For any flag simplicial complex $K$, we describe the multigraded Poincare series, minimal number of relations and degrees these in Pontryagin algebra corresponding moment-angle $Z_K$. We compute LS-category $Z_K$ for complexes give a lower bound general case. The key observation is that Milnor-Moore spectral sequence collapses at second sheet $K$. also show results Panov Ray about algebras Davi...

Given a linear space L in affine space A, we study its closure L̃ in the product of projective lines (P). We show that the degree, multigraded Betti numbers, defining equations, and universal Gröbner basis of its defining ideal I(L̃) are all combinatorially determined by the matroid M of L. We also prove I(L̃) and all of its initial ideals are Cohen-Macaulay with the same Betti numbers, and can be...

Let M be a monomial ideal in the polynomial ring S = k[x1, . . . , xn] over a field k. We are interested in the problem, first posed by Kaplansky in the early 1960’s, of finding a minimal free resolution of S/M over S. The difficulty of this problem is reflected in the fact that the homology of arbitrary simplicial complexes can be encoded (via the Stanley-Reisner correspondence [BH,Ho,St]) int...

We show that given a finitely generated standard graded algebra of dimension d over an infinite field, its graded Noether normalizations obey a certain kind of ‘generic exchange’, allowing one to pass between any two of them in at most d steps. We prove analogous generic exchange theorems for minimal reductions of an ideal, minimal complete reductions of a set of ideals, and minimal complete re...

The second Veronese ideal In contains a natural complete intersection Jn generated by the principal 2-minors of a symmetric (n× n)-matrix. We determine subintersections of the primary decomposition of Jn where one intersectand is omitted. If In is omitted, the result is the other end of a complete intersection link as in liaison theory. These subintersections also yield interesting insights int...

The diagonal in a product of projective spaces is cut out by the ideal of 2×2-minors of a matrix of unknowns. The multigraded Hilbert scheme which classifies its degenerations has a unique Borel-fixed ideal. This Hilbert scheme is generally reducible, and its main component is a compactification of PGL(d)/PGL(d). For n = 2 we recover the manifold of complete collineations. For projective lines ...