Intersection theories of coherent sheaf stacks and virtual pull-backs via semi-perfect obstruction theories

TL;DR

This paper develops intersection theories for coherent sheaf stacks and introduces a virtual pull-back framework using semi-perfect obstruction theories, extending existing concepts to broader stack contexts.

Contribution

It constructs proper pushforwards and flat pullbacks in Chow groups of coherent sheaf stacks and defines a new virtual pull-back via semi-perfect obstruction theories for DM stacks.

Findings

Defines proper pushforwards and flat pullbacks in Chow groups

Introduces virtual pull-back as a bivariant class for DM stacks

Extends virtual pull-back concepts to semi-perfect obstruction theories

Abstract

In this paper, we construct proper pushforwards and flat pullbacks in Chow groups of coherent sheaf stacks over a Deligne-Mumford(DM) stack. When there is a relative semi-perfect obstruction theory for a DM-type morphism $X \to Y$, $X$ is a DM stack and $Y$ is a DM stack or a smooth Artin stack, we define a virtual pull-back as a bivariant class. This is an analogue of virtual pull-backs defined by Manolache.