Kay Rulling: The cohomology of reciprocity sheaves (Part 3)

We start by introducing the theory of reciprocity sheaves and the necessary background of modulus sheaves with transfers, as developed by B. Kahn, H. Miyazaki, S. Saito, and T. Yamazaki.

Then we will explain some basic examples of reciprocity sheaves with a special emphasis on - and an introduction to the de Rham-Witt complex. After an overview of some fundamental results, we will proceed to recent joint work of F. Binda, S. Saito, and myself on the cohomology of reciprocity sheaves. In particular, we prove a projective bundle formula, a blow-up formula as well as a Gysin sequence, generalizing work of Voevodsky on homotopy invariant sheaves with transfers. From this we construct pushforwards under a projective morphism and an action of projective Chow correspondences on the cohomology of reciprocity sheaves. This generalizes several such constructions which originally relied on Grothendieck duality for coherent sheaves and gives a motivic view towards these results.

We use this to prove the birational invariance of the cohomology of certain classes of reciprocity sheaves in a systematic way, many of which were not considered before.