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Publications and source records attributed to Andrew R. Kustin.
Let $\pmb k$ be an arbitrary field and $A$ be a standard graded Artinian Gorenstein $\pmb k$-algebra of embedding dimension four and socle degree three. Then, except for exactly one exception, $A$ has the weak Lefschetz property. Furthermore, the exception occurs only in characteristic two.
Let k be an arbitrary field, A be a standard graded Artinian Gorenstein k-algebra of embedding dimension four and socle degree three, and pi from P to A be a surjective graded homomorphism from a polynomial ring with four variables over k onto A. We give the minimal generators of the kernel of pi and the minimal homogeneous resolution of A by free P-modules. We give formulas for the entries in the matrices in the resolution in terms of the coefficients of the Macaulay inverse system for A. We have implemented these formulas in Macaulay2 scripts. The kernel of pi has either 6, 7, or 9 minimal generators. The number of minimal generators and the precise form of the minimal resolution are determined by the rank of a 3 by 3 symmetric matrix of constants that we call SM. If the kernel of pi requires more than six generators, then we prove that the kernel of pi is the sum of two linked perfect ideals of grade three. If the the kernel of pi is six-generated, then we prove that A is a hypersurface section of a codimension three Gorenstein algebra. Our approach is based on the structure of Gorenstein-linear resolutions and the theorem that, except for exactly one exception, A has the weak Lefschetz property.
Let $J$ be a quadratically presented grade three Gorenstein ideal in the standard graded polynomial ring $R= k[x,y,z]$, where $k$ is a field. Assume that $R/J$ satisfies the weak Lefschetz property. We give the presentation matrix for $J$ in terms of the coefficients of a Macaulay inverse system for $J$. (This presentation matrix is an alternating matrix and $J$ is generated by the maximal order Pfaffians of the presentation matrix.) Our formulas are computer friendly; they involve only matrix multiplication; they do not involve multilinear algebra or complicated summations. As an application, we give the presentation matrix for $J_1=(x^{n+1},y^{n+1},z^{n+1}):(x+y+z)^{n+1}$, when $n$ is even and the characteristic of $k$ is zero. Generators for $J_1$ had been identified previously; but the presentation matrix for $J_1$ had not previously been known. The first step in our proof is to give improved formulas for the presentation matrix of a linearly presented grade three Gorenstein ideal $I$ in terms of the coefficients of the Macaulay inverse system for $I$.
In 2018 Celikbas, Laxmi, Kraśkiewicz, and Weyman exhibited an interesting family of perfect ideals of codimension three, with five generators, of Cohen-Macaulay type two with trivial multiplication on the Tor algebra. All previously known perfect ideals of codimension three, with five generators, of Cohen-Macaulay type two had been found by Brown in 1987. Brown's ideals all have non-trivial multiplication on the Tor algebra. We prove that all of the ideals of Brown are obtained from the ideals of Celikbas, Laxmi, Kraśkiewicz, and Weyman by (non-homogeneous) specialization. We also prove that both families of ideals, when built using power series variables over a field, define rigid algebras in the sense of Lichtenbaum and Schlessinger.
Let R be a non-negatively graded Cohen-Macaulay ring with R_0 a Cohen-Macaulay factor ring of a local Gorenstein ring. Let d be the dimension of R, m be the maximal homogeneous ideal of R, and M be a finitely generated graded R-module. It has long been known how to read information about the socle degrees of the local cohomology module H_m^0(M) from the twists in position d in a resolution of M by free R-modules. It has also long been known how to use local cohomology to read valuable information from complexes which approximate resolutions in the sense that they have positive homology of small Krull dimension. The present paper reads information about the maximal generator degree (rather than the socle degree) of H_m^0M from the twists in position d-1 (rather than position d) in an approximate resolution of M. We apply the local cohomology results to draw conclusions about the maximum generator degree of the second symbolic power of the prime ideal defining a monomial curve and the second symbolic power of the ideal defining a finite set of points in projective space. There is an application to general hyperplane sections of subschemes of projective space over an infinite field. There is an application of the local cohomology techniques to partial Castelnuovo-Mumford regularity. An application to the ideals generated by the lower order Pfaffians of an alternating matrix will appear in a future paper. One additional application to the study of blow-up algebras appears in a separate paper.
Let P be a commutative Noetherian ring, K be an ideal of P which is generated by a regular sequence of length four, f be a regular element of P, and Pbar be the hypersurface ring P/(f). Assume that K:f is a grade four Gorenstein ideal of P. We give a resolution N of Pbar/K Pbar by free Pbar-modules. The resolution N is built from a Differential Graded Algebra resolution of P/(K:f) by free P-modules, together with one homotopy map. In particular, we give an explicit form for the matrix factorization which is the infinite tail of the resolution N.
