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REPORT
ALEXANDER I. EFIMOV

1. New results In 2013, the following results were obtained. 1.1. MacLane (co)homology of the second kind and Wieferich primes. We show that for any number field K and an element w Ok , there is a direct connection between Maclane (co)homology of the second kind of OK with curvature w and the "critical" points of w in Spec OK . Our basic reference for MacLane (co)homology is [L], chapter 13. MacLane cohomology of an associative ring R with coefficients in R - R -bimodule M is defined as Hoschschild cohomology of the cubical Maclane construction of R :
· H M L· (R, M ) := H HZ (Q(R), M ).

Here, for any abelian group A, Q(A) is a functorial nonnegative chain complex of abelian groups which computes stable homology of Eilenberg-MacLane spaces: Hn (Q(A)) H =
n+k

(K (A, k )),
2

k > n. (Q(A)) is torsion for any abelian

In particular, H0 (Q(A)) = A, H1 (Q(A)) = 0, and H

group A. Also, we have that H>0 (A) = 0 if A is a Q -vector space. The functor Q(-) is (nonunital) monoidal (but not symmetric monoidal!), so that we have natural maps of complexes Q(A) Q(B ) Q(A B ), with natural associativity isomorphisms, satisfying cocycle condition. This makes Q(R) into a DG ring whenever R is a ring. Similarly, Maclane homology is defined as Hochschild homology: H M L· (R, M ) := H H· (Q(R), M ). For simplicity, we will restrict ourselves to Maclane cohomology. We write H M L(R) instead of H M L(R, R). The classical computations shows that for a finite field Fq we have H M L2n (Fq ) = Fq , n 0,
1

H M Lodd (Fq ) = 0.


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ALEXANDER I. EFIMOV

For the ring of integers, one has

H M Ln (Z) =

Z Z/lZ 0

for n = 0; for n = 2l > 0; otherwise.

We would like to illustrate our results for the ring Z. Using the notion of Hochschild (co)homology of the second kind, introduced by A. Polishchuk and L. Positselski [PP], one can define MacLane (co)homology of the second kind. Namely, let w R be a central element. Then define Rw to be a Z/2 -graded curved DG ring R with curvature w. We define the Maclane cohomology of the second kind by the formula H M L·
,I I

(Rw ) := H H

·,I I

(Q(R)

[w]

, R),

where [w] Q(R)0 is the natural cycle associated to w (by definition, the Z -basis of Q(R)0 is formed by non-zero elements of R ). Our main result for the ring of integers states that for any w Z we have H M L·,I I (Zw ) = Z[{p where S Spec
m -1

}

pS

]
pS,n>0

Z/pp

(n)

Z,

Z is the set of those primes p for which wp w mod p2 . If moreover

p does not divide w, then such p is called a base w Wieferich prime. It is natural to consider S as the set of critical points of w on Spec Z. Op en problem. For any w Z the set of base w Wieferich primes is infinite. Our result (and its obvious grneralization for localizations of Z ) shows that this open problem is equivalent to the following statement. Conjecture 1.1. For any positive integer n, and any w Z, we have that H M L· Z 1 n! Q. =
w

Our result may be considered as an arithmetic analogue of the following geometric statement. Let A be a smooth finitely generated commutative algebra over a field k of characteristic 0 or greater than dim A, and put X := Spec A. Let w A be an element. Then we have HH
·,I I

(Aw ) H · (· TX , [w, -]), =

where [-, -] is the Schouten-Nijenhuis bracket. So this cohomology is a coherent sheaf on X, and its support is precisely the critical locus of w.


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1.2. DG categorical dinamics. Another result is about DG categorical dynamics. Let T be a triangulated DG category generated by one ob ject G, and F : T T an endoufunctor. We consider (T , F ) as a DG-categorical discrete dynamical system. In [DHKK], the authors defined an entropy for the endofunctor F, which is a function h(F ) : R R, t ht (F ). In the case when T is smooth and proper over C, one of the equivalent definitions is the following: ht (F ) = lim
n

1 log( n

k

Z

dim Extk (G, F n (G))e

-k t

).

