Some Microlocal Computations

A couple of posts ago, I mentioned the “microlocal” category $D^b(X;\Omega)$, 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 $D^b(X;\Omega)$

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), $\Omega$ a subset of the cotangent bundle $T^*X$.  Then, the category $D^b(X;\Omega)$ is defined to be the localization of the bounded, derived category $D^b(X)$ by the null system

$Ob(D_{T^*X \backslash \Omega}^b(X)) = \{ F \in D^b(X) | SS(F) \cap \Omega = \emptyset \}$.

We then say that a morphism $u: F \to G$ in $D^b(X)$ is an isomorphism on  $\Omega$ (or, an isomorphism in $D^b(X;\Omega)$ ) if there exists a distinguished triangle $F \overset{u}{\to} G \to H \overset{+1}{\to}$ in $D^b(X)$ with $SS(H) \cap \Omega = \emptyset$.  If $\Omega = \{p\}$ for some point $p \in T^*X$, we write $D^b(X;p)$ instead of $D^b(X;\{p\})$ (because we’re lazy).

I won’t be using this property, but it’s pretty neat, and is solely a consequence of $D^b(X;\Omega)$ being a localization:

$Hom_{D^b(X;\Omega)}(F,G) \cong \varinjlim_{F' \to F} Hom_{D^b(X)}(F',G) \cong \varinjlim_{G \to G'} Hom_{D^b(X)}(F,G')$

where the limit is indexed over those morphisms $F' \to F$ with target $F$ (resp., $G \to G'$ with source $G$) in $D^b(X)$ that are isomorphisms on $\Omega$.

Lastly, before I delve into the examples, we’ll need the following facts. First, for X a topological space, $Z \subseteq X$ a closed subset, and A a commutative, unital ring with finite global dimension (say, $\mathbb{Z}$), there is a short exact sequence

$0 \to A_{X \backslash Z} \to A_X \to A_Z \to 0$

in the category of sheaves of  $A_X$-modules.  For us, this translates to: there is a distinguished triangle

$A_{X \backslash Z} \to A_X \to A_Z \overset{+1}{\to}$

in the (bounded) derived category $D^b(X) = D^b(A_X)$.

Second, for $E$ a finite dimensional real vector space, and $\gamma \subseteq E$ a closed, convex cone, we set $\gamma^\circ = \{ \xi \in E^* | \langle v, \xi \rangle \geq 0 \text{ for any} v \in \gamma\}$.  In this case, we have the equality $SS(A_\gamma) \cap \pi^{-1}(0) = \gamma^\circ$ (where $\pi : T^*E \to E$ is the canonical projection). Generalizing this, if $M$ is a closed submanifold of the manifold $X$, then $SS(A_M) = T_M^*X$ is the conormal bundle to $M$ in X.

EXAMPLES!

Example 1.

If we’re going to understand this at all, we should start in the easiest possible (not stupid) case: when $X= \mathbb{R}$.  While we’re at it, let’s revisit the example I mentioned in the previous post, in which we cared about the closed subset $Z = \{x \geq 0\}$ in $\mathbb{R}$, and the subset $\Omega = \{(x;\xi) \in T^*\mathbb{R} | \xi > 0\}$ of the cotangent bundle of $\mathbb{R}$.  The relevant distinguished triangles to keep in our heads are then:

1. $A_{\{x < 0\}} \to A_{\mathbb{R}} \to A_{\{x \geq 0\}} \overset{+1}{\to}$
2. $A_{\{x > 0\}} \to A_{\{x \geq 0\}} \to A_{\{0\}} \overset{+1}{\to}$.

Using these, I want to show the isomorphisms $A_{\{x \geq 0\}} \cong A_{\{x < 0\}}[1] \cong A_{\{0\}}$ in $D^b(\mathbb{R};\Omega)$.

We should start by calculating all those microsupports! For simplicity, we use the isomorphism $T^*\mathbb{R} \cong \mathbb{R} \times \mathbb{R}^* \cong \mathbb{R}^2$.

• $SS(A_\mathbb{R}) = \mathbb{R} \times \{0\}$ (the zero section of the cotangent bundle).  If you think of $\mathbb{R}*$ as a (closed, convex) cone in itself, its polar set is just the zero vector.  That’s the basic idea.
• $SS(A_{\{0\}}) = T_0^*\mathbb{R}$ (just think of $\{0\}$ as a closed submanifold of $\mathbb{R}$, and use fact 2 above).
• $SS(A_{\{x \geq 0\}}) = \{(x,y) \in \mathbb{R}^2 | y x = 0; y , x \geq \}$ (you’ll have to think about this one, sorry.  It’s not too bad…).
• $SS(A_{\{x > 0\}}) = \{(x,y) \in \mathbb{R}^2 | yx = 0; y \leq 0, x \geq 0\}$ (remember, $SS(F)$ is always a closed subset!).
• $SS(A_{\{x < 0\}}) = \{(x,y) \in \mathbb{R}^2 | yx = 0; y \geq 0, x \leq 0\}$ (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

$A_{\{x \geq 0\}} \to A_{\{x < 0\}}[1] \to A_\mathbb{R}[1] \overset{+1}{\to}$)

we see that $A_{\{x \geq 0\}} \to A_{\{x < 0\}}[1]$ is an isomorphism on $\Omega$, since $SS(A_\mathbb{R}) \cap \Omega = \emptyset$! One down.

Similarly, if we rotate triangle 2, we see that $A_{\{x \geq 0\}} \to A_{\{0\}}$ is an isomorphism on $\Omega$, as $SS(A_{\{x > 0\}}) \cap \Omega = \emptyset$.  Easy!

Example 2.

Okay, now we’ll consider the subset $\Omega = \{(0,y) \in \mathbb{R}^2 | y > 0\}$ and $D^b(\mathbb{R}; \Omega)$

Right off the bat, we note that if $F \in D^b(\mathbb{R})$ is locally constant in an open neighborhood U of 0, $SS(F|_U) \subseteq U \times \{0\}$ (so $SS(F) \cap \Omega = \emptyset$), implying $F \cong 0$ in $D^b(\mathbb{R};\Omega)$.  I’ll leave you guys (okay, I know that nobody actually reads this) with a cliffhanger:

For all $a,b \in \mathbb{R}$ with $-\infty \leq b < 0 \leq a \leq +\infty$, we have

$A_{[0,a)} \cong A_{[0,a]} \cong A_{[b,0)}[1] \cong A_{(b,0)}[1]$ in $D^b(\mathbb{R};\Omega)$.

Why?