Let $P$ be a commutative Noetherian ring and $F$ be a self-dual acyclic complex of finitely generated free $P$-modules. Assume that $F$ has length four and $F_0$ has rank one. We prove that $F$ can be given the structure of a Differential Graded Algebra with Divided Powers; furthermore, the multiplication on $F$ exhibits Poincaré duality. This result is already known if $P$ is a local Gorenstein ring and $F$ is a minimal resolution. The purpose of the present paper is to remove the unnecessary hypotheses that $P$ is local, $P$ is Gorenstein, and $F$ is minimal.
We prove that every quasi-complete intersection ideal is obtained from a pair of nested complete intersection ideals by way of a flat base change. As a by-product we establish a rigidity statement for the minimal two-step Tate complex associated to an ideal $I$ in a local ring $R$. Furthermore, we define a minimal two-step complete Tate complex $T$ for each ideal $I$ in a local ring $R$; and prove a rigidity result for it. The complex $T$ is exact if and only if $I$ is a quasi-complete intersection ideal; and in this case, $T$ is the minimal complete resolution of $R/I$ by free $R$-modules.
We define a notion of compressed local Artinian ring that does not require the ring to contain a field. Let $(R,\mathfrak m)$ be a compressed local Artinian ring with odd top socle degree $s$, at least five, and $\operatorname{socle}(R)\cap \mathfrak m^{s-1}=\mathfrak m^s$. We prove that the Poincaré series of all finitely generated modules over $R$ are rational, sharing a common denominator, and that there is a Golod homomorphism from a complete intersection onto $R$.
Let $S$ be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if $R$ is a non-Gorenstein quotient of $S$ of small colength, then every totally reflexive $R$-module is free. Indeed, the second syzygy of the canonical module of $R$ has a direct summand $T$ which is a test module for freeness over $R$ in the sense that if $\mathrm{Tor}_+^R(T,N)=0$, for some finitely generated $R$-module $N$, then $N$ is free.
We find the defining equations of Rees rings of linearly presented height three Gorenstein ideals. To prove our main theorem we use local cohomology techniques to bound the maximum generator degree of the torsion submodule of symmetric powers in order to conclude that the defining equations of the Rees algebra and the special fiber ring have the same image in the symmetric algebra. We show that this image is the unmixed part of the ideal generated by the maximal minors of a matrix of linear forms which is annihilated by a vector of indeterminates, and otherwise has maximal possible grade. An important step of the proof is the calculation of the degree of the variety parametrized by the forms generating the grade three Gorenstein ideal.
Let $\mathbf X$ be a generic alternating matrix, $\mathbf t$ be a generic row vector, and $J$ be the ideal $\operatorname{Pf}_4({\mathbf X})+I_1({\mathbf {t X}})$. We prove that $J$ is a perfect Gorenstein ideal of grade equal to the grade of $\operatorname{Pf}_4({\mathbf X})$ plus two. This result is used by Ramos and Simis in their calculation of the Aluffi algebra of the module of derivations of the homogeneous coordinate ring of a smooth projective hypersurface. We also prove that $J$ defines a domain, or a normal ring, or a unique factorization domain if and only if the base ring has the same property. The main object of study in the present paper is the module $\mathcal N$ which is equal to the column space of $\mathbf X$, calculated mod $\operatorname{Pf}_4({\mathbf X})$. The module $\mathcal N$ is a self-dual maximal Cohen-Macaulay module of rank two; furthermore, $J$ is a Bourbaki ideal for $\mathcal N$. The ideals which define the homogeneous coordinate rings of the Plücker embeddings of the Schubert subvarieties of the Grassmannian of planes are used in the study of the module $\mathcal N$.
