Last update: 22 October 2012
To any reader that has not met sheaves before: I suggest that you don't read this section, only refer to it a few times while you are reading Chapter VIII of these notes. The most important thing, from the point of view of these notes, is to understand the basic structures given in Chapter VIII; anyone who is going to study these topics in more depth can always come back and learn these definitions later.
A large part of the material in this section is basic material about derived categories. This material can usually be found in texts which treat homological algebra. Everything in this section, except the definition and properties of perverse sheaves given in §3 can be found in [KSc1980] Chapt. I-III. The definition of a perverse sheaf is in [BBD1992] 4.0 and the proof of Theorem (3.1) is in [BBD1992] Theorem 1.3.6. The Theorems in (3.2) are proved in [BBD1992] 2.1.9-2.1.11 and Theorem 4.3.1, respectively. We shall not review the definition of sheaves, it can be found in many textbooks, see [KSc1980] Chapt. II.
Let be an algebraic variety. A complex of sheaves on is a sequence of sheaves on and morphisms of sheaves
The morphisms are called the differentials of the complex Let and be complexes of sheaves. A morphism is a set of maps such that the diagram
The cohomology sheaf of a complex is the sheaf
We have a well defined complex of sheaves given by
A quasi-isomorphism is a morphism such that the induced morphism is an isomorphism. Note that every isomorphism is a quasi-isomorphism but not the other way around (even though the notation may be confusing).
Let be an algebraic variety. Let and be complexes of sheaves on Two morphisms and are homotopic is there is a collection of morphisms such that
The motivation for this definition is that if and are homotopic then
Define to be the category given by
This just means that, in the category we identify homotopic morphisms.
Let be a complex of sheaves on An injective resolution of is a quasi-isomorphism such that is injective (an injective object in the category of sheaves on for all Let denote the category of sheaves on and let be a functor. The right derived functor of is the functor given by
where is an injective resolution of The ith derived functor of is the functor given by
where is an injective resolution of In other words is the cohomology sheaf of the complex
A complex of sheaves is bounded if there exists a positive integer such that and for all
An algebraic stratification of an algebraic variety is a finite partition of into strata such that
Let be a prime number and let be the algebraic closure of the field of numbers. A sheaf on is if there is an algebraic stratification such that, for each the restriction of to is a locally constant sheaf of finite dimensional vector spaces over A complex is if is for all
Let be a variety. Let and be complexes of sheaves on Define an equivalence relation on diagrams
in which have and as end points by saying that the diagram is equivalent to the diagram if there exists a commutative diagram
The notation denotes that the map is a quasi-isomorphism. The bounded derived category of sheaves on is the category given by
This definition of morphisms is a formal mechanism that inverts all quasi-isomorphisms. It ensures (in a coherent way) that "inverses" of quasi-isomorphisms are morphisms, i.e. that is a morphism from to
Given two morphisms and in one can show that there always exists a commutative diagram
and one defines the composition of the two morphisms and to be the morphism defined by the diagram
A map between locally compact algebraic varieties is compact if the inverse image of every compact subset of is a compact subset of
Let be a morphism of locally compact algebraic varieties. Let be a sheaf on The support, of a section of on an open set is the complement in of the union of open sets such that
The direct image with compact support sheaf is the sheaf on defined by setting
for every open set in (For a sheaf on and an open set in This defines a functor where denotes the category of sheaves on
Let be a morphism of locally compact algebraic varieties. The direct image with compact support functor is given by
so that is the right derived functor of the functor
Let be a morphism of algebraic varieties. Let be a sheaf on The inverse image sheaf is the sheaf on associated to the presheaf
where the limit is over all open sets in which contain This defines a functor
Let be a morphism of algebraic varieties. Let Then is the unique (up to isomorphism) complex on such that
Actually, I have cheated here: We can only be sure that the complex is well defined if is a locally trivial principal is a semisimple complex on and we require to be a semisimple complex on see [Lus1993] 8.1.7 and 8.1.8 for definitions and details.
Let be a complex of sheaves on For each integer define a new complex with differentials by
The shift functor is the functor
This definition is too involved for us to take the energy to repeat it here, we shall refer the reader to [KSc1980] §3.1. The main thing that we will need to know is that this functor exists.
Let be an algebraic variety. The support, of a sheaf on is the complement of the union of open sets such that
A complex is a perverse sheaf if
where is the Verdier dual of
An abelian category is a category which has a direct sum operation and for which every morphism has a kernel and a cokernel. See [Ksc1980] I §1.2 for a precise definition.
The full subcategory of whose objects are perverse sheaves on is an abelian category.
Let be a smooth locally closed subvariety of complex dimensional and let be a locally constant sheaf on There is a unique complex in such that
The complexes are the intersection cohomology complexes and an explicit construction of these complexes is given in [BBD1992] Prop. 2.1.11.
The simple objects of the category of sheaves are the intersection complexes as runs through the irreducible locally constant sheaves on various smooth locally closed subvarieties
This is an excerpt from a paper entitled Quantum groups: A survey of definitions, motivations and results by Arun Ram. Research and writing supported in part by an Australian Research Council fellowship and a National Science Foundation grant DMS-9622985.