Today, the goal is to define terms arising in the statement of the Morse Lemma for sheaf cohomology.
1. Definitions, examples
1.1. Propagation and micro-support
Given a sheaf , for every open set , is defined via injective resolutions. The stalk at point is the set of germs of sections of
Definition 1 Let be a manifold, , . Say that propagates at in the direction of if for all functions such that and , for all , the natural map
is an isomorphism.
Patrick Massot prefers an alternative definition. The above definition includes the case , i.e. the support of . The following avoids this.
Say that propagates at through , a co-oriented hyperplane in , for all domains touching and contained on the preferred side of , for all , the map
is an isomorphism.
For , it means that one-sided germs of sections uniquely extend to ordinary germs.
Definition 2 The micro-support (often called singular support) of is the closure in of the set of where does not propagate. Notation: .
- It is a cone.
- The projection of is contained in the support of .
- The intersection of with the zero section equals the support of (indeed, then one can take , then for sufficiently small ; this implies that ).
Example 1 Let be the constant sheaf. Then equals the zero section.
Indeed, one can choose a basis of neighborhoods such that and is contractible. These open sets have equal cohomologies, so the isomorphism takes place.
Example 2 Let be the constant sheaf on a closed, smooth domain . Then on the conormal to at equals the conormal of the boundary.
By definition, consists of locally constant functions on . Again, if is not conormal to at , one can choose a basis of neighborhoods such that and is contractible, the isomorphism takes place, propagation holds. Otherwise, take for a local correctly oriented equation of the boundary, and . Then, for all neighborhoods , empty, so cohomology vanishes, isomorphism fails for , no propagation.
For the next example, the following lemma will be useful.
Lemma 3 For all short exact sequences of sheaves
for all permutations of indices ,
Proof uses the long exact sequence in cohomology and Five Lemma.
Example 3 Let be the constant sheaf on an open domain with smooth boundary. Then .
Remember that sections of on are locally constant functions on that vanish near . By construction, there is a short exact sequence
The Lemma implies that is contained in and contains the union of and .
Example 4 Let be the constant sheaf on a closed submanifold. Then equals the conormal space to .
Locally on can pick a submanifold such that . Use the exact sequence
Question: are there easy examples where non propagation is seen in higher cohomology ? Consider non smooth closed sets.
The idea of conormal sets extends to mildly singular sets, like squares.
1.3. Micro-support for objects of the derived category
Notation. For a closed set, define a sheaf by
Lemma 4 With this notation, the isomorphism
holds if and only if for all ,
Indeed, there is a short exact sequence
Note that can be denoted by . This variant of the definition of propagation opens the way to generalization to the derived category of sheaves.
Definition 5 Let be an object of the bounded derived category of sheaves on , . The micro-support is the closure of all such that there exists a smooth function with , such that
Alexandru Oancea: what does exactly mean ?
Example 5 Let be a complex manifold. Let denote the sheaf of holomorphic functions on . Let be a differential operator with holomorphic coefficients and let be its principal symbol (a function on ). Consider the following complex of sheaves
This is not a easy fact. The Cauchy-Kowalewskaya theorem is used. It motivated initially the concept of micro-support.
1.4. Direct image
Let , be symplectic manifolds. Let be a Lagrangian submanifold, then, for every subset , is the projection on of the intersection of with diagonal in .
Proposition 6 Let be a map which is proper on the support of . Then
this is associated with the graph of which is a Lagrangian submanifold of .
Proof relies on , and a passage to the derived category which uses spectral sequences. Assume that propagates at , , for all . Let satisfy , . One needs to show that
Take . By assumption, for all . One first shows that
and, second, that
One uses the following fact.
Theorem 7 Let , be left-exact functors between categories having enough injectives. Assume that send injectives to -acyclic objects. Then
C’est la preuve de ce théorème qui utilise des suites spectrales. Par exemple, le théorème implique la suite spectrale de Leray pour les fibrations.