The Grothendieck complex

Yes, I’m still here. I just haven’t been in a blogging mood. I’ve been distracted a bit with the CRing project. I’ve also been writing a bunch of half-finished notes on Zariski’s Main Theorem and some of its applications, which I’ll eventually post.

I would now like to begin talking about the semicontinuity theorem in algebraic geometry, following Mumford’s Abelian Varieties. This result is used constantly throughout the book, mainly in showing that certain line bundles are trivial. Eventually, I’ll try to say something about this.

Let ${f: X \rightarrow Y}$ be a proper morphism of noetherian schemes, ${\mathcal{F}}$ a coherent sheaf on ${X}$ . Suppose furthermore that ${\mathcal{F}}$ is flat over ${Y}$ ; intuitively this means that the fibers ${\mathcal{F}_y = \mathcal{F} \otimes_Y k(y)}$ form a “nice” family of sheaves. In this case, we are interested in how the cohomology ${H^p(X_y, \mathcal{F}_y) = H^p(X_y, \mathcal{F} \otimes k(y))}$ behaves as a function of ${y}$ . We shall see that it is upper semi-continuous and, under nice circumstances, its constancy can be used to conclude that the higher direct-images are locally free.

1. The Grothendieck complex

Let us keep the hypotheses as above, but assume in addition that ${Y = \mathrm{Spec} A}$ is affine, for some noetherian ring ${A}$ . Consider an open affine cover ${\left\{U_i\right\}}$ of ${X}$ ; we know, as ${X}$ is separated, that the cohomology of ${\mathcal{F}}$ on ${X}$ can be computed using Cech cohomology. That is, there is a cochain complex ${C^*(\mathcal{F})}$ of ${A}$ -modules, associated functorially to the sheaf ${\mathcal{F}}$ , such that

$\displaystyle H^p(X, \mathcal{F}) = H^p(C^*(\mathcal{F})),$

that is, sheaf cohomology is the cohomology of this cochain complex. Furthermore, since the Cech complex is defined by taking sections over the ${U_i}$ , we see that each term in ${C^*(\mathcal{F})}$ is a flat ${A}$ -module as ${\mathcal{F}}$ is flat. Thus, we have represented the cohomology of ${\mathcal{F}}$ in a manageable form. We now want to generalize this to affine base-changes:

Proposition 1 Hypotheses as above, there exists a cochain complex ${C^*(\mathcal{F})}$ of flat ${A}$ -modules, associated functorially to ${\mathcal{F}}$ , such that for any ${A}$ -algebra ${B}$ with associated morphism ${f: \mathrm{Spec} B \rightarrow \mathrm{Spec} A}$ , we have $\displaystyle H^p(X \times_A B, \mathcal{F} \otimes_A B) = H^p(C^*(\mathcal{F}) \otimes_A B).$

Here, of course, we have abbreviated ${X \times_A B}$ for the base-change ${X \times_{\mathrm{Spec} A} \mathrm{Spec} B}$ , and ${\mathcal{F} \otimes_A B}$ for the pull-back sheaf.

Proof: We have already given most of the argument. Now if ${\left\{U_i\right\}}$ is an affine cover of ${X}$ , then ${\left\{U_i \times_A B \right\}}$ is an affine cover of the scheme ${X \times_A B}$ . Furthermore, we have that

$\displaystyle \Gamma(U_i \times_A B, \mathcal{F}\otimes_A B) = \Gamma(U_i , \mathcal{F}) \otimes_A B$

by definition of how the pull-backs are defined. Since taking intersections of the ${U_i}$ commutes with the base-change ${\times_A B}$ , we see more generally that for any finite set ${I}$ ,

$\displaystyle \Gamma\left( \bigcap_{i \in I} U_i \times_A B, \mathcal{F}\otimes_A B\right) = \Gamma\left( \bigcap_{i \in I} U_i, \mathcal{F}\right) \otimes_A B.$

It follows that the Cech complex for ${\mathcal{F} \otimes_A B}$ over the open cover ${\left\{U_i \times_A B\right\}}$ is just ${C^*(\mathcal{F}) \otimes_A B}$ . The assertion is now clear. We thus have a reasonably convenient picture of how cohomology behaves with base-change. There is, nevertheless, an objection here: the complex ${C^*(\mathcal{F})}$ can be poorly behaved as a complex of ${A}$ -modules. There is nothing to assure that the modules are finitely generated, and they need not be. It will thus be necessary to replace them with finitely generated ones. This we can do by some general homological algebra.

Proposition 2 Let ${C}$ be a finite cochain complex of flat modules over the noetherian ring ${A}$ such that each cohomology group ${H^p(C)}$ is finitely generated. Then there is a finite complex ${K}$ of finitely generated flat modules and a morphism of complexes ${K \rightarrow C}$ inducing an isomorphism in cohomology.