One can show that this limit exists and does not depend on the choice of generator G. In all examples the exponent of the entropy e
±t h(F )

is integral element (in the sense of

commutative algebra) over Z[e ]. So the natural question arises: is the (exponential of ) entropy always algebraic? We consider the generating function QF
;E1 ,E2

(q , x) :=

(

n 0 k

Z

dim Extk (E1 , F n (E2 ))q k )xn Z[q

±1

][[x]].

Clearly, in the case when E1 = E2 is a generator, after substitution q = e-t , we get that the radius of convergence of QF
;E1 ,E2

(e-t , x) is just e

-ht (F )

. is not rational, and not even

Unfortunately, in general the generating function QF

;E1 ,E2

differentially finite on x. However, we show that its "complexity" reduces to the very special case of a category generated by exceptional collection of length 2. Theorem 1.2. Let T be smooth and proper, F : T T a DG functor, E1 , E2 T two objects. Then there exists another smooth and proper DG category T , a DG functor F :T T and two objects E1 , E2 such that the fol lowing holds: (i) the category T is generated by exceptional col lection A1 , A2 ; (ii) We have F (Ai ) Wi Ai for some Wi Perf (k ); = (iii) the difference QF is a rational function. 2. Comparison with the application The following conjecture of Kontsevich was proved, as well as its generalization for coherent matrix factorizations. Theorem 2.1. Let Y be a separated scheme of finite type over a field k of characteristic
b zero. Then the DG category Dcoh (Y ) is hfp. ;E1 ,E
2

(q , (1 + q )x) - QF

;E1 ,E2

(q , x)


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ALEXANDER I. EFIMOV

It was mentioned as a plan for future work. On the other hand, I did not prove HMS conjecture for punctured spheres of genus g 1, and the conjecture about existence of full strong exceptional collections on pro jective toric DM stacks, which were also mentioned in the plan of future work. In fact I was mostly working on other topics. 3. Papers and preprints. Papers in 2013: Abouzaid, Mohammed; Auroux, Denis; Efimov, Alexander I.; Katzarkov, Ludmil; Orlov, Dmitri, Homological mirror symmetry for punctured spheres. J. Amer. Math. Soc. 26 (2013), no. 4, 10511083. Preprints in 2013: A. Efimov, Homotopy finiteness of some DG categories from algebraic geometry, arXiv:1308.0135. 4. Conferences and Schools 1) "Conference on Homological Mirror Symmetry", January 2013, Miami, USA. Talk: "Homotopy finiteness of DG categories from algebraic geometry" 2) "Third Latin Congress on Symmetries in Geometry and Physics", February, Sao Luis, Brazil. Talks: "Homotopy finiteness of DG categories from algebraic geometry"; "Derived categories of Grassmannians over integers and modular representation theory" 3) "DT-invariants in Paris", June 2013, Paris, France. Talk: "Topological Hochschild homology of the second kind and Wieferich primes" 4) "Quantum and motivic cohomology, Fano varieties and mirror symmetry", September 2013, Saint-Petersburg, Russia. Talk: "Topological Hochschild (co)homology of the second kind and Wieferich primes" 5) "Categories and Complexity", November 2013, Vienna, Austria. Talk: "Categories and Complexity" References
[DHKK] G. Dimitrov, F. Haiden, L. Katzarkov, M. Kontsevich, Dynamical systems and categories, arXiv:1307.8418 (preprint). [L] J.-L. Loday, Cyclic homology. Appendix E by Marґ O. Ronco. Second edition. Chapter 13 by the ia author in collaboration with Teimuraz Pirashvili. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 301. Springer-Verlag, Berlin, 1998. xx+513 pp.


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[PP]

A. Polishchuk, L. Positselski, Hochschild (co)homology of the second kind I. Trans. Amer. Math. Soc. 364 (2012), no. 10, 53115368.

Steklov Mathematical Institute of RAS, Gubkin str. 8, GSP-1, Moscow 119991, Russia E-mail address : efimov@mccme.ru