These are the notes accompanying three lectures given by K. Rülling at the
Motivic Geometry program at CAS, which aim to give an introduction and an
overview of some recent developments in the field of reciprocity sheaves. We
begin 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. We then explain some basic examples of
reciprocity sheaves with a special emphasis on Kähler differentials and the
de Rham–Witt complex. After an overview of some fundamental results,
we survey the recent work of F. Binda, K. Rülling, and S. Saito on the
cohomology of reciprocity sheaves. In particular, we discuss a projective
bundle formula, a blow-up formula, and a Gysin sequence, which generalizes
work of Voevodsky on homotopy invariant sheaves with transfers. From this,
pushforwards along projective morphisms can be constructed, which give rise to an
action of projective Chow correspondences on the cohomology of reciprocity
sheaves. This generalizes several constructions which originally relied on
Grothendieck duality for coherent sheaves and gives a motivic view towards these
results.
We then survey some applications which include the birational invariance of the
cohomology of certain classes of reciprocity sheaves, many of which were not
considered before. Finally, we outline some recent results which were not part of the
lecture series.
Keywords
reciprocity sheaves, cohomology, algebraic geometry,
cycles, modulus, de Rham–Witt
