ﻻ يوجد ملخص باللغة العربية
Let $R=Bbbk[x_1,dots,x_n]$ be a polynomial ring over a field $Bbbk$ and let $Isubset R$ be a monomial ideal preserved by the natural action of the symmetric group $mathfrak S_n$ on $R$. We give a combinatorial method to determine the $mathfrak S_n$-module structure of $mathrm{Tor}_i(I,Bbbk)$. Our formula shows that $mathrm{Tor}_i(I,Bbbk)$ is built from induced representations of tensor products of Specht modules associated to hook partitions, and their multiplicities are determined by topological Betti numbers of certain simplicial complexes. This result can be viewed as an $mathfrak S_n$-equivariant analogue of Hochsters formula for Betti numbers of monomial ideals. We apply our results to determine extremal Betti numbers of $mathfrak S_n$-invariant monomial ideals, and in particular recover formulas for their Castelnuovo--Mumford regularity and projective dimension. We also give a concrete recipe for how the Betti numbers change as we increase the number of variables, and in characteristic zero (or $>n$) we compute the $mathfrak S_n$-invariant part of $mathrm{Tor}_i(I,Bbbk)$ in terms of $mathrm{Tor}$ groups of the unsymmetrization of $I$.
Hochsters theta invariant is defined for a pair of finitely generated modules on a hypersurface ring having only an isolated singularity. Up to a sign, it agrees with the Euler invariant of a pair of matrix factorizations. Working over the complex
Scattered over the past few years have been several occurrences of simplicial complexes whose topological behavior characterize the Cohen-Macaulay property for quotients of polynomial rings by arbitrary (not necessarily squarefree) monomial ideals. T
Let $A = K[X_1,ldots, X_d]$ and let $I$, $J$ be monomial ideals in $A$. Let $I_n(J) = (I^n colon J^infty)$ be the $n^{th}$ symbolic power of $I$ wrt $J$. It is easy to see that the function $f^I_J(n) = e_0(I_n(J)/I^n)$ is of quasi-polynomial type, s
An explicit combinatorial minimal free resolution of an arbitrary monomial ideal $I$ in a polynomial ring in $n$ variables over a field of characteristic $0$ is defined canonically, without any choices, using higher-dimensional generalizations of com
We prove a characterization of the j-multiplicity of a monomial ideal as the normalized volume of a polytopal complex. Our result is an extension of Teissiers volume-theoretic interpretation of the Hilbert-Samuel multiplicity for m-primary monomial i