So, by now, we have some ideas about what these object are, as (co)directions of non-propagation of sections of the various cohomology sheaves of the complex
. But that’s really only scratching the surface, as all we’ve done so far is fiddle around with the definition of microsupport. For a general complex of sheaves
(X a real, smooth manifold), there isn’t much hope for mere mortals like ourselves in calculating
. But, things get a little brighter when we restrict to those objects whose cohomology sheaves are all constructible; for
, there exists a stratification
of
by smooth submanifolds such that
.
Specifically, in addition to being -conic and closed,
is also a real subanalytic, isotropic subset of
(with respect to the usual symplectic structure on the cotangent bundle).
Example:
One of the easiest examples of this is when we consider the constant sheaf on X. Suppose we’re using as our base ring for
, and let
be a closed submanifold. To
, we can associate the sheaf
(considered as a complex of sheaves concentrated in degree zero), whose stalks are
if
, and
if
. Then, it’s not too hard to show the equality
, the full conormal bundle to
in
.
Another good toy example is when , and we’re considering the constant sheaf on a closed interval
, extended by zero to all of
. What would this sheaf’s microsupport be?
Since almost all the objects we’ll be interested in are in , we should explore the microsupport of these complexes as deeply as we can, from as many perspectives as we can find (morally, the “Yoneda lemma” approach (http://mathoverflow.net/a/3223)).
The perspective I want to talk about today is from the (triangulated) derived categories , from Kashiwara and Schapira’s Sheaves on Manifolds ([K-S]), that are associated to subsets
. Briefly,
is the localization of
by the full subcategory of objects whose microsupport is disjoint from
. Then, we want to say that working “microlocally” with some complex of sheaves
on
means to consider
as an object of the category
for some subset
. This should seem pretty reasonable; If
, then
has no interesting microlocal behavior on
and can’t really interact in a meaningful way with other, non-zero, objects of
.
So, now it’s time for the actual details. Let be a subset of
. We define
to be the full subcategory of
consisting of those objects
such that
. Additionally, this is a triangulated category, with distinguished triangles (naturally) those of
whose objects belong to
. Then,
is a null-system in
, and we may define the relevant localization. That is, we set
(where
). In the case we’ll usually use,
for some covector
, we’ll be lazy and write
for the localized category.
One of the first things we can say about is: a morphism
in
becomes an isomorphism in
(or, we say, “on
“) if we can embed
in a distinguished triangle $latex \mathcal{A}^\bullet \overset{u}{\to} \mathcal{B}^\bullet \to \mathcal{C}^\bullet \overset{+1}{\to}$ with
.
Let’s play with some examples, and see how working in differs from working in
.
Example:
Let . To the closed subset
, there is a distinguished triangle in
:
from this triangle, it follows that the complexes and
are isomorphic in
(i.e., there is a morphism between the complexes that induces an isomorphism on cohomology).
Let’s now consider the same distinguished triangle, but this time in the category , where
. After a quick calculation, we obtain:
,
, and
.
whence it follows that in
, as well as
in
.
Why should you care?
Okay, that’s cute. But why are these categories useful? Why should you bother to play with at all?
Earlier in this post, I mentioned that, for constructible,
is an
-conic, subanalytic, isotropic subset of
. Similarly, using
, I want to give a “microlocal” characterization of those
whose microsupport is contained in an involutive submanifold of
. Particularly, when the microsupport is contained in the conormal bundle
to some submanifold
of
(as it turns out, the microsupport is ALWAYS an involutive subset of the cotangent bundle). Hopefully, this pursuit will tell us something new and interesting about the objects of our favorite category,
. Eventually (not this time!) we’ll put these two things together, and see what happens along a Lagrangian submanifold; this leads to Kashiwara and Schapira’s notion of pure sheaves. These are pretty neat.
Proposition 1: ([6.6.1, K-S])
Suppose is a closed submanifold of
, and let
denote the associated smooth, closed embedding. Let
(non-zero, or else this is a stupid exercise), and let
. Let
denote the projection of the cotangent bundle. Assume further that, in a neighborhood of
,
. Then, there exists a
such that
in
.
(Quick sketch: Basically, we want to successively, microlocally, cut up by restricting to various subsets of
, and throw away those pieces whose microsupport doesn’t meet
.)
Detailed Sketch:
By induction on the codimension of in
, we may assume
is a hypersurface, say given by
, with
. Set
, so we have the open embeddings
. By [6.3.1, K-S], there is an inclusion
,
where (in our case here) consists of the zero section of
, together with those covectors
where
. Vaguely, the
operation characterizes those covectors arising as limits of sequences of covectors in
(see [6.2.3, K-S]).
Since is not in the right hand side, we must have
. Hence, from the adjunction distinguished triangle
in , we see that
in
. So, we might as well have assumed from the start that
. Similarly, we find (by [6.3.1, K-S]) that
, and see that
in
, using the distinguished triangle
(where is the closed complement of
in
), and using the fact that
. All together, this gives us the desired isomorphism
in
.
Q.E.D.
Proposition 2: [6.6.1, K-S]
Same as the set up of proposition 1, but now we assume that in a neighborhood of
. Then, there exists
such that
in
.
Since , the previous proposition applies, and we may assume
in
for some
. In particular, this tells us
in a neighborhood of
. So, the assumption that
, together with [5.4.4, K-S] (this tells us that
consists of covectors
such that
, and
) give us that
is contained in the zero section,
in a neighborhood of
: Since
, the condition
implies
) that
in a neighborhood of
.
Next, let be the canonical map to a point. Since
, there exists a
such that
. Hence,
in
, and we’re done.
Q.E.D.
Now, what can we say our constructible case ? Keywords: non-degenerate covectors, and Morse modules. But that’s for another time.