An excellent source of motivation is Morel’s note. My handwritten note is here
Perverse sheaves is essentially an exotic t-structure on . The notion of t-exact functor is self-evident. The motivation for the study of one-sided t-exactness properties is that they lead to one-sided exactness properties on hearts. More precisely, if is the heart of and is a t-exact functor, then the fuctor obtained from composing inclusion and and followed by taking is exact.
In defining perverse t-structure we use the notion of modified dimension (See here for the notion of the grade of a module). This is needed when is a general Noetherian commutative ring of finite global dimension, because (Prop. A.10.6 of Achar’s book). When is a field, and iff .
When is a field, the intersection cohomology construction produces the simple objects in the category of perverse sheaves. For general noetherian , every perverse sheaf still admits a finite filtration by the intersection cohomology complexes. This fact is a step on the way to the proof of the fact that perverse sheaves form a noetherian abelian category.
The perverse t-structure has the following properties:
(Lemma 3.1.2) The categories and are closed under extension.
(Lemma 3.1.3) Let be a smooth connected variety. Then the perverse t-structure are the usual t-structure shifted by (This uses Lemma 2.8.2 describing the relationship between taking stalk and the duality functor .)
(Lemma 3.1.4) The following statements describe the interactions between perverse t-structure and open and closed embeddings and .
and are t-exact for the perverse t-structure.
and are left t-exact.
and are right t-exact (Note that , this is a non-formal statement and requires a local calculation, see Lemma 2.8.5; it is used to prove the reflexiveness of , i.e. for any variety . From this it follows that holds for any morphism of varieties and also interwines and .)
(Lemma 3.1.6) This is the definition of perverse sheaves in Wikipedia in terms of stratifications (each stratum is smooth and on which is a local system).
(Lemma 3.1.7) Let be an object such that for each , thesupport of is an algebraic variety, and that . For all ,we have (proof uses Lemma 3.1.6 and induction on the number of strata).
(Theorem 3.1.9) The perverse t-structure is a bounded t-structure on .
(Prop. 3.1.10) The functor induces an equivalence of categories with .
(Prop. 3.1.11 & 3.11.12) The functor and for finite morphism are t-exact for the perverse t-structure.
Intersection cohomology complexes: For locally closed embedding we have the following functor that “interpolates” between and : given by We emphasize that the following functor is not defined for arbitrary objects in .
The key results about intersection cohomology complexes is the following (Exercise 3.1.6):
Let be a locally closed embedding. Let be a perverse sheaf on , and let be a perverse sheaf on that is supported on . Show that and that .
This is not true for the ordinary t-structure.
(Lemma 3.3.2) Let a locally closed embedding. we have
For , there is a natural isomorphism .
The object has no nonzero subobjects or quotient objects supported on .
(Lemma 3.3.3) Characterization of by Lemma 3.3.2 (2).
(Lemma 3.3.7 & 3.3.8) Let be an irreducible variety. Let be the inclusion map of an open subset, and let be the complementary closed subset. Let be a perverse sheaf on .
If has no quotient supported on , then there is a natural short exact sequence .
If has no subobject supported on , then there is a natural short exact sequence .
This is the analogue of the open-closed SES for ordinary sheaves.
Definition 3.3.9: Intersection complexes for smooth, locally closed subvariety and local system of finite type. Special case when and denoted by .
Lemma 3.3.11: Criterion for when for a stratification w.r.t. and are constructible.
Lemma 3.3.12: For open and smooth and a local system on , we have .
The proof uses Lemma 3.3.11, as in the proof of Lemma 3.3.13 and Lemma 3.3.14 (dual and tensor product of IC).
Noetherian property for perverse sheaves:
(Prop. 3.4.1) The category for a smooth connected subvariety of dimension is a Serre subcategory of .
The key is to show this is closed under taking subobjects. The idea is to use Lemma 3.3.12.
(Theorem 3.4.2) Every perverse sheaf admits a finite filtration whose subquotients are IC complexes.
The idea is keep using Lemma 3.3.7 and 3.3.8 to break things down into IC complexes.
(Theorem 3.4.4) The category is Noetherian.
(Theorem 3.4.5) Assume is a field, then the category is also Artinian and the simple objects are where is an irreducible local system and is a smooth locally closed subvariety.
Theorem: If is a smooth morphism of relative dimension . The functor is t-exact for the perverse t-structure.
Introduce and . We have the following important theorem:
(Thm. 3.6.6) For smooth surjective morphism , the functor is faithful and if has connected fiber, then is fully faithful.
For shifted local system this can be understood in terms of restriction of representation of . See Remark 3.6.7.
(Corollary 3.6.9) If is a field, then sends any simple perverse sheaf to simple perverse sheaf.
Section 3.7: Perverse sheaves admit smooth descent.
General result on constructible sheaves:
It is easy to see that and preserve constructibility. For and , consult Theorem 2.7.1 of Achar’s book. The idea is to use Nagata’s compactification theorem to reduce to the open embedding case and the proper case. In the proper case we need to use some structure theorem like resolution of singularities. For , see Proposition 2.7.3. Finally for it is a consequence of and .
(Cohomological vanishing result) Let be a variety of dimension , and be a constructible sheaf on . The -modules and are finitely generated for all , and it vanishes unless .
Reference: Achar’s book, Goresky’s note (excellent for the motivation of perverse sheaves and the sheaf-theoretic definition)