# International Workshop on motives in Tokyo, 2017

### Organizing Committee:

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

Federico Binda (Regenburg)

James Borger (Canberra)

Patrick Brosnan (Maryland)

Matthias Flach (CALTECH)

Christian Haesemeyer (Melbourne)

Ryomei Iwasa (Tokyo)

Uwe Jannsen (Regensburg)

Wataru Kai (Duisburg-Essen)

Tohru Kohrita (Nagoya)

Shane Kelly (Berlin)

Amalendu Krishna (TIFR)

Baptiste Morin (Bordeaux)

Matthew Morrow (Paris)

Amnon Neeman (Canberra)

Hiroyasu Miyazaki (Tokyo)

### 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) 15H03607 representative Kazuhiro Fujiwara,

JSPS Grant-in-aid (C) #25400004 representative Kenichiro Kimura,

### Program

### 20(Mon)/Feb:

10:00-11:00 Jannsen, "Phi-gauges, a new cohomology theory in characteristic p."

11:15-12:15 Neemann , "Strong generation in derived categories of schemes, I."

14:00-15:00 Krishna, "Bloch's formula for 0-cycles on some singular varieties."

15:30-16:30 Brosnan, "The Shareshian-Wachs conjecture."

16:45-17:45 Kelly, "A motivic formalism in representation theory."

### 21(Tue)/Feb:

10:00-11:00 Binda, "Zero cycles on singular varieties and zero cycles with modulus."

11:15-12:15 Miyazaki, "Cube invariance of higher Chow group with modulus."

14:00-15:00 Neemann, "Strong generation in derived categories of schemes, II."

15:30-16:30 Borger, "A structural approach to Witt vectors, I."

16:45-17:45 Morin, "Weil-etale cohomology and Zeta-values of arithmetic schemes."

18:30-20:30: Reception

### 22(Wed)/Feb:

9:30-10:30 Neemann, "Strong generation in derived categories of schemes, III."

10:45-11:45 Borger, "A structural approach to Witt vectors, II."

12:00-13:00 Morrow, "Integral p-adic Hodge theory, I."

13:30-15:00 "Free discussions."

15:30-17:00 "Free discussions."

### 23(Thu)/Feb:

10:00-11:00 Iwasa, "Homology pro stability and relative algebraic K-theory."

11:15-12:15 Kai, "Chern classes with modulus."

14:00-15:00 Morrow, "Integral p-adic Hodge theory, II."

15:30-16:30 Kohrita, "Deligne-Beilinson cycle maps for Lichtenbaum cohomology."

16:45-17:45 Borger, "A structural approach to Witt vectors, III."

### 24(Fri)/Feb:

10:00-11:00 Morrow, "Integral p-adic Hodge theory, III."

11:15-12:15 Haesemeyer, "Non-commutative coefficients in Gubeladze's nilpotence conjecture."

13:30-14:30 Flach, "Lambda-operations via non-commutative motives."

### Abstract

### Jannsen:

This is joint work with Jean-Marc Fontaine. The new cohomology is constructed to measure the p-divisibility of Frobenius. I will explain the relationship with other known theories, e.g., the theory of F-zips or the theory of displays.

### Neemann:

The two fundamental papers that got the subject going is a 2003 article by Bondal and Orlov, and then a much longer 2008 article by Rouquier (which appeared in one of the first issues of the now-defunct Journal of K-Theory). We will begin by recalling what it means for a triangulated category to be strongly generated, the classical examples and what the main applications are - this concept is especially useful in noncommutative algebraic geometry.

With the background out of the way we will state the main new theorems: the category of perfect complexes is strongly generated if and only if the scheme is regular and finite dimensional, while the derived category of coherent sheaves is strongly generated under a fairly mild excellence condition.

Time permitting we will also say something about the proofs. A key to everything is an approximation theorem, telling us that we can approximate bounded-above complexes of quasicoherent sheaves arbitrarily well, in an effectively bounded number of steps, using a compact generator. This key lemma turns out to have applications having nothing to do with strong generation: for example it allows us to prove, easily and directly, that the derived pushforward of a map of noetherian schemes respects perfect complexes if and only if the map is proper and of finite Tor-dimension.

### Krishna:

Bloch's formula describes the Chow group of 0-cycles on smooth surfaces in terms of the cohomology of certain K-theory sheaves. This formula was generalized to all smooth varieties by Quillen. However, the existence of such a formula for the cohomological Chow group of 0-cycles on singular varieties is known in only very few cases. In this talk, we shall present such a formula for the the 0-cycles on certain class of singular varieties.