Proof: Let us suppose, if necessary by re-indexing, that ${C^i \neq 0}$ only for ${0 \leq i \leq n}$ . We shall construct a ${K}$ such that ${K^i \neq 0}$ only for those ${i}$ as well. We shall do this by descending induction on ${i}$ : for ${i}$ large, we just take ${K^i = 0}$ .

Suppose, inductively, that we have finitely generated free ${A}$ -modules ${K^p, K^{p+1}, K^{p+2}, \dots}$ with boundary maps ${K^i \rightarrow K^{i+1}}$ ( ${i \geq p}$ ) which are differentials and morphisms ${K^i \rightarrow C^i}$ ( ${i \geq p}$ ) such that the induced morphism ${H^i(K) \rightarrow H^i(C)}$ is an isomorphism for ${i \geq p+1}$ , and furthermore such that the map

$\displaystyle \ker(K^p \rightarrow K^{p+1}) \rightarrow H^p(C)$

is surjective. We can draw a commutative diagram of modules

where the problem is to extend this diagram. That is, we need to construct a ${K^{p-1}}$ with maps ${K^{p-1} \rightarrow K^p, K^{p-1} \rightarrow C^{p-1}}$ such that the two initial conditions are also satisfied for ${p-1}$ .

The first condition is that the map ${\ker(K^{p-1} \rightarrow K^p) \rightarrow H^{p-1}(C)}$ is surjective. To do this, we simply take a finite free module ${F}$ surjecting onto ${H^{p-1}(C)}$ , and lift this to a map ${F \rightarrow C^{p-1}}$ (or more precisely into the module of ${p-1}$ -cycles). We then define ${F \rightarrow K^{p}}$ to be zero. This is one part of the definition of ${K^{p-1}}$ ; it is clear that the diagram

commutes.

For the other part, we have to kill the kernel of ${\ker(K^p \rightarrow K^{p+1}) \rightarrow H^p(C)}$ . To do this, note that this kernel is finitely generated. So we can find a finitely generated free module ${F'}$ surjecting onto this kernel. Define ${F' \rightarrow C^{p-1}}$ to be zero and ${F' \rightarrow K^p}$ to be the composite of the aforementioned surjection and the imbedding ${\ker(K^p \rightarrow K^{p+1}) \hookrightarrow K^p}$ . We then set ${K^{p-1} = F \oplus F'}$ and define the two maps as in the diagram to be the direct sum of the maps out of ${F, F'}$ .

It is then clear that we have continued the construction of ${K}$ by another step. We can inductively repeat this until we reach ${K^0}$ . Here, we have to do something different because we want ${K^{-1} = 0}$ . Consider the diagram

where we know (by the inductive construction above) that the map on ${H^1}$ is an isomorphism while the map ${\ker(f) \rightarrow \ker(g)}$ is surjective. We then replace ${K^0}$ by ${K^0/(\ker(f) \cap \ker(g))}$ and stop the complex here. Then the cohomology in dimension zero of ${0 \rightarrow K^0/(\ker(f) \cap \ker(g)) \rightarrow K^1}$ is ${\ker(f)/(\ker(g) \cap \ker(f))}$ , which is the same as that of ${C}$ .

Nonetheless, the proof is still incomplete, as we have not shown that ${K}$ is a flat complex. We do know that ${K}$ is flat (even free) in positive dimensions by construction, but in dimension zero the argument took a quotient. Thus, we shall need a lemma.

Lemma 3 Suppose ${K, C}$ are finite complexes in nonnegative dimensions such that:

${K^i}$ is flat for ${i>0}$ .

${C}$ is flat (in all dimensions).

There is a morphism ${K \rightarrow C}$ inducing isomorphisms on cohomology.

Then ${K}$ is flat in all dimensions.

Proof: Consider the mapping cylinder ${L}$ of the morphism ${\phi: K \rightarrow C}$ . By definition, ${L^p = C^p \oplus K^{p+1}}$ , and the coboundary morphism ${\delta^L}$ is defined by ${\delta^L(c, k) = (\delta^C c -\phi(k), \delta^K k) }$ . There is a short exact sequence of complexes

$\displaystyle 0 \rightarrow C \rightarrow L \rightarrow \widetilde{K} \rightarrow 0$

where ${\widetilde{K}}$ denotes ${K}$ shifted by a degree. In the long exact sequence of chain complexes

$\displaystyle H^p(L) \rightarrow H^p(\widetilde{K}) \rightarrow H^{p+1}(C) \rightarrow H^{p+1}(L) \rightarrow \dots$

one can easily check that ${H^p(\widetilde{K}) \rightarrow H^{p+1}(C)}$ is identified with ${\phi^*: H^{p+1}(K) \rightarrow H^{p+1}(C)}$ . Thus all the connecting homomorphisms are isomorphisms, so exactness implies that ${L}$ is acyclic. However, we know that ${L^i}$ is flat for ${i \geq 0}$ , while we need to show that ${L^{-1}}$ is flat. The exactness of the complex

$\displaystyle 0 \rightarrow L^{-1} \rightarrow L^0 \rightarrow \dots \rightarrow L^N \rightarrow 0,$

where ${N \gg 0}$ , shows that ${L^{-1}}$ is flat (by induction or “dimension-shifting”). Thus ${K^0}$ , as a direct factor, is also flat.

