Local cohomology III

This post continues the series on local cohomology.

Let ${A}$ be a noetherian ring, ${\mathfrak{a} \subset A}$ an ideal. We are interested in the category ${\mathrm{QCoh}(\mathrm{Spec} A \setminus V(\mathfrak{a}))}$ of quasi-coherent sheaves on the complement of the closed subscheme cut out by ${\mathfrak{a}}$ . When ${\mathfrak{a} = (f)}$ for ${f \in A}$ , then

$\displaystyle \mathrm{Spec} A \setminus V(\mathfrak{a}) = \mathrm{Spec} A_f,$

and so ${\mathrm{QCoh}(\mathrm{Spec} A \setminus V(\mathfrak{a}))}$ is the category of modules over ${A_f}$ . When ${\mathfrak{a}}$ is not principal, the open subschemes ${\mathrm{Spec} A \setminus V(\mathfrak{a})}$ are generally no longer affine, but understanding quasi-coherent sheaves on them is still of interest. For instance, we might be interested in studying vector bundles on projective space, which pull back to vector bundles on the complement ${\mathbb{A}^{n+1} \setminus \left\{(0, 0, \dots, 0)\right\}}$ . This is not affine once ${n > 0}$ .

In order to do this, let’s adopt the notation

$\displaystyle X' = \mathrm{Spec} A \setminus V(\mathfrak{a}) , \quad X = \mathrm{Spec} A,$

and let ${i: X' \rightarrow X}$ be the open imbedding. This induces a functor

$\displaystyle i_* : \mathrm{QCoh}(X') \rightarrow \mathrm{QCoh}(X)$

which is right adjoint to the restriction functor ${i^* : \mathrm{QCoh}(X) \rightarrow \mathrm{QCoh}(X')}$ . Since the composite ${i^* i_* }$ is the identity on ${\mathrm{QCoh}(X')}$ , we find by a formal argument that ${i_*}$ is fully faithful.

Fully faithful right adjoint functors have a name in category theory: they are localization functors. In other words, when one sees a fully faithful right adjoint ${\mathcal{C} \rightarrow \mathcal{D}}$ , one should imagine that ${\mathcal{C}}$ is obtained from ${\mathcal{D}}$ by inverting various morphisms, say a collection ${S}$ . The category ${\mathcal{C}}$ sits inside ${\mathcal{D}}$ as the subcategory of ${S}$ -local objects: in other words, those objects ${x}$ such that ${\hom(\cdot, x)}$ turns morphisms in ${S}$ into isomorphisms.

1. Localization

In particular, we should think of the category ${\mathrm{QCoh}(X')}$ as being obtained from ${\mathrm{QCoh}(X)}$ (which is the very concrete category of ${A}$ -modules) by inverting a collection of morphisms; the functor ${i_*}$ corresponds to the inclusion of the local objects. Specifically, the morphisms we are inverting are all morphisms ${\mathcal{F} \rightarrow \mathcal{G}}$ in ${\mathrm{QCoh}(X)}$ which are isomorphisms when restricted to ${X'}$ . This point of view will explain some of the results on the relations between these two categories.

For instance, we might ask the following.

Question: When does an ${A}$ -module arise as the global sections of a quasi-coherent sheaf on ${X'}$ ? In other words, what is the image of ${i_*: \mathrm{QCoh}(X') \rightarrow \mathrm{QCoh}(X)}$ ?

The categorical point of view gives an answer. An ${A}$ -module ${M}$ is in the image precisely if it is a local object for this class of morphisms to be inverted. In other words, ${M}$ is local if and only if

$\displaystyle \hom(F, M) \simeq \hom(E, M)$

whenever ${E \rightarrow F}$ is a homomorphism of ${A}$ -modules inducing an isomorphism of sheaves on ${X'}$ .

Theorem 12 (Horrocks) An ${A}$ -module ${M}$ arises as ${\Gamma(X', \mathcal{F})}$ for ${\mathcal{F} \in \mathrm{QCoh}(X')}$ if and only if

$\displaystyle \hom(A/\mathfrak{a}, M) = \mathrm{Ext}^1(A/\mathfrak{a}, M) = 0.$

Proof: We know that an ${A}$ -module ${M}$ is in the image of ${i_*}$ (i.e., is local) if and only if for every map ${E \rightarrow F}$ of modules inducing an isomorphism of sheaves restricted to ${X'}$ , we have ${\hom(F, M) \simeq \hom(E, M)}$ . Let’s suppose that this is the case, and test out the conditions.

