In this post we will be working with the etale topology once again, so we start by formalizing some concepts. We want to first define the category (see Category Theory) called or in the literature. The objects of this category are etale morphisms (see Cohomology in Algebraic Geometry) of schemes to , while the morphisms are etale morphisms such that if is another etale morphism to , then .
Presheaves of sets, abelian groups, etc. on the category are defined as contravariant functors from the category to sets, abelian groups, etc. They are sheaves if they satisfy the sheaf conditions commonly referred to as local identity and gluing (see Even More Category Theory: The Elementary Topos). We will refer to presheaves (resp. sheaves) on as etale presheaves (resp. etale sheaves) or simply as presheaves (resp. sheaves) on .
Let be a morphism of schemes, and let be a sheaf on . There is a sheaf on determined by the and , called the direct image sheaf, written and defined by
for an etale morphism .
The direct image functor is the functor that assigns to a sheaf the direct image sheaf . The derived functor (see More on Chain Complexes and The Hom and Tensor Functors) of is called the higher direct image functor and written .
On the other hand, a morphism of schemes and a sheaf on also determine a sheaf on called the inverse image sheaf, written and obtained via the following construction:
Let be an etale morphism of schemes. The presheaf is given by
where the direct limit (see Etale Cohomology of Fields and Galois Cohomology) is taken over all such that the morphisms commute (i.e. the composition of morphisms and is equal to the composition of morphisms and ).
We then define the sheaf as the sheaf associated to the presheaf (the process of associating a sheaf to a presheaf, also known as sheafification, is left to the references for now).
We now introduce the notions of open subschemes and closed subschemes, and open immersions and closed immersions. We quote directly from the book Algebraic Geometry by Robin Hartshorne:
An open subscheme of a scheme is a scheme whose topological space is an open subset of , and whose structure sheaf is isomorphic to the restriction of the structure sheaf of . An open immersion is a morphism which induces an isomorphism of with an open subscheme of .
Note that every open subset of a scheme carries a unique structure of open subscheme.
A closed immersion is a morphism of schemes such that induces a homeomorphism of onto a closed subset of and furthermore the induced map of sheaves on is surjective. A closed subscheme of a scheme is an equivalence class of closed immersions, where we say and are equivalent if there is an isomorphism such that .
Now it may happen that given an open immersion and a sheaf on , the stalks of may not be zero for points outside . Therefore we define another sheaf on , given by the following construction:
Given a sheaf on , and an etale morphism , let
Once again, is a presheaf on , but it need not be a sheaf, therefore we define instead to be the sheaf associated to the presheaf .
One concept related to this “extension by zero” functor is cohomology with compact support, written .
The functors , , and the generalization of , called the direct image with compact support and denoted , are part of the so-called “six operations” which play an important role in modern algebraic geometry.
Image Functors for Sheaves on Wikipedia
Direct Image Functor on Wikipedia
Inverse Image Functor on Wikipedia
Direct Image with Compact Support on Wikipedia
Lectures on Etale Cohomology by J.S. Milne
Algebraic Geometry by Robin Hartshorne
Etale Cohomology and the Weil Conjecture by Eberhard Freitag and Reinhardt Kiehl
2 thoughts on “Direct Images and Inverse Images of Sheaves”
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