115 research outputs found
Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura
We describe the Voevodsky's category of motives in terms of
Suslin complexes of smooth projective varieties. This shows that Voeovodsky's
is anti-equivalent to Hanamura's one. We give a description of any
triangulated subcategory of (including the category of
effective mixed Tate motives). We descibe 'truncation' functors for
. generalizes the weight complex of Soule and Gillet; its target
is ; it calculates , and checks whether a
motive is a mixed Tate one. 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 we
have a nice description of in terms of .
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 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'
there exists a motivic -structure for the category of Voevodsky's
-motives (as constructed by Cisinski and Deglise). If is 'very
reasonable' (for example, of finite type over a field) then the heart of this
-structure (the category of mixed motivic sheaves over ) 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 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 -structure is transversal to the Chow weight
structure for -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
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