The Auslander–Reiten translate on monomial rings

For t in Nn, E. Miller has defined a category of positively t-determined modules over the polynomial ring S in n variables. We consider the Auslander–Reiten translate, Nt, on the (derived) category of such modules. A monomial ideal I is t-determined if each generator xa has a?t. We compute the multigraded cohomology and Betti spaces of for every iterate k and also the S-module structure of these cohomology modules. This comprehensively generalizes results of Hochster and Gräbe on local cohomology of Stanley–Reisner rings.     

The Newton Procedure for several variables

Let us consider an equation of the form

     

On the Arithmetical Rank of the Edge Ideals of Forests

We show that for the edge ideals of a certain class of forests, the arithmetical rank equals the projective dimension.     

Serre's Condition Rℓ for Affine Semigroup Rings

In this note we characterize the affine semigroup rings K[S] over an arbitrary field K that satisfy condition R_? of Serre. Our characterization is in terms of the face lattice of the positive cone pos(S) of S. We start by reviewing some basic facts about the faces of pos(S) and consequences for the monomial primes of K[S]. After proving our characterization we turn our attention to the Rees algebras of a special class of monomial ideals in a polynomial ring over a field. In this special case, some of the characterizing criteria are always satisfied. We give examples of non-normal affine semigroup rings that satisfy R₂.     

Upper and Lower Bounds for the Stanley Depth of Certain Classes of Monomial Ideals and Their Residue Class Rings

We give an upper bound for the Stanley depth of the edge ideal of a complete k-partite hypergraph and as an application we give an upper bound for the Stanley depth of a monomial ideal in a polynomial ring S. We also give a lower and an upper bound for the cyclic module S/I associated to the complete k-partite hypergraph.     

Resolutions of 2 and 3 Dimensional Rings of Invariants for Cyclic Groups

Let G be the cyclic group of order n, and suppose F is a field containing a primitive nth root of unity. We consider the ring of invariants F[W] ^G of a three dimensional representation W of G where G ? SL(W). We describe minimal generators and relations for this ring and prove that the lead terms of the relations are quadratic. These minimal generators for the relations form a Gröbner basis with a surprisingly simple combinatorial structure. We describe the graded Betti numbers for a minimal free resolution of F[W] ^G . The case where W is any two dimensional representation of G is also handled.     

Asymptotic Growth of Associated Primes of Certain Graph Ideals

We specify a class of graphs, H _t , and characterize the irreducible decompositions of all powers of the cover ideals. This gives insight into the structure and stabilization of the corresponding associated primes; specifically, providing an answer to the question “For each integer t ≥ 0, does there exist a (hyper) graph H _t such that stabilization of associated primes occurs at n ≥ (χ(H _t ) ?1) + t?” [4 Francisco , C. A. , Hà , H. T. , Van Tuyl , A. ( 2011 ). Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals . J. Algebra 331 : 224 – 242 .[Crossref], [Web of Science ®] , [Google Scholar]]. For each t, H _t has chromatic number 3 and associated primes that stabilize at n = 2 + t.