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Register a new userLinear syzygies, flag complexes, and regularity
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We show that for every positive integer R there exist monomial ideals generated in degree two, with linear syzygies, and regularity of the quotient equal to R. Such examples can not be found among Gorenstein ideals since the regularity of their quotients is at most four. We also show that for most monomial ideals generated in degree two and with linear syzygies the regularity grows at most doubly logarithmically in the number of variables.
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We show that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley--Reisner ring of its nerve. Using this connection between geometric group theory and commutative algebra, as well as techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo--Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily many steps. We give a doubly logarithmic bound on the regularity as a function of the number of variables if these ideals are Cohen--Macaulay.
These are the notes of the lectures of the author at the 2013 CIME/CIRM summer school on Combinatorial Algebraic Geometry. Koszul algebras, introduced by Priddy, are positively graded K-algebras R whose residue field K has a linear free resolution as an R-module. The first part of the notes is devoted to the introduction of Koszul algebras and their characterization in terms of Castelnuovo-Mumford regularity. In the second part we discuss recernt results on the syzygies of Koszul algebras. Finally in the last part we discuss the Koszul property of Veronese algebras and of algebras associated with collections of hyperspaces.
Let $Delta$ be a one-dimensional simplicial complex. Let $I_Delta$ be the Stanley-Reisner ideal of $Delta$. We prove that for all $s ge 1$ and all intermediate ideals $J$ generated by $I_Delta^s$ and some minimal generators of $I_Delta^{(s)}$, we have $${rm reg} J = {rm reg} I_Delta^s = {rm reg} I_Delta^{(s)}.$$
Let $A$ be a Noetherian standard $mathbb{N}$-graded algebra over an Artinian local ring $A_0$. Let $I_1,ldots,I_t$ be homogeneous ideals of $A$ and $M$ a finitely generated $mathbb{N}$-graded $A$-module. We prove that there exist two integers $k$ and $k$ such that [ mathrm{reg}(I_1^{n_1} cdots I_t^{n_t} M) leq (n_1 + cdots + n_t) k + k quadmbox{for all }~n_1,ldots,n_t in mathbb{N}. ]
Let $R=S/I$ be a graded algebra with $t_i$ and $T_i$ being the minimal and maximal shifts in the minimal $S$ resolution of $R$ at degree $i$. In this paper we prove that $t_nleq t_1+T_{n-1}$, for all $n$ and as a consequence, we show that for Gorenstein algebras of codimension $h$, the subadditivity of maximal shifts $T_i$ in the minimal resolution holds for $i geq h-1$, i.e, we show that $T_i leq T_a+T_{i-a}$ for $igeq h-1$.
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