# International Workshop on motives in Tokyo, 2016

### Organizing Committee:

Thomas Geisser (Rikkyo University), Tomohida Terasoma (Tokyo University), Shuji Saito (TIT)### Speakers

Aravind Asok (Los Angeles)

Denis-Charles Cisinski (Toulouse)

Willie Cortinas (Buenos Aires)

Christian Haesemeyer (Melbourne)

Jens Hornbostel (Wuppertal)

Uwe Jannsen (Regensburg)

Peter Jossen (Lausanne)

Wataru Kai (Tokyo)

Shane Kelly (Freiburg)

Tohru Kohrita (Nagoya)

Marc Levine (Duisburg-Essen)

Stephen Lichtenbaum (Brown University)

Gereon Quick (Trondheim)

Oliver Roendigs (Osnabrueck)

Andreas Rosenschon (Munich)

Tamas Szamuely (Budapest)

Goncalo Tabuada (MIT)

Georg Tamme (Regensburg)

Takao Yamazaki (Tohoku)

### This workshop is supported by

Tokyo Institute of Technology, Graduate School of Science, International Education and Research Center of Science,

Graduate School of Mathemtaical Sciences, the University of Tokyo, Leading Graduate Course for Frontiers of Mathematical Sciences and Physics

JSPS Grant-in-aid (A) #15H02048 representative Tomohide Terasoma,

JSPS Grant-in-aid (B) #23340004 representative Thomas Geisser,

JSPS Grant-in-aid (B) 15H03607 representative Kazuhiro Fujiwara,

### Program

### 15(Mon)/Feb:

### 15(Mon)/Feb:

10:00-11:00 Lichtenbaum, "Motives, periods, and special values of zeta functions."

11:15-12:15 Cisinski, "On the motivic nature of sheaves."

14:00-15:00 Quick, "On some invariants for algebraic cobordism cycles."

15:30-16:30 Yamazaki, "Motives with modulus."

16:45-17:45 Kelly, "Motivic homology theories."

### 16(Tue)/Feb:

10:00-11:00 Tamme, "Secondary invariants via differential algebraic K-theory."

11:15-12:15 Asok, "On the A^1-homotopy classification of vector bundles."

14:00-15:00 Cisinski, "Weil-etale cohomology and Weil sheaves."

15:30-16:30 Tabuada, "Noncommutative Artin motives."

16:45-17:45 Levine, "On the stable motivic homotopy groups of spheres."

18:30-20:30: Reception

### 17(Wed)/Feb:

9:30-10:30 Szamuely , "Cohomology and Chow groups over the maximal cyclotomic extension."

10:45-11:45 Cortinas, "Cyclic homology and rigid cohomology in characteristic p>0."

12:00-13:00 Rosenschon, "Rost Nilpotence and etale motivic cohohmology."

Free afternoon

### 18(Thu)/Feb:

10:00-11:00 Kai, "Chow's moving lemma with modulus."

11:15-12:15 Hornbostel, "On non-nilpotent self maps in stable motivic homotopy theory ."

14:00-15:00 Roendigs, "Motivic stable homotopy groups of spheres."

15:30-16:30 Kohrita, "Algebraic part of motivic cohomology with compact supports."

16:45-17:45 Haesemeyer, "Some constructions in the world of monoid schemes and the K-theory of toric schemes in mixed characteristic."

### 19(Fri)/Feb:

10:00-11:00 Cortinas, "Borel regulator and K-theory of group algebras."

11:15-12:15 Jossen, "On the relation between Galois groups and Motivic Galois groups."

13:30-14:30 Jannsen, "Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes."

### Abstract

### Lichtenbaum:

In this talk we we discuss a very general conjectured formula for special values of zeta-functions in terms of Weil-etale motivic cohomology, Betti cohomology, and the Hodge decomposition of de Rham cohomology. We also discuss.the compatibility of this formula with Serre's conjectured functional equation. The most elegant way of stating this formula involves modifying the period map from the Betti cohomology of a motive M to the de Rham cohomology of M by multiplying it by special values of the Gamma-function.

### Cisinski:

Talk I: Cohomology theories always come with an appropriate notion of sheaves on which we can perform standard operations (the six operations of Grothendieck): e.g. (algebraic) de Rham cohomology correpond to (algebraic) D-modules, Betti cohomology correspond to sheaves of abelian groups on the underlying space. Similarly, motivic cohomology correspond to motivic sheaves, i.e. objects of the triangulated categories of mixed motives introduced by Voevodsky, Morel and others. If we restrict to coefficients of geometric origin in an appropriate sense, one can see that sheaves are completely determined by the representation of the corresponding cohomology theory in motives; e.g. one can reconstruct the categories of D-modules from the motive representing de Rham cohomology, or constructible torsion etale sheaves from etale cohomology of smooth schemes with constant finite coefficients.

Talk II: There are two ways to define constructible Weil sheaves of geometric origin, whether we speak the language of triangulated categories or of abelian categories. The fact that these two points of view coincide is non-trivial: although the reconstruction of a triangulated category of Weil sheaves out of l-adic Weil-etale cohomology comes from general principles (as explained in the previous lecture), the reconstruction of the usual abelian category of Weil sheaves of geometric origin is another story; it uses Deligne's theory of weights, as developed by Beilinson, Bernstein and Deligne, together with nice results of Keller and Nicola on the existence of t-structures, which generalize previous results of Rickard and Rouquier. This gives a new description of the abelian category of l-adic Weil sheaves, from which we can formulate a conjecture related to a categorical version of independence of l over a finite field.

### Quick:

There are many interesting invariants of cycles on a smooth complex variety. I will give a survey on how some of these invariants can be generalised to algebraic cobordism cycles, and will discuss some facts we know about them.

