𝒪_{{x}̄, y}.

Now ${\mathcal{O}}_{X,y}$ is a quotient of a regular local ring by hypothesis, hence

dim 𝒪_{{s}̄, y} = dim 𝒪_{{s}̄, x} + dim 𝒪_{{x}̄, y} > n.

QED.

Bibliography

1. [M] H. Cartan and S. Eilenberg, Homological Algebra, Princeton Math. Series vol. 19, Princeton University Press, 1956.

Translation ledger — Exposé VIII

Terms confirmed or first activated in this Exposé (consult glossary.md for the volume-wide list):

Note on the $\Gamma Z$/${\mathcal{H}}_Y^{\bullet }$ typographic convention: in this Exposé, sheafified local cohomology is rendered ${\mathcal{H}}_Y^i(F)$ (script-H) and global sections with support remain $H_Y^i(X,F)$. The underlined section functor of the source is, when it appears, written ${\Gamma }_Y$ in line with the volume-wide convention recorded in the introduction.