On Unimodality of Hilbert Functions of Gorenstein Artin Algebras of Embedding Dimension Four

We prove that the Hilbert functions of Gorenstein Artin algebras R/I of embedding dimension four are unimodal provided I has a minimal generator in degree less than five. It is still an open question as to whether all Gorenstein Hilbert functions in codimension four are SI-sequences. It is not even known if they are all unimodal. In this article, we prove that Hilbert functions of all Gorenstein Artin algebras starting with (1, 4, 10, 20, h ₄,…), where h ₄ = 34 are unimodal. Combining this with previously known results, we obtain that all Gorenstein Hilbert functions (1, 4, h ₂, h ₃, h ₄,…4, 1) are unimodal if h ₄ ≤ 34.     

Gorenstein代数上的Hopf代数作用

令H是有限维Hopf代数，A是左H-模代数。本文证明了A是Gorenstein代数的充分必要条件。A^H也是Gorenstein代数的条件。它是Enochs EE,GarciaJJ和del RioA关于群作用相应的理论的推广，同时给出A/A^H是Frobenius扩张的条件。     

Isomorphism Types of Artinian Gorenstein Local Algebras of Multiplicity at Most 9

Let k be an algebraically closed field of characteristic 0. We give here a complete classification of local commutative, associative, Artinian, Gorenstein k-algebras with unit, up to multiplicity 9.     

Gorenstein Dimension and AS-Gorenstein Algebras

The purpose of this paper is to connect the notion of Gorenstein dimension with AS-Gorenstein algebras. In particular, we show that a noetherian connected graded algebra having a balanced dualizing complex is AS-Gorenstein if the balanced dualizing complex has finite Gorenstein dimension. As a preparation, we generalize the Auslander–Bridger formula to the class of noncommutative noetherian connected graded algebras having balanced dualizing complexes.     

An Improved Multiplicity Conjecture for Codimension 3 Gorenstein Algebras

The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree) of a Cohen–Macaulay standard graded algebra with certain extremal graded Betti numbers in its minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a stronger conjecture. We prove this conjecture in the Gorenstein case.     

Frobenius Full Matrix Algebras and Gorenstein Tiled Orders

It is well-known that if R is a left Noetherian ring, then there is a bijective correspondence between minimal prime ideals of R and maximal torsion radicals of R-Mod. Using the notion of a prime M-ideal, it is shown that this correspondence can be extended to the category σ[M] of modules subgenerated by a module M, provided that M is a Noetherian quasi-projective generator in σ[M]. Furthermore, under this hypothesis the prime M-ideals are the fully invariant submodules P of M such that M/P is semi-compressible.     

Hilbert functions of Gorenstein monomial curves

It is a conjecture due to M. E. Rossi that the Hilbert function of a one-dimensional Gorenstein local ring is non-decreasing. In this article, we show that the Hilbert function is non-decreasing for local Gorenstein rings with embedding dimension four associated to monomial curves, under some arithmetic assumptions on the generators of their defining ideals in the non-complete intersection case. In order to obtain this result, we determine the generators of their tangent cones explicitly by using standard basis computations under these arithmetic assumptions and show that the tangent cones are Cohen-Macaulay. In the complete intersection case, by characterizing certain families of complete intersection numerical semigroups, we give an inductive method to obtain large families of complete intersection local rings with arbitrary embedding dimension having non-decreasing Hilbert functions.

     

Level Algebras,Lex Segments,and Minimal Hilbert Functions

Abstract

In this paper we prove the existence of minimal level artinian graded algebras having socle degree r and type t and describe their h-vector in terms of the r-binomial expansion of t. We also investigate the graded Betti numbers of such algebras and completely describe their extremal resolutions. We also show that any set of points in ? ⁿ whose Hilbert function has first difference as described above, must satisfy the Cayley-Bacharach property.     

连通微分分次代数的Gorenstein性质

证明了由两个同调光滑的,Koszul,Gorenstein连通微分分次代数作张量得到的连通微分分次代数仍为同调光滑的,Koszul,Gorenstein连通微分分次代数;假设A是同凋光滑的连通微分分次代数使得H(A)是Koszul连通分次代数,则A是Gorenstein连通微分分次代数当且仅当H(A)是Gorenstein连通分次代数.     