### Yamazaki:

We explain our attempt to extend Voevodsky's triangulated category of geometric motives in such a way as to encompass non-homotopy invariant phenomena. This is a joint work with B. Kahn and S. Saito.

### Kelly:

This is joint work with Shuji Saito. Using work of Bondarko, we identify additive functors on Chow motives with a class of homological functors on Voevodsky's motives. Using this identification, we define a "weight" homology functor, which in special cases, recovers Gillet-Soules weight homology, and Geisser's Kato-Suslin homology. We also mention the "field of constants" part of a motive as defined by Ayoub and Barbieri-Viale, and the canonical homology theory which calculates the difference between the motivic homology and etale motivic homology of a motive, and explain how these are all connected over a finite field.

### Tamme:

The Beilinson regulator is a map from the algebraic K-theory of a variety defined over a number field to its Deligne-Beilinson cohomology. The latter one can be computed by a complex of differential forms on the complex valued points of the variety. Differential algebraic K-theory is a refinement of algebraic K-theory, which combines information about K-theory classes with differential forms representing their images under Beilinson's regulator. It can be used as a tool to construct "new" K-theory classes out of "old" ones as secondary invariants. I will explain this in one example. This is joint work with Ulrich Bunke.

### Asok:

I will try to outline the proof that if k is a Dedekind domain, then for any smooth affine k-scheme X, the set of isomorphism classes of rank r vector bundles on X is in bijection with the set of morphisms in the Morel-Voevodsky A^1-homotopy category from X to Gr_r the usual ``infinite Grassmanian". When k is a perfect field, this is a result of Morel, with important simplifications due to Schlichting. In the generality stated, the result is based on joint work with M. Hoyois and M. Wendt.

### Levine:

Understanding the stable homotopy groups of spheres is a central problem in stable homotopy theory, and although there are still quite a lot that is not known, much progress has been made in terms of seeing some basic underlying structures. The situation in the motivic setting is much more complicated, but some of the methods and structures in the classical case carry over. We will discuss connections between motivic and classical techniques and structures, as well as pointing out significant differences, and give a report on recent progress understanding the rational stable homotopy groups, due to Cisinski-Deglise and Ananyevskiy-Levine-Panin.

### Szamuely:

Some 35 years ago, Ken Ribet proved that an abelian variety defined over the maximal cyclotomic extension K of a number field has only finitely many torsion points. In joint work with Damian Roessler, we show that Ribet's theorem is an instance of a general cohomological statement about smooth projective varieties over K. We also present a largely conjectural generalization to torsion cycles of higher codimension, as well as an analogue in positive characteristic.

### Cortinas:

Talk I: I will present a result expressing rigid cohomology of commutative finite type algebras over a field k of characteristic p in terms of periodic cyclic homology. This result suggests a possible new definition of periodic cyclic homology for general, not necessarily commutative k-algebras, which I will discuss.

Talk II: We shall see how an arithmetic invariant, such as the Borel regulator, can be used to relate algebraic and analytic conjectures on the K- theory of group algebras.

### Rosenschon:

We show that an etale motivic analogue of the Rost nilpotence principle holds for all smooth projective schemes over a perfect field. This provides a new approach to the question of Rost nilpotence and allows us to show that Rost nilpotence holds for surfaces, as well as for birationally ruled 3-folds over a field of characteristic 0.

### Kai:

The higher Chow group with modulus, introduced by Binda-Saito in 2014, is a candidate of the cycle-theoretic counterpart of the relative K-theory of pairs of a smooth variety and a (possibly non-reduced) Cartier divisor. In this talk, we establish a new moving lemma `with modulus' to show that the higher Chow group with modulus has the expected contravariant functoriality if we replace its definition with the Nisnevich-local one.

### Hornbostel:

We discuss some implications of the failure of Nishida nilpotence in the motivic setting over the complex numbers. Our results include an integral comparison result of the motivic sphere spectrum and the spectrum representing Witt groups.

### Roendigs:

In joint work with Markus Spitzweck and Paul Arne Ostvar, we study the spectral sequence based on Voevodsky's slice filtration. Work of Levine and Voevodsky shows that the slices of the motivic sphere spectrum are determined by the second page of the topological MU-based Adams-Novikov spectral sequence. We use this information to compute the first stable motivic homotopy groups of spheres over fields of characteristic zero, at least up to a completion with respect to the first algebraic Hopf map.

### Kohrita:

As a version of algebraic part of Chow groups, we define the algebraic part of motivic cohomology with compact supports for arbitrary smooth varieties over an algebraically closed field. We study these algebraic parts in relation to the universal regular homomorphisms with targets in the category of semi-abelian varieties.

### Haesemeyer:

We explain some recent joint work with Cortinas, Walker and Weibel concerning flatness in the world of monoids and how it is applied in the resolution of (the remaining case of) Gubeladze's nilpotence conjecture about the K-theory of monoid algebras.

### Jossen:

The usual absolute Galois group of a field with respect to some fixed algebraic closure, and its Motivic Galois group with respect to some fixed fibre functor, are expected to be closely related. Indeed, the usual Galois group should be the group of connected components of its motivic companion. l will explain this relationship in the framework of Nori's tannakian category of mixed motives.

### Jannsen:

This is joint work with Vincent Cossart and Shuji Saito. Resolution of singularities is known in characteristic zero by work of Hironaka. In positive and mixed characteristic very little is known: Abhyankar proved resolution of surfaces embedded in a threefold over algebraically closed fields of characteristic p > 0, and Lipman proved resolution of excellent schenes of dimension 2 by a process alternating normalization and blowing up closed point. However, this does not give embedded desinguarization, and the calculation of a normalization is difficult in parctice. Our method uses permissible blow-ups and I will describe the explicit strategy for identifying the centers where to blow up.