### Brosnan:

The Shareshian-Wachs conjecture relates a combinatorial object, the Shareshian-Wachs chromatic quasi-symmetric function, to an action of the symmetric group on the cohomology of certain varieties known as Hessenberg varieties. The main motivation for the conjecture is to use it to study another older conjecture of Stanley and Stembridge on the positivity of a related symmetric function, Stanley's chromatic symmetric function. About a year ago, Tim Chow and I proved the Shareshian-Wachs conjecture by relating it to the monodromy a family of sheaves on the space of regular semi-simple matrices. A couple of months later, M. Guay-Paquet gave a completely different proof using a new Hopf algebra. In my talk, I will explain the first proof. If time permits, I will also say a little bit about the second one as well as other recent developments.

### Kelly:

This is joint work with Jens Niklas Eberhardt. A fundamental problem in representation theory is to determine the characters of all simple rational SLn(Fp)-modules. Unlike in the characteristic zero case, this is still wide open and the subject of ongoing research. Fifteen years ago, Soergel proposed a strategy using geometric methods: He translates the problem at least for some of the simple modules into a question about certain sheaves on a flag variety. In this talk we present a way to enhance his statements by replacing sheaves with motives and applying work of Ayoub, Cisinski-Deglise, and Geisser-Levine.

### Binda:

Given a smooth variety X and an effective divisor D on it, we show that the Levine-Weibel cohomological Chow group of zero cycles of the double of X along D admits a canonical decomposition in terms of the Kerz-Saito Chow group of zero cycles CH_0(X|D) and the Chow group CH_0(X). This decomposition allow us to deduce a number of new results for the Chow groups with modulus from known results on the Chow groups of points of singular varieties. This includes a Rojtman-type torsion theorem when X is projective, and a relative Bloch formula when X is affine and of dimension two. This is a joint work with Amalendu Krishna.

### Miyazaki:

The higher Chow group with modulus, introduced by Binda-Saito, is a common generalization of Bloch's higher Chow group, the additive higher Chow group and the Chow group with modulus of zero cycles. In this talk, we establish a modulus-theoretic generalization of homotopy invariance of Bloch's higher Chow group, which we call cube invariance, and discuss its applications.

### Borger:

This lecture series will give an introduction to the basic theory of Witt vectors. The most common approaches rely on explicit formulas, but there is another, less well-known approach (first discovered by Joyal) which allows for a natural and completely transparent development of the theory. I'll explain this, and perhaps some further developments.

### Morin:

We give a description of the vanishing order and the special value of the Zeta function of a proper regular arithmetic arithmetic scheme at any integer argument in terms of Weil-etale cohomology with compact support and derived de Rham cohomology. We also explain a conjectural cohomological formalism for arithmetic schemes suggesting such a description, and state some conditional results supporting these conjectures. This is joint work with Matthias Flach.

### Morrow:

I will give an introduction to the joint work ``Integral p-adic Hodge theoryff with Bhargav Bhatt and Peter Scholze. The first talk will present an overview of the main results, including the existence of a cohomology theory integrally interpolating crystalline, p-adic etale, and de Rham cohomology, and the remaining talks will enter into details of some proofs. If there is some time remaining I will mention new results, such as Poincare duality, the existence of Chern classes, and a calculation of p-adic vanishing cycles.

### Iwasa:

Let $A$ be a pro unital commutative ring of finite stable range (e.g. an ideal of a noetherian commutative ring of finite Krull dimension). We prove that the pro systems of the integral group homologies $\{H_l(GL_n(A^m)\}_m$ stabilize as pro systems for $n$ larger enough than $l$. This phenomena has a strong relation with the efK-theory pro excisionff. By using the pro stability, we give a specific description of the pro system of the relative K-groups $\{K_n(R,A^m)\}_m$ which does not depend on $R$, a unital ring which contains $A$ as a two-sided ideal.

### Kai:

This is joint work with Ryomei Iwasa. We explain how to construct Chern class maps from the relative K-groups K_n(X,D) of an algebraic scheme X and an effective Cartier divisor D --- where we assume the complement X-D is smooth --- to their relative motivic cohomology groups (a la Binda-Saito). Via homotopy theory of relative Volodin space presheaf, the problem boils down to an explicit construction of a universal cycle class with modulus.

### Kohrita:

We define Deligne-Beilinson cycle maps for Lichtenbaum cohomology of arbitrary complex varieties using an eh-topological method. We show that these cycle maps are always surjective on torsion, and in some degrees, injective on torsion as well.

### Haesemeyer:

this is a report on joint work with Cortinas, Walker and Weibel. If M is the monoid of lattice points in a rational polyhedral cone, and R is a regular commutative ring then Gubeladze conjectured that the dilation operators act nil potently on the reduced K-theory of the monoid algebra R[M]. He proved his conjecture when R contains the rational numbers. In this talk we discuss how our approach to proving the general case generalises (somewhat surprisingly) to K-regular associative rings flat over either the integers or over a product of fields. Possible coefficient rings include commutative and stable C^* - algebras.

### Flach:

We discuss the construction of Lambda-operations on the algebraic K-theory space of a symmetric monoidal, stable Q-linear infinity-category using representability of the algebraic K-theory functor in a suitable category of noncommutative motives.