Constructible Sheaves

TL;DR

This paper proves a fundamental lemma related to constructible sheaves, providing a motivic proof of a classical theorem, and clarifies the categorical structure and properties of constructible sheaves in algebraic geometry.

Contribution

It establishes the basic lemma's proof, connects constructible sheaves with the derived category, and demonstrates key properties like effaceability within the category.

Findings

Proof of the basic lemma by Beilinson

Equivalence of categories for constructible sheaves

Effaceability of higher direct images and Ext groups

Abstract

This article contains a proof of the basic lemma. This lemma, discovered by Beilinson, yields a motivic proof of the Andreotti-Frankel theorem for affine varieties. Next, it is shown that the category of Cohomologically Constructible Sheaves (as it is referred to in the Riemann-Hilbert correspondence) coincides with the derived category of bounded complexes of constructible sheaves. It is also shown that higher direct images and the sheaf-Ext groups are effaceable in the category of constructible sheaves.