This post continues the series on local cohomology.
Let be a noetherian ring,
an ideal. We are interested in the category
of quasi-coherent sheaves on the complement of the closed subscheme cut out by
. When
for
, then
and so is the category of modules over
. When
is not principal, the open subschemes
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
. This is not affine once
.
In order to do this, let’s adopt the notation
and let be the open imbedding. This induces a functor
which is right adjoint to the restriction functor . Since the composite
is the identity on
, we find by a formal argument that
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 , one should imagine that
is obtained from
by inverting various morphisms, say a collection
. The category
sits inside
as the subcategory of
-local objects: in other words, those objects
such that
turns morphisms in
into isomorphisms.
1. Localization
In particular, we should think of the category as being obtained from
(which is the very concrete category of
-modules) by inverting a collection of morphisms; the functor
corresponds to the inclusion of the local objects. Specifically, the morphisms we are inverting are all morphisms
in
which are isomorphisms when restricted to
. 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 -module arise as the global sections of a quasi-coherent sheaf on
? In other words, what is the image of
?
The categorical point of view gives an answer. An -module
is in the image precisely if it is a local object for this class of morphisms to be inverted. In other words,
is local if and only if
whenever is a homomorphism of
-modules inducing an isomorphism of sheaves on
.
Theorem 12 (Horrocks) An
-module
arises as
for
if and only if
Proof: We know that an -module
is in the image of
(i.e., is local) if and only if for every map
of modules inducing an isomorphism of sheaves restricted to
, we have
. Let’s suppose that this is the case, and test out the conditions.
Take for instance
then both and
are supported on the closed subscheme defined by
, and we conclude that
Similarly, we can take
in this case the kernel and cokernel of are both supported on the closed subscheme
, so that
is required to be an equivalence. Looking at the long exact sequence in , we find that the cokernel of this map is
, which must therefore vanish.
Conversely, suppose satisfies the above two vanishing conditions; we need to show that
is local in the above sense. Observe that condition
implies, by taking extensions and direct limits, that
for any module which is entirely
-power torsion. The condition
implies that
for any finitely generated
-module
which is
-power torsion. Using a version of the Milnor exact sequence and the assumption
, we can deduce this for any
-module
which is
-power torsion.
Now, if is a morphism of
-modules which induces an isomorphism of sheaves on
, then the kernel and cokernel of
are
-power torsion. This, together with the previous paragraph, implies that
is local with respect to this class of maps, as desired.
Remark: It is possible to have a class in (where
are modules over some ring) which is nonzero, but such that the restriction to
, for
a finitely generated submodule, always vanishes. For instance, any nontrivial extension of
as a
-module fulfills this condition, since
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 is finitely generated, the conditions
precisely state that
In other words, a coherent sheaf on
is such that
is an isomorphism precisely when the depth at points of
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 . In the categorical picture, this is the localization functor; in the geometric picture, it is
Proposition 13 The localization functor can be described as
.
This is related to the formula for local cohomology as . 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
is an isomorphism of sheaves on , and that the target is local. The first observation is evident, since each map
is an isomorphism away from
. For the second assertion, we need to check that
To prove this, we note first that
and the successive maps in the direct limit are all zero. We can make the same argument for , once we note that
commutes with filtered colimits of modules, because of the description
and the fact that and
are finitely presented
-modules. In general, though,
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 be a two-dimensional regular local ring. Let
denote the punctured spectrum
. Then there is an equivalence of categories
In particular, every vector bundle on the punctured spectrum is trivial.
In order to prove this, let us first show that the restriction functor is fully faithful.
We will show in fact that if (in fact,
is necessarily trivial), then
is an isomorphism, for the inclusion. This reduces to the case of
. But we saw earlier that a coherent sheaf on
, corresponding to an
-module
, is the push-forward of its restriction to
if and only if
. This is true for
, by assumption on
.
We next have to show that the restriction functor is essentially surjective. Let be a vector bundle on
. If we can show that
is a coherent sheaf on
, then it will correspond (as above) to an
-module of depth
, which is projective by the Auslander-Buchsbaum formula: in particular,
will then extend to a vector bundle on
. 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
be a regular local ring (or more generally, a quotient of one). Let
be an
-module. Then
is a finitely generated
-module if and only if
for each
.
In other words, we get a condition for when is finitely generated in terms of the local cohomology groups of
at points of
. When we can say something about the depth of
at non-maximal primes, for instance, we may be able to apply this criterion.
Proof: We will only prove the “if” direction.
Assume regular local, of dimension
. Then
for the Matlis duality functor on
-modules, given by
for
an injective envelope of the residue field. The duality functor preserves finite length modules. So, if
is finite length, then
is too.
Now is a finitely generated module over the local ring
; to say that it is of finite length is to say that it is, as a coherent sheaf, supported at the closed point of
. In other words, we want
Or, equivalently, for each , we require that
Here we can apply local duality for (which is also regular), and the fact that
: that is,
is catenary. We find:
where, by abuse of notation, also refers to the corresponding duality functor for
-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
vanish under the hypotheses of the result.
Let’s now finish the proof of the earlier example, classifying vector bundles on for
two-dimensional and regular. If
, we had to show that the push-forward
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 extending
. As such,
corresponds to an
-module
, not necessarily projective: it could even have summands of
, but
always exists.
The claim is that is finitely generated. To see this, observe that we need to show that
for
. Observe that
only depends on
and not on
, and it vanishes because
is a free module over
for each
. In particular, given the exact sequence
we find that the module is finitely generated. But by the formula above, this is precisely the module corresponding to the push-forward of
. As we saw above, this is enough to conclude the proof.
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