A couple of posts ago, I mentioned the “microlocal” category , and I talked about a couple of neat things you could do with it. Looking back, I’m finding myself unsatisfied with the level of detail in my examples (as well as the number of examples!), so I figured I’ll make a post solely dedicated to examples of .
So, let’s recap the construction. Let X be a smooth manifold (fine for our purposes here; usually we’d take X to be a real analytic manifold), a subset of the cotangent bundle . Then, the category is defined to be the localization of the bounded, derived category by the null system
We then say that a morphism in is an isomorphism on (or, an isomorphism in ) if there exists a distinguished triangle in with . If for some point , we write instead of (because we’re lazy).
I won’t be using this property, but it’s pretty neat, and is solely a consequence of being a localization:
where the limit is indexed over those morphisms with target (resp., with source ) in that are isomorphisms on .
Lastly, before I delve into the examples, we’ll need the following facts. First, for X a topological space, a closed subset, and A a commutative, unital ring with finite global dimension (say, ), there is a short exact sequence
in the category of sheaves of -modules. For us, this translates to: there is a distinguished triangle
in the (bounded) derived category .
Second, for a finite dimensional real vector space, and a closed, convex cone, we set . In this case, we have the equality (where is the canonical projection). Generalizing this, if is a closed submanifold of the manifold , then is the conormal bundle to in X.
If we’re going to understand this at all, we should start in the easiest possible (not stupid) case: when . While we’re at it, let’s revisit the example I mentioned in the previous post, in which we cared about the closed subset in , and the subset of the cotangent bundle of . The relevant distinguished triangles to keep in our heads are then:
Using these, I want to show the isomorphisms in .
We should start by calculating all those microsupports! For simplicity, we use the isomorphism .
- (the zero section of the cotangent bundle). If you think of as a (closed, convex) cone in itself, its polar set is just the zero vector. That’s the basic idea.
- (just think of as a closed submanifold of , and use fact 2 above).
- (you’ll have to think about this one, sorry. It’s not too bad…).
- (remember, is always a closed subset!).
- (see above, and use the triangle inequality for the microsupport).
Okay! Now we can show those isomorphisms. If we rotate the triangle in 1, (i.e., the distinguished triangle
we see that is an isomorphism on , since ! One down.
Similarly, if we rotate triangle 2, we see that is an isomorphism on , as . Easy!
Okay, now we’ll consider the subset and .
Right off the bat, we note that if is locally constant in an open neighborhood U of 0, (so ), implying in . I’ll leave you guys (okay, I know that nobody actually reads this) with a cliffhanger:
For all with , we have