Rational singularities, de Rham-Witt sheaves and reciprocity functors
Final Report Abstract
I planed to consider projects centered around the following topics during the Heisenberg fellowship: (a) Rational singularities on excellent schemes (b) Vanishing results _a la Grauert-Riemenschneider in positive characteristics and de Rham-Witt sheaves (c) Reciprocity sheaves, motives with modulus, and rami_cation phenomena The problem suggested in (a) was essentially solved by Andre Chatzistamatiou and myself even before the Heisenberg fellowship has begun, and was later completely settled by S_andor Kov_acs. I worked extensively on the problems around (b) and discussed this with Srinivas but so far without much progress. However there is the hope that further progress in (c) might shed some light on this problem. There was great progress in (c). In joint work with Takao Yamazaki we computed Suslin homology with modulus of a relative curve in [RY16]; a result which turns out to be very useful in the theory. In joint work with Shuji Saito [RS18] we relate the Nisnevich cohomology of relative Milnor K-theory, a version of which was introduced by Kato-Saito in the 1980's, with motivic cohomology with modulus as introduced by Binda-Saito relying on ideas of Bloch-Esnault. Results and techniques of this paper were used by several authors in subsequent articles. In joint work with Shuji Saito [RS] we introduce a new notion of conductor and relate it to reciprocity sheaves as de_ned by Kahn-Saito-Yamazaki. Furthermore, we prove that this notion of conductor uni_es many classical (abelian) conductors which come up in very di_erent contexts, such as the Artin conductor of lisse _Q`-sheaves of rank 1 and the irregularity of integrable rank 1 connections (in characteristic 0). This work is also used in a joint work with Sugiyama and Yamazaki [RSY] in which, building on work of Kahn-Miyazaki-Saito-Yamazaki, constructions and computations of Kato-Somekawa, Kahn-Yamazaki, and Ivorra-Rulling are generalized and clari_ed and in which we in particular obtain new descriptions of Kahler di_erentials and related objects. Finally in joint work with Binda and Saito [BRS] we use all the above and more to prove various general properties of the Nisnevich cohomology of reciprocity sheaves, such as projective bundle - and blow-up formulas, Gysin sequences and actions of proper Chow correspondences. These results yield, e.g., new relative birational invariants and new obstructions for the existence of zero-cycles of degree 1 in the generic _ber of a projective and dominant morphism between smooth varieties. This work generalizes and uni_es many results which were proven either in the context of A1-invariant sheaves or in the context of coherent sheaves by completely di_erent methods and various authors. Most of the points which were part of the program around (c) were settled and those which are still open are part of various works in progress.
Publications
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Suslin homology of relative curves with modulus. Journal of the London Mathematical Society, Vol. 93. 2016, Issue 3, pp. 567-589.
Kay Rülling, Takao Yamazaki
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Higher Chow groups with modulus and relative Milnor K-theory.
Transactions of the American Mathematical Society, Vol. 370. 2018, pp. 987-1043.
Kay Rülling, Shuji Saito
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Reciprocity sheaves and abelian ramification theory, Preprint 2019, 83 pages.
Kay Rülling, Shuji Saito
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Tensor structures in the theory of modulus presheaves with transfers, Preprint 2019, 60 pages.
Kay Rülling, Rin Sugiyama, Takao Yamazaki
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Introductory course on l-adic sheaves and their ramification theory on curves. Clay Mathematics Proceedings 21 (2020), 103-229.
Lars Kindler, Kay Rülling
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On the cohomology of reciprocity sheaves, Preprint 2020, 116 pages.
Federico Binda, Kay Rülling, Shuji Saito
