Vladimir Voevodsky

Bloch-Kato Conjecture and Motivic Cohomology with Finite Coefficients

In this paper we show that the Beilinson-Lichtenbaum Conjecture which describes motivic cohomology of (smooth) varieties with finite coefficients is equivalent to the Bloch-Kato Conjecture, relating Milnor K-theory to Galois cohomology. The latter conjecture is known to be true in weight 2 for all primes [M-S] and in all weights for the prime 2 [V 3].

An exact sequence for K^M_*/2 with applications to quadratic forms

We construct a four-term exact sequence which provides information on the kernel and cokernel of the multiplication by a pure symbol in Milnors K-theory mod 2 of fields of characteristic zero. As an application we establish, for fields of characteristics zero, the validity of three conjectures in the theory of quadratic forms - the Milnor conjecture on the structure of the Witt ring, the Khan-Rost-Sujatha conjecture and the J-filtration conjecture. The first version of this paper was written in the spring of 1996.

Motivic Cohomology Groups Are Isomorphic to Higher Chow Groups in Any Characteristic

In this short paper we show that the motivic cohomology groups defined in [3] are isomorphic to the motivic cohomology groups defined in [1] for smooth schemes over any field. In view of [1, Proposition 12.1] this implies that motivic cohomology groups of [3] are isomorphic to higher Chow groups. This fact was previously known only under the resolution of singularities assumption. The new element in the proof is Proposition 4. The motivic complex Z(q) of weight q was defined in [3] as C∗(Ztr(G m ))[−q]. In [1, Section 8] Friedlander and Suslin defined complexes, which we will denote Z tr (q), as C∗(zequi(A , 0))[−2q] where zequi(X, 0) is the sheaf of equidimensional cycles on X of relative dimension zero. In this paper we prove the following result.

Drawing Curves Over Number Fields

This paper develops some of the ideas outlined by Alexander Grothendieck in his unpublished Esquisse d’un programme [0] in 1984.

A nilpotence theorem for cycles algebraically equivalent to zero

In this paper we prove that a correspondence from a smooth projective variety over a field to itself which is algebraically equivalent to zero is a nilpotent in the ring of correspondences modulo rational equivalence (with rational coefficients). We also show that a little more general result holds, namely that for any algebraic cycle Z on a smooth projective X which is algebraically equivalent to zero there exists N > 0 such that the cycle Z on X is zero in the corresponding Chow group of X with rational coefficients. In the first section we remind the definition and some elementary properties of the additive category of Chow motives over a field. In the second one we prove our nilpotence theorems for cycles algebraically equivalent to zero. Finally in the third section we formulate a very strong Nilpotence Conjecture for algebraic cycles and explain how it is related to some other known conjectures. We did not try to give very accurate proofs in this last section since almost all the statements there are conditional anyway and the only reason to incude this section at all was to illustrate the imporatance of nilpotence results for the theory of algebraic cycles. Everywhere in this paper we consider the Q-linear situation i.e. we completely ignore all torsion and cotorsion effects.

Braided monoidal 2-categories and Manin-Schechtman higher braid groups

Abstract We study a certain coherence problem for braided monoidal 2-categories. For ordinary braided monoidal categories such a problem is well known to lead to braid groups: If we denote by T(n) the pure braid group on n strands then this group acts naturally on each product A1 ⊗ · ⊗ An. It turns out that in the 2-categorical case we have to consider the so-called higher braid group T(2,n) introduced by Manin and Schechtman. The main result is that T(2,n) naturally acts by 2-automorphisms on the canonical 1-morphism A1 ⊗ · ⊗ An → An⊗ · ⊗ A1 for any objects A1,…, An.

An experimental library of formalized Mathematics based on the univalent foundations

This is a short overview of an experimental library of Mathematics formalized in the Coq proof assistant using the univalent interpretation of the underlying type theory of Coq. I started to work on this library in February 2010 in order to gain experience with formalization of Mathematics in a constructive type theory based on the intuition gained from the univalent models (see Kapulkin et al. 2012).

A univalent formalization of the p -adic numbers

The goal of this paper is to report on a formalization of the p -adic numbers in the setting of the second authors univalent foundations program. This formalization, which has been verified in the Coq proof assistant, provides an approach to the p -adic numbers in constructive algebra and analysis.

Categorical Structures for Type Theory in Univalent Foundations.

In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: categories with families, split type-categories, and representable maps of presheaves. We study these in the setting of univalent foundations, where the relationships between them can be stated more transparently. Specifically, we construct maps between the different structures and show that these maps are equivalences under suitable assumptions. We then analyze how these structures transfer along (weak and strong) equivalences of categories, and, in particular, show how they descend from a category (not assumed univalent/saturated) to its Rezk completion. To this end, we introduce relative universes, generalizing the preceding notions, and study the transfer of such relative universes along suitable structure. We work throughout in (intensional) dependent type theory; some results, but not all, assume the univalence axiom. All the material of this paper has been formalized in Coq, over the UniMath library.

Univalent semantics of constructive type theories

In this talk I will outline a new semantics for dependent polymorphic type theories with Martin-Lof identity types. It is based on a class of models which interpret types as simplicial sets or topological spaces defined up to homotopy equivalence. The intuition based on the univalent semantics leads to new answers to some long standing questions of type theory providing in particular well-behaved type theoretic definitions of sets and set quotients. So far the main application of these ideas has been to the development of “native” type-theoretic foundations of mathematics which are implemented in a growing library of mathematics for proof assistant Coq. On the other hand the computational issues raised by the univalent semantics may lead in the future to a new class of programming languages.