Koszulity of Algebras with Nonpure Resolutions

We discuss certain homological properties of graded algebras whose trivial modules admit nonpure resolutions. Such algebras include both of Artin–Schelter regular algebras of types (12221) and (13431). Under certain conditions, a module with nonpure resolution is decomposed to form an extension by two modules with pure resolutions.     

Maximal families of Gorenstein algebras

The purpose of this paper is to study maximal irreducible families of Gorenstein quotients of a polynomial ring . Let be the scheme parametrizing graded quotients of with Hilbert function . We prove there is a close relationship between the irreducible components of , whose general member is a Gorenstein codimension quotient, and the irreducible components of , whose general member is a codimension Cohen-Macaulay algebra of Hilbert function related to . If the Castelnuovo-Mumford regularity of the Gorenstein quotient is large compared to the Castelnuovo-Mumford regularity of , this relationship actually determines a well-defined injective mapping from such ``Cohen-Macaulay' components of to ``Gorenstein' components of , in which generically smooth components correspond. Moreover the dimension of the ``Gorenstein' components is computed in terms of the dimension of the corresponding ``Cohen-Macaulay' component and a sum of two invariants of . Using linkage by a complete intersection we show how to compute these invariants. Linkage also turns out to be quite effective in verifying the assumptions which appear in a generalization of the main theorem.

     

Norm Estimates for Multiplication Operators in Hilbert Algebras

In this paper, it is proved that for the bilinear operator defined by the operation of multiplication in an arbitrary associative algebra with unit over the fields or , the infimum of its norms with respect to all scalar products in this algebra (with ) is either infinite or at most . Sufficient conditions for this bound to be not less than are obtained. The finiteness of this bound for infinite-dimensional Grassmann algebras was first proved by Kupsh and Smolyanov (this was used for constructing a functional representation for Fock superalgebras).     

The Lattice of Deductive Systems on Hilbert Algebras

The concept of deductive system on a Hilbert algebra was introduced by A. Diego. We show that the set Ded A of all deductive systems on a Hilbert algebra A forms an algebraic lattice which is distributive.AMS Classification (2000): 06F35, 03G25, 08A30     

Gorenstein Injective and Gorenstein Flat Resolution of Modules Over Gorenstein Rings

Abstract

Let R be a commutative Noetherian ring. There are several characterizations of Gorenstein rings in terms of classical homological dimensions of their modules. In this paper, we use Gorenstein dimensions (Gorenstein injective and Gorenstein flat dimension) to describe Gorenstein rings. Moreover a characterization of Gorenstein injective (resp. Gorenstein flat) modules over Gorenstein rings is given in terms of their Gorenstein flat (resp. Gorenstein injective) resolutions.     

Gorenstein algebras of finite Cohen-Macaulay type

An Artin algebra A is said to be CM-finite if there are only finitely many isomorphism classes of indecomposable finitely generated Gorenstein-projective A-modules. Inspired by Auslander's idea on representation dimension, we prove that for 2?n<∞, A is a CM-finite n-Gorenstein algebra if and only if there is a resolving Gorenstein-projective A-module E such that gl.dimEndAop(E)?n.     

A Class of Gorenstein Artin Algebras of Embedding Dimension Four

In this article, we study height four graded Gorenstein ideals I in k[x, y, z, w] such that I ₂ is of height one and generated by three quadrics. After a suitable linear change of variables, I ∩ k[x, y, z] is either Gorenstein or of type two. The former case was studied by Iarrobino and Srinivasan [8 Iarrobino , A. , Srinivasan , H. ( 2005 ). Artininan Gorenstein algebras of embedding dimension four: components of ? Gor (H) for H = (1, 4, 7,…, 1) . Journal of Pure and Applied Algebra 201 : 62 – 96 .[Crossref], [Web of Science ®] , [Google Scholar]] where they give the structure of the ideal and its resolution. We study the latter case and give the structure of these ideals and their minimal resolution. We also explicitly write the form of the generators of I and the maps in the free resolution of R/I.