Take for instance

$\displaystyle E = A/\mathfrak{a}, \quad F = 0;$

then both ${E }$ and ${F}$ are supported on the closed subscheme defined by ${\mathfrak{a}}$ , and we conclude that

$\displaystyle \hom(A/\mathfrak{a}, M) = 0.$

Similarly, we can take

$\displaystyle E = \mathfrak{a}, \quad F = A;$

in this case the kernel and cokernel of ${E \rightarrow F}$ are both supported on the closed subscheme ${V(\mathfrak{a})}$ , so that

$\displaystyle \hom(R, M) \rightarrow \hom(\mathfrak{a}, M)$

is required to be an equivalence. Looking at the long exact sequence in ${\mathrm{Ext}}$ , we find that the cokernel of this map is ${\mathrm{Ext}^1(R/\mathfrak{a}, M)}$ , which must therefore vanish.

Conversely, suppose ${M}$ satisfies the above two vanishing conditions; we need to show that ${M}$ is local in the above sense. Observe that condition ${\hom(A/\mathfrak{a}, M) =0 }$ implies, by taking extensions and direct limits, that

$\displaystyle \hom(T, M) = 0$

for any module ${T}$ which is entirely ${\mathfrak{a}}$ -power torsion. The condition ${\mathrm{Ext}^1(A/\mathfrak{a}, M) =0}$ implies that ${\mathrm{Ext}^1(T, M) =0}$ for any finitely generated ${A}$ -module ${T}$ which is ${\mathfrak{a}}$ -power torsion. Using a version of the Milnor exact sequence and the assumption ${\hom(A/\mathfrak{a}, M) =0}$ , we can deduce this for any ${A}$ -module ${T}$ which is ${\mathfrak{a}}$ -power torsion.

Now, if ${E \rightarrow F}$ is a morphism of ${A}$ -modules which induces an isomorphism of sheaves on ${X'}$ , then the kernel and cokernel of ${E \rightarrow F}$ are ${\mathfrak{a}}$ -power torsion. This, together with the previous paragraph, implies that ${M}$ is local with respect to this class of maps, as desired. $\Box$

Remark: It is possible to have a class in ${\mathrm{Ext}^1(M, N)}$ (where ${M, N}$ are modules over some ring) which is nonzero, but such that the restriction to ${\mathrm{Ext}^1(M', N)}$ , for ${M' \subset M}$ a finitely generated submodule, always vanishes. For instance, any nontrivial extension of ${\mathbb{Z}[2^{-1}]}$ as a ${\mathbb{Z}}$ -module fulfills this condition, since ${\mathbb{Z}[2^{-1}]}$ is not projective while every finitely generated submodule is. In other words, the phenomenon of phantom maps appears in an algebraic context, too.

2. Describing the localization functor

Let’s look back at the conditions in Horrocks’s theorem above. If ${M}$ is finitely generated, the conditions ${\hom(A/\mathfrak{a}, M) = \mathrm{Ext}^1(A/\mathfrak{a}, M) = 0}$ precisely state that

$\displaystyle \mathrm{depth}_{\mathfrak{a}}(M) \geq 2.$

In other words, a coherent sheaf ${\mathcal{F}}$ on ${\mathrm{Spec} A}$ is such that ${\mathcal{F} \rightarrow i_* i^* \mathcal{F}}$ is an isomorphism precisely when the depth at points of ${V(\mathfrak{a})}$ is at least two. This is a statement that can be interpreted in terms of local cohomology.

We can also try to understand the functor ${i_* i^*: \mathrm{QCoh}(X) \rightarrow \mathrm{QCoh}(X)}$ . In the categorical picture, this is the localization functor; in the geometric picture, it is

$\displaystyle M \mapsto \Gamma( \mathrm{Spec} A \setminus V(\mathfrak{a}), \widetilde{M}).$

Proposition 13 The localization functor can be described as ${M \mapsto \varinjlim \hom( \mathfrak{a}^n, M)}$ .

This is related to the formula for local cohomology as ${\varinjlim \mathrm{Ext}^i( R/\mathfrak{a}^n, M)}$ . This particular formula was apparently first observed by Deligne, and is an exercise in Hartshorne that I unsuccessfully tried to solve a while back. It can be solved directly enough, but we can also interpret it via this categorical framework.

