arXiv subjects
Publications and source records attributed to Winfried Bruns.
We describe some of the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski's theory of complete ideals and of representation theory.
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).
Let $R$ be a noetherian ring and $M$ a finite $R$-module. With a linear form $χ$ on $M$ one associates the Koszul complex $K(χ)$. If $M$ is a free module, then the homology of $K(χ)$ is well-understood, and in particular it is grade sensitive with respect to $\Imχ$. In this note we investigate the case of a module $M$ of projective dimension 1 (more precisely, $M$ has a free resolution of length 1) for which the first non-vanishing Fitting ideal $\I_M$ has the maximally possible grade $r+1$, $r=\rank M$. Then $h=\grade \Imχ\le r+1$ for all linear forms $χ$ on $M$, and it turns out that $H_{r-i}(K(χ))=0$ for all even $i<h$ and $H_{r-i}(K(χ))\iso \SS^{(i-1)/2}(C)$ for all odd $i<h$ where $\SS$ denotes symmetric power and $C=\Ext_R^1(M,R)$, in other words, $C=\Cokψ^*$ for a presentation $$ 0\to F\stackrelψ{\to} G \to M\to 0. $$ Moreover, if $h\le r$, then $H_{r-h}(K(χ))$ is neither 0 nor isomorphic to a symmetric power of $C$, so that it is justified to say that $K(χ)$ is grade sensitive for the modules $M$ under consideration. We furthermore show that the maximally possible value $\grade \Imχ=r+1$ can only occur in two extreme cases: (i) $r=1$ or (ii) $\rank F=1$ and $r$ is odd.
The first sections contain a survey of the application of the Knuth-Robinson-Schensted corerspondence to the computation of Groebner bases of determinantal ideals. We also set up a conceptual framework for this application in terms of so-called "KRS invariants". Then we show that the initial ideal of a determinantal ideal "defined by shape" is given by its KRS image. We furthermore characterize those among these ideals that even have a Groebner basis of products of minors, and show that they can be characterized in terms of Greene's KRS invariants. Furthermore it is shown that for the ideal generated by all t-minors the formation of initial ideal and symbolic power commutes. The last section contains a discussion of potential KRS invariants related to so-called 1-cogenerated ideals.
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).