Let us return to the algebro-geometric situation of earlier. There, we had associated to each flat, coherent sheaf ${\mathcal{F}}$ on ${X}$ a finite (the complex is finite since we can choose the affine cover finite) complex ${C^*(\mathcal{F})}$ of flat ${A}$ -modules whose cohomologies were ${H^p(X, \mathcal{F})}$ . By the proper mapping theorem, these are finitely generated ${A}$ -modules. Thus, by \cref{}, we deduce that there is a finite complex of finitely generated flat modules ${K^*}$ such that

$\displaystyle H^p(X, \mathcal{F}) = H^p(K)$

for all ${\mathcal{F}}$ , and furthermore such that there is a map ${K \rightarrow C^*(\mathcal{F})}$ inducing an isomorphism on cohomology (which is stronger than the displayed statement). This by itself does not help if we wish to study cohomology and base-change, however, as it does not give information on ${H^p(X \times_A B, \mathcal{F} \otimes_A B)}$ for an ${A}$ -algebra ${B}$ . We would expect and hope that this could be obtained via ${H^p(K \otimes_A B)}$ . The next result will imply that.

Proposition 4 Suppose ${K \stackrel{\phi}{\rightarrow} C}$ is a morphism of finite flat complexes of ${A}$ -modules\footnote{Here “finite” means that there are finitely many terms; the actual modules need not be finitely generated.} inducing an isomorphism in cohomology. Then for any ${A}$ -algebra ${B}$ , the map ${K \otimes_A B \rightarrow C \otimes_A B}$ induces an isomorphism in cohomology.

Proof: Indeed, consider the mapping cylinder ${L}$ of ${\phi}$ . There is a short exact sequence ${0 \rightarrow \widetilde{C} \rightarrow L \rightarrow K \rightarrow 0}$ , whose long exact sequence shows as earlier that ${L}$ is acyclic. In addition, ${L}$ is obviously flat. Since ${L}$ is flat, we find that ${L \otimes_A B}$ is also exact. Indeed, we draw the complex

$\displaystyle 0 \rightarrow L^{-1} \rightarrow L^0 \rightarrow \dots \rightarrow L^N \rightarrow 0,$

and note that, by induction, the kernels and cokernels at each step are also flat. So we can split this into several short exact sequences

$\displaystyle 0 \rightarrow Z^i \rightarrow L^i \rightarrow Z^{i-1} \rightarrow 0$

with each ${Z^i}$ (the group of cycles) flat. Tensoring with any ${B}$ produces short exact sequences

$\displaystyle 0 \rightarrow Z^i \otimes_A B \rightarrow L^i \otimes_B \rightarrow Z^{i-1} \otimes_A B \rightarrow 0 ,$

which show that

$\displaystyle 0 \rightarrow L^{-1} \otimes_A B \rightarrow L^0 \otimes_A B \rightarrow \dots \rightarrow L^N \otimes_A B \rightarrow 0$

is exact. It follows that the mapping cylinder of ${\phi \otimes 1_B: K \otimes_A B \rightarrow C \otimes_A B}$ is acyclic, so ${\phi \otimes 1_B }$ induces an isomorphism in cohomology. We know see that the finite flat replacement ${K}$ for the Cech complex ${C^*(\mathcal{F})}$ has the convenient property that ${H^p(X \times_A B, \mathcal{F} \otimes_A B) = H^p(C^*(\mathcal{F}) \otimes_A B) = H^p(K \otimes_A B)}$ . We have thus proved:

Theorem 5 (The Grothendieck Complex) Let ${f: X \rightarrow Y}$ be a proper morphism of noetherian schemes for ${Y = \mathrm{Spec} A}$ affine. Let ${\mathcal{F}}$ be a coherent sheaf on ${X}$ , flat over ${Y}$ . Then there is a finite complex ${K}$ of finitely generated ${A}$ -modules such that $\displaystyle H^p(K\otimes_A B ) = H^p(X \times_A B, \mathcal{F}\otimes_A B)$

for any ${A}$ -algebra ${B}$ .

The semicontinuity theorem « Climbing Mount Bourbaki Says:

January 10, 2011 at 9:38 am

[…] Last time, we proved an important theorem. Namely, for a proper morphism of noetherian schemes , we showed that the cohomology of a coherent sheaf on , flat over the base, could be described as the cohomology of a finite complex of flat, finitely generated modules, and moreover that if we base-changed to some other scheme , we just had to compute the cohomology of to get the cohomology of the base extension of the initial sheaf. […]

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