In fact, we need to check that the natural map

$\displaystyle M \rightarrow \varinjlim \hom( \mathfrak{a}^n, M)$

is an isomorphism of sheaves on ${X' = \mathrm{Spec} A \setminus V(\mathfrak{a})}$ , and that the target is local. The first observation is evident, since each map ${\hom(R, M) \rightarrow \hom(\mathfrak{a}^n, M)}$ is an isomorphism away from ${V(\mathfrak{a})}$ . For the second assertion, we need to check that

$\displaystyle \hom(R/\mathfrak{a}, \varinjlim \hom(\mathfrak{a}^n, M)) = \mathrm{Ext}^1(R/\mathfrak{a}, \varinjlim \hom(\mathfrak{a}^n, M)) = 0.$

To prove this, we note first that

$\displaystyle \hom(R/\mathfrak{a}, \varinjlim \hom(\mathfrak{a}^n, M)) = \varinjlim \hom(R/\mathfrak{a} \otimes \mathfrak{a}^n, M)$

and the successive maps in the direct limit are all zero. We can make the same argument for ${\mathrm{Ext}^1}$ , once we note that ${\mathrm{Ext}^1(R/\mathfrak{a}, \cdot)}$ commutes with filtered colimits of modules, because of the description

$\displaystyle \mathrm{Ext}^1(R/\mathfrak{a} , \cdot) = \mathrm{coker}( \hom(R, \cdot) \rightarrow \hom(\mathfrak{a}, \cdot)),$

and the fact that ${R}$ and ${\mathfrak{a}}$ are finitely presented ${R}$ -modules. In general, though, ${R/\mathfrak{a}}$ need not be a compact object in the derived category—that is, it is probably not a perfect complex.

3. Sheaves on a punctured spectrum

We are going to apply this formula below to deduce finiteness criteria for push-forwards of sheaves along open immersions.

Example 3 Let ${(A, \mathfrak{m})}$ be a two-dimensional regular local ring. Let ${\mathrm{PSpec} A}$ denote the punctured spectrum ${\mathrm{Spec} A \setminus \left\{\mathfrak{m}\right\}}$ . Then there is an equivalence of categories

$\displaystyle \mathrm{Vect}(\mathrm{Spec} A) \simeq \mathrm{Vect}(\mathrm{PSpec} A).$

In particular, every vector bundle on the punctured spectrum ${\mathrm{PSpec} A}$ is trivial.

In order to prove this, let us first show that the restriction functor ${\mathrm{Vect}(\mathrm{Spec} A) \rightarrow \mathrm{Vect}(\mathrm{PSpec} A)}$ is fully faithful.

We will show in fact that if ${\mathcal{E} \in \mathrm{Vect}(\mathrm{Spec} A)}$ (in fact, ${\mathcal{E}}$ is necessarily trivial), then

$\displaystyle \mathcal{E} \rightarrow i_* i^* \mathcal{E}$

is an isomorphism, for ${i: \mathrm{PSpec} A \rightarrow \mathrm{Spec} A}$ the inclusion. This reduces to the case of ${\mathcal{E} = \mathcal{O}_{\mathrm{Spec} A}}$ . But we saw earlier that a coherent sheaf on ${\mathrm{Spec} A}$ , corresponding to an ${A}$ -module ${M}$ , is the push-forward of its restriction to ${\mathrm{PSpec} A}$ if and only if ${\mathrm{depth}_{\mathfrak{m}} M \geq 2}$ . This is true for ${M = A}$ , by assumption on ${A}$ .

We next have to show that the restriction functor is essentially surjective. Let ${\mathcal{E}'}$ be a vector bundle on ${\mathrm{PSpec} A}$ . If we can show that ${i_* \mathcal{E}'}$ is a coherent sheaf on ${\mathrm{Spec} A}$ , then it will correspond (as above) to an ${A}$ -module of depth ${2}$ , which is projective by the Auslander-Buchsbaum formula: in particular, ${\mathcal{E}'}$ will then extend to a vector bundle on ${\mathrm{Spec} A}$ . In order to do this, we need a general result which tells us when push-forwards along open immersions preserve coherence.

Here is the main finiteness result we need.

Theorem 14 (Finiteness condition) Let ${A}$ be a regular local ring (or more generally, a quotient of one). Let ${M}$ be an ${A}$ -module. Then $H^i_{\mathfrak{m}}(M)$ is a finitely generated ${A}$ -module if and only if

$\displaystyle H^{i - \dim A/\mathfrak{q}}_{\mathfrak{q}}(M_{\mathfrak{q}}) = 0,$

for each ${\mathfrak{q} \in \mathrm{PSpec} A}$ .

In other words, we get a condition for when ${H^i_{\mathfrak{m}}(M)}$ is finitely generated in terms of the local cohomology groups of ${\widetilde{M}|_{\mathrm{PSpec} A}}$ at points of ${\mathrm{PSpec} A}$ . When we can say something about the depth of ${M}$ at non-maximal primes, for instance, we may be able to apply this criterion.

