Ehrhart clutters: regularity and max-flow min-cut.
Existence of Gorenstein projective resolutions and Tate cohomology
Existence of proper Gorenstein projective resolutions and Tate cohomology is proved over rings with a dualizing complex. The proofs are based on Bousfield Localization which is originally a method from algebraic topology.
Extensions of toric varieties.
Families of reduced zero-dimensional schemes.
A great deal of recent activity has centered on the question of whether, for a given Hilbert function, there can fail to be a unique minimum set of graded Betti numbers, and this is closely related to the question of whether the associated Hilbert scheme is irreducible or not. We give a broad class of Hilbert functions for which we show that there is no minimum, and hence that the associated Hilbert sheme is reducible. Furthermore, we show that the Weak Lefschetz Property holds for the general element...
Finiteness aspects of Gorenstein homological dimensions
We present an alternative way of measuring the Gorenstein projective (resp., injective) dimension of modules via a new type of complete projective (resp., injective) resolutions. As an application, we easily recover well known theorems such as the Auslander-Bridger formula. Our approach allows us to relate the Gorenstein global dimension of a ring R to the cohomological invariants silp(R) and spli(R) introduced by Gedrich and Gruenberg by proving that leftG-gldim(R) = maxleftsilp(R), leftspli(R),...
Free minimal resolutions and the Betti numbers of the suspension of an -gon.
Free Resolutions of The Defining Ideal of Certain Rational Surfaces.
G-dimension over local homomorphisms with respect to a semi-dualizing complex
We study the G-dimension over local ring homomorphisms with respect to a semi-dualizing complex. Some results that track the behavior of Gorenstein properties over local ring homomorphisms under composition and decomposition are given. As an application, we characterize a dualizing complex for in terms of the finiteness of the G-dimension over local ring homomorphisms with respect to a semi-dualizing complex.
Generalized tilting modules over ring extension
Let be a ring extension of . We show the left -module with the endmorphism ring End is a generalized tilting module when is a generalized tilting module under some conditions.
Graded Betti numbers of some embedded rational n-folds.
Homological algebra and divergent series.
Initially Koszul algebras.
Intertwining numbers; the -rowed shapes
A fairly old problem in modular representation theory is to determine the vanishing behavior of the groups and higher groups of Weyl modules and to compute the dimension of the -vector space for any partitions , of , which is the intertwining number. K. Akin, D. A. Buchsbaum, and D. Flores solved this problem in the cases of partitions of length two and three. In this paper, we describe the vanishing behavior of the groups and provide a new formula for the intertwining number for any...
Koszul homology and Lie algebras with application to generic forms and points.
La conjecture de Green générique
Une courbe projective et lisse de genre , non hyperelliptique, admet un plongement canonique dans un espace projectif . Un résultat classique affirme que l’idéal gradué des équations de dans est engendré par ses éléments de degré , sauf si admet certains systèmes linéaires très particuliers. Mark Green en a proposé il y a vingt ans une vaste généralisation, qui décrit la résolution minimale de en fonction de l’existence de systèmes linéaires spéciaux sur . Claire Voisin vient de...
Linear forms on modules of projective dimension one.
Minimal resolutions of lattice ideals and integer linear programming.
A combinatorial description of the minimal free resolution of a lattice ideal allows us to the connection of Integer Linear Programming and Al1gebra. The non null reduced homology spaces of some simplicial complexes are the key. The extremal rays of the associated cone reduce the number of variables.
Monomial ideals with 3-linear resolutions
In this paper, we study the Castelnuovo-Mumford regularity of square-free monomial ideals generated in degree . We define some operations on the clutters associated to such ideals and prove that the regularity is preserved under these operations. We apply these operations to introduce some classes of ideals with linear resolutions and also show that any clutter corresponding to a triangulation of the sphere does not have linear resolution while any proper subclutter of it has a linear resolution....
Multigraded modules.
Multigraded regularity: syzygies and fat points.