Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura

We describe the Voevodsky's category $DM^{eff}_{gm}$ of motives in terms of Suslin complexes of smooth projective varieties. This shows that Voeovodsky's $DM_{gm}$ is anti-equivalent to Hanamura's one. We give a description of any triangulated subcategory of $DM^{eff}_{gm}$ (including the category of effective mixed Tate motives). We descibe 'truncation' functors $t_N$ for $N>0$. $t=t_0$ generalizes the weight complex of Soule and Gillet; its target is $K^b(Chow_{eff})$; it calculates $K_0(DM^{eff}_{gm})$, and checks whether a motive is a mixed Tate one. $t_N$ give a weight filtration and a 'motivic descent spectral sequence' for a large class of realizations, including the 'standard' ones and motivic cohomology. This gives a new filtration for the motivic cohomology of a motif. For 'standard realizations' for $l,s\ge 0$ we have a nice description of $W_{l+s}H^i/W_{l-1}H^i(X)$ in terms of $t_s(X)$. We define the 'length of a motif' that (modulo standard conjectures) coincides with the 'total' length of the weight filtration of singular cohomology. Over a finite field $t_0Q$ is (modulo Beilinson-Parshin conjecture) an equivalence.Comment: Several linguistic corrections made; section 2.3 was corrected als

Motivic Eilenberg-Maclane spaces

This paper is the second one in a series of papers about operations in motivic cohomology. Here we show that in the context of smooth schemes over a field of characteristic zero all the bi-stable operations can be obtained in the usual way from the motivic reduced powers and the Bockstein homomorphism.Comment: This version is very close to the final version accepted to the publication in Publ. IHE

Mixed motivic sheaves (and weights for them) exist if 'ordinary' mixed motives do

The goal of this paper is to prove: if certain 'standard' conjectures on motives over algebraically closed fields hold, then over any 'reasonable' $S$ there exists a motivic $t$-structure for the category of Voevodsky's $S$-motives (as constructed by Cisinski and Deglise). If $S$ is 'very reasonable' (for example, of finite type over a field) then the heart of this $t$-structure (the category of mixed motivic sheaves over $S$) is endowed with a weight filtration with semi-simple factors. We also prove a certain 'motivic decomposition theorem' (assuming the conjectures mentioned) and characterize semi-simple motivic sheaves over $S$ in terms of those over its residue fields. Our main tool is the theory of weight structures. We actually prove somewhat more than the existence of a weight filtration for mixed motivic sheaves: we prove that the motivic $t$-structure is transversal to the Chow weight structure for $S$-motives (that was introduced previously and independently by D. Hebert and the author; weight structures and their transversality with t-structures were also defined by the author in recent papers). We also deduce several properties of mixed motivic sheaves from this fact. Our reasoning relies on the degeneration of Chow-weight spectral sequences for 'perverse 'etale homology' (that we prove unconditionally); this statement also yields the existence of the Chow-weight filtration for such (co)homology that is strictly restricted by ('motivic') morphisms.Comment: a few minor corrections mad

Primary chemical composition from simultaneous recording of muons induced cascades and accompanying muon group underground

A new method to estimate the mean atomic number of primary cosmic rays in energy range 10 to the 3rd power to 10 to the 5th power Gev/nucleon is suggested. The Baksan underground scintillation telescope data are used for this analysis. The results of 7500 h run of this experiment are presented

Muon groups and primary composition at 10 to the 13th power to 10 to the 15th power eV

The data on muon groups observed at Baksan underground scintillation telescope is analyzed. In this analysis we compare the experimental data with calulations, based on a superposition model in order to obtain the effective atomic number of primary cosmic rays in the energy range 10 to the 13th power to 10 to the 15th power eV

Galois cohomology of certain field extensions and the divisible case of Milnor-Kato conjecture

We prove the "divisible case" of the Milnor-Bloch-Kato conjecture (which is the first step of Voevodsky's proof of this conjecture for arbitrary prime l) in a rather clear and elementary way. Assuming this conjecture, we construct a 6-term exact sequence of Galois cohomology with cyclotomic coefficients for any finite extension of fields whose Galois group has an exact quadruple of permutational representations over it. Examples include cyclic groups, dihedral groups, the biquadratic group Z/2\times Z/2, and the symmetric group S_4. Several exact sequences conjectured by Bloch-Kato, Merkurjev-Tignol, and Kahn are proven in this way. In addition, we introduce a more sophisticated version of the classical argument known as "Bass-Tate lemma". Some results about annihilator ideals in Milnor rings are deduced as corollaries.Comment: LaTeX 2e, 17 pages. V5: Updated to the published version + small mistake corrected in Section 5. Submitted also to K-theory electronic preprint archives at http://www.math.uiuc.edu/K-theory/0589

Equivariant pretheories and invariants of torsors

In the present paper we introduce and study the notion of an equivariant pretheory: basic examples include equivariant Chow groups, equivariant K-theory and equivariant algebraic cobordism. To extend this set of examples we define an equivariant (co)homology theory with coefficients in a Rost cycle module and provide a version of Merkurjev's (equivariant K-theory) spectral sequence for such a theory. As an application we generalize the theorem of Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a G-equivariant pretheory we associate a graded ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information concerning the motivic J-invariant of E and in the case of Grothendieck's K_0 -- indexes of the respective Tits algebras.Comment: 23 pages; this is an essentially extended version of the previous preprint: the construction of an equivariant cycle (co)homology and the spectral sequence (generalizing the long exact localization sequence) are adde

Univalent Foundations and the UniMath Library

We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3). On the way our odyssey from the foundations to the "horizon" of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander

Tetrahedron and 3D reflection equations from quantized algebra of functions

Soibelman's theory of quantized function algebra A_q(SL_n) provides a representation theoretical scheme to construct a solution of the Zamolodchikov tetrahedron equation. We extend this idea originally due to Kapranov and Voevodsky to A_q(Sp_{2n}) and obtain the intertwiner K corresponding to the quartic Coxeter relation. Together with the previously known 3-dimensional (3D) R matrix, the K yields the first ever solution to the 3D analogue of the reflection equation proposed by Isaev and Kulish. It is shown that matrix elements of R and K are polynomials in q and that there are combinatorial and birational counterparts for R and K. The combinatorial ones arise either at q=0 or by tropicalization of the birational ones. A conjectural description for the type B and F_4 cases is also given.Comment: 26 pages. Minor correction