Proof: We will only prove the “if” direction.

Assume ${A}$ regular local, of dimension ${d}$ . Then

$\displaystyle H^i_{\mathfrak{m}}(M) = D ( \mathrm{Ext}^{d-i}_A(M, A)),$

for ${D}$ the Matlis duality functor on ${A}$ -modules, given by ${\hom(\cdot, Q)}$ for ${Q}$ an injective envelope of the residue field. The duality functor preserves finite length modules. So, if ${\mathrm{Ext}_A^{d-i}(M, A)}$ is finite length, then ${H^i_{\mathfrak{m}}(M)}$ is too.

Now ${\mathrm{Ext}_A^{d-i}(M, A)}$ is a finitely generated module over the local ring ${A}$ ; to say that it is of finite length is to say that it is, as a coherent sheaf, supported at the closed point of ${\mathrm{Spec} A}$ . In other words, we want

$\displaystyle \widetilde{\mathrm{Ext}_A^{d-i}(M, A)}|_{\mathrm{PSpec} A} = 0.$

Or, equivalently, for each ${\mathfrak{q} \in \mathrm{PSpec} A}$ , we require that

$\displaystyle \mathrm{Ext}^{d-i}_{A_{\mathfrak{q}}}(M_{\mathfrak{q}}, A_{\mathfrak{q}}) = 0.$

Here we can apply local duality for ${A_{\mathfrak{q}}}$ (which is also regular), and the fact that ${\dim A = \dim A_{\mathfrak{q}} + \dim A/\mathfrak{q}}$ : that is, ${A}$ is catenary. We find:

$\displaystyle H^{i - \dim A/\mathfrak{q} }_{\mathfrak{q}}(M_{\mathfrak{q}}) = D( \mathrm{Ext}_{A_{\mathfrak{q}}} ^{d-i}(M_{\mathfrak{q}}, A_{\mathfrak{q}})),$

where, by abuse of notation, ${D}$ also refers to the corresponding duality functor for ${A_{\mathfrak{q}}}$ -modules. The duality functor is conservative: in particular, a module is zero if and only if its Matlis dual is zero. It follows that, in particular, the groups ${\mathrm{Ext}_{A_{\mathfrak{q}}}^{d-i}(M_{\mathfrak{q}}, A_{\mathfrak{q}})}$ vanish under the hypotheses of the result. $\Box$

Let’s now finish the proof of the earlier example, classifying vector bundles on ${\mathrm{PSpec} A}$ for ${A}$ two-dimensional and regular. If ${\mathcal{E}' \in \mathrm{Vect}(\mathrm{PSpec} A)}$ , we had to show that the push-forward

$\displaystyle i_* \mathcal{E} ' \in \mathrm{QCoh}(\mathrm{Spec} A)$

was actually coherent; we saw earlier that the Auslander-Buchsbaum formula would imply that this would be an actual vector bundle. In order to do this, first choose anycoherent sheaf ${\mathcal{F} \in \mathrm{QCoh}(\mathrm{Spec} A)}$ extending ${\mathcal{E}'}$ . As such, ${\mathcal{F}}$ corresponds to an ${A}$ -module ${M}$ , not necessarily projective: it could even have summands of ${A/\mathfrak{m}}$ , but ${\mathcal{F}}$ always exists.

The claim is that $H^1_{\mathfrak{m}}(M)$ is finitely generated. To see this, observe that we need to show that ${H^{1 - \dim A/\mathfrak{q}}_{\mathfrak{q}}(M_{\mathfrak{q}}) = 0}$ for ${\mathfrak{q} \in \mathrm{PSpec} A}$ . Observe that ${M_{\mathfrak{q}}}$ only depends on ${\mathcal{E}'}$ and not on ${M}$ , and it vanishes because ${M_{\mathfrak{q}}}$ is a free module over ${A_{\mathfrak{q}}}$ for each ${\mathfrak{q} \in \mathrm{PSpec} A}$ . In particular, given the exact sequence

$\displaystyle 0 \rightarrow H^0_{\mathfrak{m}}(M) \rightarrow \hom(A, M) \rightarrow \varinjlim \hom(\mathfrak{m}^n, M) \rightarrow \varinjlim \mathrm{Ext}^1(A/\mathfrak{m}^n, M) = H^1_{\mathfrak{m}}(M)$

we find that the module ${\varinjlim \hom( \mathfrak{m}^n, M)}$ is finitely generated. But by the formula above, this is precisely the module corresponding to the push-forward of ${\mathcal{E}'}$ . As we saw above, this is enough to conclude the proof.

Climbing Mount Bourbaki