Higher Intersection Theory on Algebraic Stacks II, by Roy Joshua

This is the second part of our work on the intersection theory of algebraic stacks. Here we establish the existence of Chern classes and Chern character for all Artin stacks of finite type over a field with values in our Chow groups. We also extend these to higher Chern classes and a higher Chern character for {\it perfect complexes} on an algebraic stack, taking values in cohomology theories of algebraic stacks that are defined with respect to complexes of sheaves on the big smooth site. We also provide an integral intersection pairing for all smooth Artin stacks which we show reduces to the known intersection pairing on the Chow groups of smooth Deligne-Mumford stacks modulo torsion. This involves showing the existence of Adams operations on the rational \'etale K-theory of all smooth Deligne-Mumford stacks. As a by-product of our techniques we also provide an extension of higher intersection theory to all schemes of finite type over a field.

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