We consider the exact computation of matrix eigenproblems in residue class rings for solving systems of algebraic equations. We construct multiplication tables using a Gröbner basis of a zero-dimensional ideal. Then, we analyze the tables by exactly computing their Frobenius normal forms. The derogatoriness and the diagonalizability are determined by the normal forms, and the problem is divided...

A partition of a finite abelian group gives rise to a dual partition on the character group via the Fourier transform. Properties of the dual partitions are investigated and a convenient test is given for the case that the bidual partition coincides the primal partition. Such partitions permit MacWilliams identities for the partition enumerators of additive codes. It is shown that dualization c...

We develop a theory of G-dimension over local homomorphisms which encompasses the classical theory of G-dimension for finitely generated modules over local rings. As an application, we prove that a local ring R of characteristic p is Gorenstein if and only if it possesses a nonzero finitely generated module of finite projective dimension that has finite G-dimension when considered as an R-modul...

We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a “nice” Garside element, certain monoids S with quadratic relations, whose monoidal algebra A = kS has a Frobenius Koszul dual A with regular socle, the monoids of skew-polynomial type (or equivalently, binomial skew-polynomial rings) which were i...

The values of the normalized homogeneous weight are determined for arbitrary finite Frobenius rings and expressed in a form that is independent from a generating character and the Möbius function on the ring. The weight naturally induces a partition of the ring, which is invariant under left or right multiplication by units. It is shown that the character-theoretic left-sided dual of this parti...

We associate to every equicharacteristic zero Noetherian local ring R a faithfully flat ring extension which is an ultraproduct of rings of various prime characteristics, in a weakly functorial way. Since such ultraproducts carry naturally a non-standard Frobenius, we can define a new tight closure operation on R by mimicking the positive characteristic functional definition of tight closure. T...

We introduce the notion of relative singularity category with respect to any self-orthogonal subcategory ω of an abelian category. We introduce the Frobenius category of ω-Cohen-Macaulay objects, and under some reasonable conditions, we show that the stable category of ω-Cohen-Macaulay objects is triangle-equivalent to the relative singularity category. As applications, we relate the stable cat...

Much of the wonderful invariant theory of the 19 century worked over fields of characteristic zero, while the theory for prime characteristic lagged behind. However, the Frobenius (p power) map in characteristic p > 0 leads to a rich theory of invariants in prime characteristic. This theory is closely bound up with the Steenrod Algebra, which allows us to derive new invariants from known invari...

In this paper we prove that if G is a connected, simplyconnected, semi-simple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra (Ug)0 of the Lie algebra g of G can be endowed with a Koszul grading (extending results of Andersen, Jantzen, Soergel). We also give information about the Koszul dual rings....