A weaker form of the multiplicity conjecture of Herzog, Huneke, and Srinivasan is proven for two classes of monomial ideals: quadratic monomial ideals and squarefree monomial ideals with sufficiently many variables relative to the Krull dimension. It is also shown that tensor products, as well as Stanley-Reisner ideals of certain unions, satisfy the multiplicity conjecture if all the components...

When do syzygies depend on the characteristic of field? Even for well-studied families examples, very little is known. For a family random monomial ideals, namely Stanley--Reisner ideals flag complexes, we prove that Betti numbers asymptotically almost always characteristic. Using this result, also develop heuristic dependence asymptotic algebraic varieties.

Let (an)n≥0 be a sequence of complex numbers such that its generating series satisfies ∑ n≥0 ant n = h(t) (1−t)d for some polynomial h(t). For any r ≥ 1 we study the transformation of the coefficient series of h(t) to that of h〈r〉(t) where ∑ n≥0 anrt n = h 〈r〉(t) (1−t)d . We give a precise description of this transformation and show that under some natural mild hypotheses the roots of h〈r〉(t) c...

Given a permutation w ∈ Sn, we consider a determinantal ideal Iw whose generators are certain minors in the generic n × n matrix (filled with independent variables). Using ‘multidegrees’ as simple algebraic substitutes for torus-equivariant cohomology classes on vector spaces, our main theorems describe, for each ideal Iw: • variously graded multidegrees and Hilbert series in terms of ordinary ...

Introduction: For a standard graded k-algebra A = ⊕ n≥0 An let H(A, t) = ∑ n≥0 dimk Ant n be its Hilbert-series. By classical results H(A, t) = hA(t) (1−t)d is a rational function, where d is the Krull dimension of A and hA(t) = h0 + · · ·+hrt a polynomial with integer coefficients with h0 = 1. Since the work of Stanley (see e.g. [41], [37]) in the 70’s enumerative properties of the coefficient...

Let ZK C be the moment angle complex associated to a simplicial complex K on Œm , together with the natural action of the torus T D U.1/ . Let G T be a (possibly disconnected) closed subgroup and R WD T=G . Let ZŒK be the Stanley– Reisner ring of K and consider ZŒR  WD H .BRIZ/ as a subring of ZŒT  WD H .BTIZ/ . We prove that H G .ZK IZ/ is isomorphic to Tor ZŒR .ZŒK;Z/ as a graded module...

For an arbitrary simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose integral cohomology rings are isomorphic to the Stanley-Reisner algebra of K. Subsequently, Buchstaber and Panov gave an alternative construction, which they showed to be homotopy equivalent to Davis and Januszkiewicz’s examples. It is therefore natural to investigate t...