Let R = k[x 1 , · · · , x n ] be a polynomial ring and let I ⊂ R be a graded ideal. In [16], Römer asked whether under the Cohen-Macaulay assumption the i-th Betti number β i (R/I) can be bounded above by a function of the maximal shifts in the minimal graded free R-resolution of R/I as well as bounded below by a function of the minimal shifts. The goal of this paper is to establish such bounds...

We introduce the notion of E-depth graded modules over polynomial rings to measure depth certain Ext modules. First, we characterize with (sufficiently) large as those whose partial general initial submodules preserve Hilbert function local cohomology supported at irrelevant maximal ideal, extending a result Herzog and Sbarra on sequentially Cohen-Macaulay Second, describe cone tables sufficien...

We show that any quasi-Gorenstein deformation of a 3-dimensional Buchsbaum local ring with I-invariant 1 admits maximal Cohen-Macaulay module, provided it is quotient Gorenstein ring. Such class rings includes two instances unique factorization domains constructed by Marcel-Schenzel and Imtiaz-Schenzel, respectively. Apart from this result, motivated the small conjecture in prime characteristic...

In this paper we introduce some monomial orders for the class of closed path polyominoes and prove that set generators polyomino ideal attached to a forms reduced Gröbner basis with respect these orders. It is known containing an $L$-configuration or ladder at least three steps, equivalently having no zig-zag walks, prime. As consequence, obtain coordinate ring walks normal Cohen-Macaulay domain.

an isomorphism? These questions have interest partly because if S and .S’(J, S) are Cohen-Macaulay, then so is gr, S := S/J@ J/J’... [ 16 1 and under these hypotheses if N is perfect and R @ .S(J, S) = .$(I, R), then R and .7(1. R) are Cohen-Macaulay too. Thus, gr,R is Cohen-Macaulay, and its torsion freeness and normality, for exmple, can be characterized in terms of analytic spreads, as in [ ...

We prove that a Cohen-Macaulay normal variety X has Du Bois singularities if and only if π∗ωX′(G) ≃ ωX for a log resolution π : X ′ → X , where G is the reduced exceptional divisor of π. Many basic theorems about Du Bois singularities become transparent using this characterization (including the fact that Cohen-Macaulay log canonical singularities are Du Bois). We also give a straightforward an...