Related papers: Betti numbers of Z^n-graded modules

Let $(A,\mathfrak{m})$ be a short Artin local ring (i.e., $\mathfrak{m}^3 = 0$ and $\mathfrak{m}^2 \neq 0$). Assume $A$ is not a hypersurface ring. We show there exists $c_A \geq 2$ such that if $M$ is any finitely generated module with…

A $\mathbb{Z}^d$-graded differential $R$-module is a $\mathbb{Z}^d$-graded $R$-module $D$ equipped with an endomorphism, $\delta$, that squares to zero. For $R=k[x_1,\ldots,x_d]$, this paper establishes a lower bound on the rank of such a…

We study the linear syzygies of a homogeneous ideal I in a polynomial ring S, focusing on the graded betti numbers b_(i,i+1) = dim_k Tor_i(S/I, k)_(i+1). For a variety X and divisor D with S = Sym(H^0(D)*), what conditions on D ensure that…

We survey results related to the magnitude of the Betti numbers of numerical semigroup rings and of their tangent cones.

For every $n\ge 6$, we give an example of a finite subset of $\mathbb{P}^{2}$ of degree $n$ which does not descend to any Brauer-Severi surface over the field of moduli. Conversely, for every $n\le 5$ we prove that a finite subset of degree…

In this note we prove the relation between Betti numbers of an Arf semigroup $S$ and its blowup $S'$ in the case when they have the same multiplicity $n$. The relation is then $\beta_{i,s}(S')=\beta_{i,{s+(i+1)n}}(S)$.

Let $S \subset \R^{k + m}$ be a compact semi-algebraic set defined by a system of $\ell$ polynomial inequalities of degree at most 2. $ Let $\pi$ denote the standard projection from $\R^{k + m}$ onto $\R^m$. We prove that for any $q >0$,…

We obtain polylogarithmic bounds in the polynomial Szemer\'{e}di theorem when the polynomials have distinct degrees and zero constant terms. Specifically, let $P_1, \dots, P_m \in \mathbb Z[y]$ be polynomials with distinct degrees, each…

In the first part of the paper we answer (positively) a question raised by the first author which has to do with some sort of rigity of the tail of resolution of an ideal. Let $I$ be a homogeneous ideal in a polynomial ring over a field of…

Let X be a smooth quasiprojective subscheme of P^n of dimension m >= 0 over F_q. Then there exist homogeneous polynomials f over F_q for which the intersection of X and the hypersurface f=0 is smooth. In fact, the set of such f has a…

We introduce a quantitative version of polynomial cohomology for discrete groups and show that it coincides with usual group cohomology when combinatorial filling functions are polynomially bounded. As an application, we show that Betti…

Following the well-established terminology in commutative algebra, any (not necessarily commutative) finite-dimensional local algebra $A$ with radical $J$ will be said to be short provided $J^3 = 0$. As in the commutative case, also in…

We prove a smooth analogue of the classical Thom-Milnor bound, showing that the Betti numbers of the zero set of a smooth map on a compact Riemannian manifold can be controlled by a condition number computed from its first jet. This extends…

Let $S=K[x_1,\dots,x_n]$ be a polynomial ring in $n$ variables with coefficients over a field $K$. A $t$-spread lexsegment ideal $I$ of $S$ is a monomial ideal generated by a $t$-spread lexsegment set. We determine all $t$-spread lexsegment…

This paper aims to provide several relations between Bass and Betti numbers of a given module and its deficiency modules. Such relations and the tools used throughout allow us to generalize some results of Foxby, characterize Cohen-Macaulay…

Inductive formulas for the Betti numbers of the moduli spaces of stable holomorphic vector bundles of coprime rank and degree over a fixed Riemann surface of genus at least two were obtained by Harder, Narasimhan, Desale and Ramanan using…

We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set is bounded from below (up to a multiplicative constant) by the logarithm of m-th Betti number (with respect to singular homology) of…

We derive polynomial identities of arbitrary degree $n$ for syzygies degrees of numerical semigroups S_m=<d_1,...,d_m> and show that for n>=m they contain higher genera G_r=\sum_{s\in Z_>\setminus S_m}s^r of S_m. We find a number…

Consider a non-trivial fiber product $R=S\times_kT$ of local rings $S$, $T$ with common residue field $k$. Given two finitely generate $R$-modules $M$ and $N$, we show that if $\operatorname{Tor}^R_i(M,N)=0=\operatorname{Tor}^R_{i+1}(M,N)$…

A subset $R$ of integers is a set of Bohr recurrence if every rotation on $\mathbb{T}^d$ returns arbitrarily close to zero under some non-zero multiple of $R$. We show that the set $\{k!\, 2^m3^n\colon k,m,n\in \mathbb{N}\}$ is a set of…