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Register a new userLinear flags and Koszul filtrations
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We show that the graded maximal ideal of a graded $K$-algebra $R$ has linear quotients for a suitable choice and order of its generators if the defining ideal of $R$ has a quadratic Grobner basis with respect to the reverse lexicographic order, and show that this linear quotient property for algebras defined by binomial edge ideals characterizes closed graphs. Furthermore, for algebras defined by binomial edge ideals attached to a closed graph and for join-meet rings attached to a finite distributive lattice we present explicit Koszul filtrations.
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This is a survey paper on commutative Koszul algebras and Castelnuovo-Mumford regularity. We describe several techniques to establish the Koszulness of algebras. We discuss variants of the Koszul property such as strongly Koszul, absolutely Koszul and universally Koszul. We present several open problems related with these notions and their local variants.
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.
In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, generalizing the classical theory for $m_R$-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of $R$ is less than the dimension of $R$, which holds for instance if $R$ is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of $m_R$-primary ideals hold for filtrations, including the famous Minkowski inequalities of Teissier, and Rees and Sharp.
We give a two step method to study certain questions regarding associated graded module of a Cohen-Macaulay (CM) module $M$ w.r.t an $mathfrak{m}$-primary ideal $mathfrak{a}$ in a complete Noetherian local ring $(A,mathfrak{m})$. The first step, we call it complete intersection approximation, enables us to reduce to the case when both $A$, $ G_mathfrak{a}(A) = bigoplus_{n geq 0} mathfrak{a}^n/mathfrak{a}^{n+1} $ are complete intersections and $M$ is a maximal CM $A$-module. The second step consists of analyzing the classical filtration ${Hom_A(M,mathfrak{a}^n) }_{mathbb{Z}}$ of the dual $Hom_A(M,A)$. We give many applications of this point of view. For instance let $(A,mathfrak{m})$ be equicharacteristic and CM. Let $a(G_mathfrak{a}(A))$ be the $a$-invariant of $G_mathfrak{a}(A)$. We prove: 1. $a(G_mathfrak{a}(A)) = -dim A$ iff $mathfrak{a}$ is generated by a regular sequence. 2. If $mathfrak{a}$ is integrally closed and $a(G_mathfrak{a}(A)) = -dim A + 1$ then $mathfrak{a}$ has minimal multiplicity. We extend to modules a result of Ooishi relating symmetry of $h$-vectors. As another application we prove a conjecture of Itoh, if $A$ is a CM local ring and $mathfrak{a}$ is a normal ideal with $e_3^mathfrak{a}(A) = 0$ then $G_mathfrak{a}(A)$ is CM.
Given a Serre class $mathcal{S}$ of modules, we compare the containment of the Koszul homology, Ext modules, Tor modules, local homology, and local cohomology in $mathcal{S}$ up to a given bound $s geq 0$. As some applications, we give a full characterization of noetherian local homology modules. Further, we establish a comprehensive vanishing result which readily leads to the formerly known descriptions of the numerical invariants width and depth in terms of Koszul homology, local homology, and local cohomology. Also, we immediately recover a few renowned vanishing criteria scattered about the literature.
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