arXiv subjects
Publications and source records attributed to Joseph Gubeladze.
This is an overview of results from our experiment of merging two seemingly unrelated disciplines - higher algebraic K-theory of rings and the theory of lattice polytopes. The usual K-theory is the ``theory of a unit simplex''. A conjecture is proposed on the structure of higher polyhedral K-groups for certain class of polytopes for which the coincidence of Quillen's and Volodin's theories is known.
In each dimension n>2 there are many projective simplicial toric varieties whose Grothendieck groups of vector bundles are at least as big as the ground field. In particular, the conjecture that the Grothendieck groups of locally trivial sheaves and coherent sheaves on such varieties are rationally isomorphic fails badly.
A natural higher K-theoretic analogue of the triviality of vector bundles on affine toric varieties is the conjecture on nilpotence of the multiplicative action of the natural numbers on the K-theory of these varieties. This includes both Quillen's fundamental result on K-homotopy invariance of regular rings and the stable version of the triviality of vector bundles on affine toric varieties. Moreover, it yields a similar behavior of not necessarily affine toric varieties and, further, of their equivariant closed subsets. The conjecture is equivalent to the claim that the relevant admissible morphisms of the category of vector bundles on an affine toric variety can be supported by monomials not in a non-degenerate corner subcone of the underlying polyhedral cone. We prove that one can in fact eliminate all lattice points in such a subcone, except maybe one point. The elimination of the last point is also possible in 0 characteristic if the action of the big Witt vectors satisfies a very natural condition. A partial result on this in the arithmetic case provides first non-simplicial examples -- actually, an explicit infinite series of combinatorially different affine toric varieties, verifying the conjecture for all higher groups simultaneously.
We define higher polyhedral K-groups for commutative rings, starting from the stable groups of elementary automorphisms of polyhedral algebras. Both Volodin's theory and Quillen's + construction are developed. In the special case of algebras associated with unit simplices one recovers the usual algebraic K-groups, while the general case of lattice polytopes reveals many new aspects, governed by polyhedral geometry. This paper is a continuation of [BrG5] (math.KT/0104206) which is devoted to the study of polyhedral aspects of the classical Steinberg relations. The present work explores the polyhedral geometry behind Suslin's well known proof of the coincidence of the classical Volodin's and Quillen's theories. We also determine all K-groups coming from 2-dimensional polytopes.
Let P be a d-dimensional lattice polytope. We show that there exists a natural number c_d, only depending on d, such that the multiples cP have a unimodular cover for every natural number c >= c_d. Actually, a subexponential upper bound for c_d is provided, together with an analogous result for unimodular covers of rational cones.
The main result of the work ``The nilpotence conjecture in K-theory of toric varieties'' is extended to all coefficient fields of characteristic 0, thus covering the class of genuine toric varieties.
Using elementary graded automorphisms of polytopal algebras (essentially the coordinate rings of projective toric varieties) polyhedral versions of the group of elementary matrices and the Steinberg and Milnor groups are defined. They coincide with the usual K-theoretic groups in the special case when the polytope is a unit simplex and can be thought of as compact/polytopal substitutes for the tame automorphism groups of polynomial algebras. Relative to the classical case, many new aspects have to be taken into account. We describe these groups explicitly when the underlying polytope is 2-dimensional. Already this low-dimensional case provides interesting classes of groups.
In this article we overview those aspects of the theory of affine semigroups and their algebras that have been relevant for our own research, and pose several open problems. Answers to these problems would contribute substantially to the further development of the theory. The paper treats two main topics: (1) affine semigroups and several covering properties for them and (2) algebraic properties for the corresponding rings (Koszul, Cohen-Macaulay, different ``sizes'' of the defining binomial ideals). We emphasize the special case when the initial data are encoded into lattice polytopes. The related objects -- polytopal semigroups and algebras -- provide a link with the classical theme of triangulations into unimodular simplices. We have also included an algorithm for checking the semigroup covering property in the most general setting. Our counterexample to certain covering conjectures was found by the application of a small part of this algorithm. The general algorithm could be used for a deeper study of affine semigroups.
We investigate similarities between the category of vector spaces and that of polytopal algebras, containing the former as a full subcategory. In Section 2 we introduce the notion of a polytopal Picard group and show that it is trivial for fields. The coincidence of this group with the ordinary Picard group for general rings remains an open question. In Section 3 we survey some of the previous results on the automorphism groups and retractions. These results support a general conjecture proposed in Section 4 about the nature of arbitrary homomorphisms of polytopal algebras. Thereafter a further confirmation of this conjecture is presented by homomorphisms defined on Veronese singularities. This is a continuation of the project started in our papers "Polytopal linear groups" (J. Algebra 218 (1999), 715--737), "Polytopal linear retractions" preprint, math.AG/0001049) and "Polyhedral algebras, arrangements of toric varieties, and their groups" (preprint, http://www.mathematik.uni-osnabrueck.de/K-theory/0232/index.html). The higher $K$-theoretic aspects of polytopal linear objects will be treated in "Polyhedral $K$-theory" (in preparation).
We investigate graded retracts of polytopal algebras (essentially the homogeneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are again polytopal algebras and that codimension 1 retractions factor through retractions preserving the semigroup structure. We expect that these results hold in general. This paper is a part of the project started in our paers "Polytopal linear groups" (J. Algebra 219 (1999), 715-737) and "Polyhedral algebras, toric arrangements, and their groups" (preprint), where we have investigated the graded automorphism groups of polytopal algebras. Part of the motivation comes from the observation that there is a reasonable `polytopal' generalization of linear algebra (and, subsequently, that of algebraic $K$-theory).