Gorenstein Algebras Research Articles

On the intrinsic complexity of the arithmetic Nullstellensatz

We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorithmic procedure computing the polynomials and constants occurring in a Bézout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the first time height estimates of intrinsic type for the polynomials and constants appearing, again polynomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function π S(x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteristic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.

Gorenstein algebras with nonunimodal Betti numbers

Gorenstein algebras with nonunimodal Betti numbers

A note on Artinian Gorenstein algebras of codimension three

In this paper, using a standard fact in linkage theory, we give a new construction of Artinian Gorenstein algebras achieving all possible sets of graded Betti numbers for codimension three. Furthermore, as an application, we give another proof of Stanley's well-known characterization theorem for the Hilbert functions of codimension three Artinian Gorenstein algebras.

A Note on the Multiplicities of Gorenstein Algebras

In this note we give a formula for the multiplicities of homogenous Gorenstein algebras. Herzog, Huneke, and Srinivasan have conjectured bounds for the multiplicities of homogeneous Cohen–Macaulay algebras. Herzog and Srinivasan have proved this conjecture for C-M algebras with quasi-pure resolutions. Using this formula for the multiplicities of Gorenstein algebras, we prove a stronger bound for the multiplicities of Goerenstein algebras with quasi-pure resolutions.

Gorenstein Graded Algebras and the Evaluation Map

AbstractWe consider graded connected Gorenstein algebras with respect to the evaluation map . We prove that if evG ≠ 0, then the global dimension of G is finite.

Smooth points of [formula omitted] or( T)

In this paper we investigate some relationships between codimension 2 Cohen-Macaulay graded rings and codimension three Artinian Gorenstein graded rings and use these to make some comments about the classifying spaces of such Gorenstein algebras having a fixed Hubert function.

Lower bounds for diophantine approximations

We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new procedure for the decision of consistency of polynomial equation systems whose bit complexity is subexponential, too. As a byproduct, we analyze the division of a polynomial modulo a reduced complete intersection ideal and from this, we obtain an intrinsic lower bound for the logarithmic height of diophantine approximations to a given solution of a zero-dimensional polynomial equation system. This result represents a multivariate version of Liouville's classical theorem on approximation of algebraic numbers by rationals. A special feature of our procedures is their polynomial character with respect to the mentioned geometric invariants when instead of bit operations only arithmetic operations are counted at unit cost. Technically our paper relies on the use of straight-line programs as a data structure for the encoding of polynomials, on a new symbolic application of Newton's algorithm to the Implicit Function Theorem and on a special, basis independent trace formula for affine Gorenstein algebras.

Connected Graded Gorenstein Algebras with Enough Normal Elements

We generalize [12, 1.1 and 1.2] to the following situation.Theorem 1.Let A be a connected graded noetherian algebra of injective dimension d such that every nonsimple graded prime factor ring of A contains a homogeneous normal element of positive degree. Then:(1)A is Auslander–Gorenstein and Cohen–Macaulay.(2)A has a quasi-Frobenius quotient ring.(3)Every minimal prime ideal P is graded andGKdimA/P=d.(4)If, moreover, A has finite global dimension, then A is a domain and a maximal order in its quotient division ring.To prove the above we need the following result, which is a generalization of [3, 2.46(ii)].Theorem 2.Let A be a connected graded noetherian AS-Gorenstein algebra of injective dimension d. Then:(1)The last term of the minimal injective resolution of AAis isomorphic to a shift of A*.(2)For every noetherian graded A-module M,Extd(M,A)is finite dimensional over k.

Gorenstein algebras, symmetric matrices, self-linked ideals, and symbolic powers

Inspired by recent work in the theory of central projections onto hypersurfaces, we characterize self-linked perfect ideals of grade 2 2 as those with a Hilbert–Burch matrix that has a maximal symmetric subblock. We also prove that every Gorenstein perfect algebra of grade 1 1 can be presented, as a module, by a symmetric matrix. Both results are derived from the same elementary lemma about symmetrizing a matrix that has, modulo a nonzerodivisor, a symmetric syzygy matrix. In addition, we establish a correspondence, roughly speaking, between Gorenstein perfect algebras of grade 1 1 that are birational onto their image, on the one hand, and self-linked perfect ideals of grade 2 2 that have one of the self-linking elements contained in the second symbolic power, on the other hand. Finally, we provide another characterization of these ideals in terms of their symbolic Rees algebras, and we prove a criterion for these algebras to be normal.

Koszul Modules and Gorenstein Algebras

I first define Koszul modules, which are a generalization to arbitrary rank of complete intersections. After a study of some of their properties, it is proved that Gorenstein algebras of codimension one or two over a local or graded CM ring are Koszul modules, thus generalizing a well known statement for rank one modules. The general techniques used to describe Koszul modules are then used to obtain a structure theorem for Gorenstein algebras in codimension one and two, over a local or graded CM ring.

Irreducibility and dimension theorems for families of height 3 Gorenstein algebras

Irreducibility and dimension theorems for families of height 3 Gorenstein algebras

A Grade Five Gorenstein Algebra with No Minimal Algebra Resolutions

A Grade Five Gorenstein Algebra with No Minimal Algebra Resolutions

A note on generators of least degree in Gorenstein ideals

Assume R R is a polynomial ring over a field and I I is a homogeneous Gorenstein ideal of codimension g ≥ 3 g\ge 3 and initial degree p ≥ 2 p\ge 2 . We prove that the number of minimal generators ν ( I p ) \nu (I_p) of I I that are of degree p p is bounded above by ν 0 = ( p + g − 1 g − 1 ) − ( p + g − 3 g − 1 ) \nu _0=\binom {p+g-1}{g-1}-\binom {p+g-3}{g-1} , which is the number of minimal generators of the defining ideal of the extremal Gorenstein algebra of codimension g g and initial degree p p . Further, I I is itself extremal if ν ( I p ) = ν 0 \nu (I_p)=\nu _0 .

The evaluation map of some Gorenstein algebras

We define the Ext-Milnor–Moore spectral sequence and prove that, over anyfield K , a differential graded algebra, with a finite-dimensional Lie algebra, is Gorenstein, and it has finite-dimensional cohomology if and only if it has non-zero evaluation map.

A Note on Artinian Gorenstein Algebras Defined by Monomials

A Note on Artinian Gorenstein Algebras Defined by Monomials

Gorenstein algebras and algebras with dominant dimension at least 2

Gorenstein algebras and algebras with dominant dimension at least 2

Gorenstein algebras with unimodalh-sequences

Gorenstein algebras with unimodal<i>h</i>-sequences

A nonunimodal graded gorenstein artin algebra in codimension five

A graded standard Gorenstein Artin algebra quotient of the polynomial ring R over k can be viewed as the algebra Af of partial differential operators of all degrees on a form F. The algebra A is unimodal if the Hilbert function has a single local maximum. We use the theory of compressed algebras to construct forms F in five or more variables whose Gorenstein algebras Af are not unimodal.

On the number of branches for real and complex weighted homogeneous curve singularities

A FORMULA for the number of real branches for a curve singularity defined by a smooth germI: R”+‘, 0 -+ R”, 0 was discovered by Aoki. Fukuda. and Nishimura [Z]. They showed that this number can be computed as the degree of an associated mapping. This degree can, in turn, be computed algebraically by the formula of Eiscnbud -Levine [6]. This has recently been extended to more general real curve singularities by Montaldi and van Stratcn [I I]. In the case of complex curve singularities the original formula of Milnor [ IOJ for plant curve singularities p = 26 r + I has been gcncralized by Buchweitz ~Grcucl[3J to gcncral curve singularities. As is often the case a gcncral formula about invariants of singularities can bc rcplaccd by a much simpler formula in the wcightcd homogcncous cast. In this note wc give such a formula for the number of branches for real or complex curves in terms of the Jacobian algebra of the curve singularity. Recall that for/above, the Jacobian algebra J(j) = I’$ + , /J (1). whcrc E, + , dcnotcs the algchra of smooth germs on R”+‘, 0 and J(j’) dcnotcs the Jacobian idcal gcncratcd by the coordinate functionsj; of/. together with the n x n minors of the n x n + I matrix ((lj;/?.~,). The basic observation that suggests there may be a simpler formula whcnf is wcightcd homogeneous is that in this case (with/defining an isolated singularity, so J(j) is a finite dimensional vector space) J(j) is a O-dimensional Gorenstein algebra. This means there is a linear functional 4: J(j)R such that the bilinear multiplication pairing J(j) x J(/) + R given by (g.h.)H b(g.h) is nonsingular. For complete intersection curves this was shown in [4] to be equivalent to a duality result for surface singularities, which was subsequently proven by Wahl [ 133. Later, Wahl defined a Jacobian algebra J(X) for arbitrary weighted homogeneous complex curve singularities (X. 0) which are Gorenstein and proved quite generally that the J(X) is O-dimensional Gorenstein [I43 (also see [93). This Jacobian algebra has a middle weight s so that 2s is the maximal nonzcro weight in J(X). We Ict J(X),, resp. J(X),,, resp. J(X),, denote the subspaces of J(X) generated by elements of weight = s, resp. s. Since the multiplication pairing is additive on weights, it restricts to a nonsingular pairing on J(X), and dually pairs J(X),, with J(X),,. The pairing in the real case has the signature as an invariant; we shall denote it by sig(J(/)) (or sig(J(X))). By the above remark sig(J(X)) = sig(J(X),); and hence, in particular, also for J(f). The original Eisenbud-Levine result computed the degree of

Gorenstein algebras of codimension four and characteristic two

Gorenstein algebras of codimension four and characteristic two