Let Phi be an f by g matrix with entries from a commutative Noetherian ring R, with g at most f. Recall the family of generalized Eagon-Northcott complexes {C^{i}} associated to Phi. (See, for example, Appendix A2 in "Commutative Algebra with a view toward Algebraic Geometry" by David Eisenbud.) For each integer i, C^i is a complex of free R-modules. For example, C^{0} is the original "Eagon-Northcott" complex with zero-th homology equal to the ring defined by the maximal order minors of Phi; and C^{1} is the "Buchsbaum-Rim" complex with zero-th homology equal to the cokernel of the transpose of Phi. If Phi is sufficiently general, then each C^{i}, with i at least -1, is acyclic; and, if Phi is generic, then these complexes resolve half of the divisor class group of R/I_g(Phi). The family {C^{i}} exhibits duality; and, if -1\le i\le f-g+1, then the complex C^{i} exhibits depth-sensitivity with respect to the ideal I_g(Phi) in the sense that the tail of C^{i} of length equal to grade(I_g(Phi)) is acyclic. The entries in the differentials of C^i are linear in the entries of Phi at every position except at one, where the entries of the differential are g by g minors of Phi. This paper expands the family {C^i} to a family of complexes {C^{i,a}} for integers i and a with 1\le a\le g. The entries in the differentials of C^{i,a} are linear in the entries of Phi at every position except at two consecutive positions. At one of the exceptional positions the entries are a by a minors of Phi, at the other exceptional position the entries are g-a+1 by g-a+1 minors of Phi. The complexes {C^i} are equal to {C^{i,1}} and {C^{i,g}}. The complexes {C^{i,a}} exhibit all of the properties of {C^{i}}. In particular, if -1\le i\le f-g and 1\le a\le g, then C^{i,a} exhibits depth-sensitivity with respect to the ideal I_g(Phi).
Let R be a standard graded polynomial ring in f variables over a field and Psi be an f by g matrix of linear forms from R, where g is positive and less than f. Assume that the row vector of variables annihilates Psi and that the ideal I generated by the g by g minors of Psi has grade exactly one short of the maximum possible grade. We resolve R/I, prove that I has a g-linear resolution, record explicit formulas for the h-vector and multiplicity of R/I, and prove that if f-g is even, then the ideal I is unmixed. Furthermore, if f-g is odd, then we identify an explicit generating set for the unmixed part, I^{unm}, of I, resolve R/I^{unm}, and record explicit formulas for the h-vector of R/I^{unm}. These results have applications to the study of the blow-up algebras associated to linearly presented grade three Gorenstein ideals.
Our object of study is a rational map Psi from projective s-1 space to projective n-1 space defined by homogeneous forms g1,...,gn, of the same degree d, in the homogeneous coordinate ring R=k[x1,...,xs] of projective s-1 space. Our goal is to relate properties of Psi, of the homogeneous coordinate ring A=k[g1,...,gn] of the variety parametrized by Psi, and of the Rees algebra R[It], the bihomogeneous coordinate ring of the graph of Psi. For a regular map Psi, for instance, we prove that R[It] satisfies Serre's condition R_i, for some positive i, if and only if A satisfies R_{i-1} and Psi is birational onto its image. Thus, in particular, Psi is birational onto its image if and only if R[It] satisfies R_1. Either condition has implications for the shape of the core, namely, the core of I is the multiplier ideal of I to the power s and the core of I equals the maximal homogeneous ideal of R to the power sd-s+1. Conversely, for s equal to two, either equality for the core implies birationality. In addition, by means of the generalized rows of the syzygy matrix of g1,...,gn, we give an explicit method to reduce the non-birational case to the birational one when s is equal to 2.
We give an explicit formula for the Hilbert Series of an algebra defined by a linearly presented, standard graded, residual intersection of a grade three Gorenstein ideal.
A quasi-complete intersection (q.c.i.) ideal of a local ring is an ideal with "free exterior Koszul homology"; the definition can also be understood in terms of vanishing of André-Quillen homology functors. Principal q.c.i. ideals are well understood, but few constructions are known to produce q.c.i. ideals of grade zero that are not principal. This paper examines the structure of q.c.i. ideals. We exhibit conditions on a ring $R$ which guarantee that every q.c.i. ideal of $R$ is principal. On the other hand, we give an example of a minimal q.c.i. deal $I$ which does not contain any principal q.c.i. ideals and is not embedded, in the sense that no faithfully flat extension of $I$ can be written as a quotient of complete intersection ideals. We also describe a generic situation in which the maximal ideal of $R$ is an embedded q.c.i. ideal that does not contain any principal q.c.i. ideals.
This is the second paper in a series of three papers. In the first paper of the series, "Artinian Gorenstein algebras with linear resolutions", (arXiv:1306.2523, J. of Algebra, to appear) we prove that it is possible to give the minimal resolution of the rings from the title in terms of the coefficients of the corresponding Macaulay inverse system. In this context, the word "give" means, "give in a polynomial manner". The first paper in the series proves, essentially, an existence theorem. The second and third papers in the series construct the explicit formulas for the resolution. The present paper is concerned with Artinian Gorenstein algebras of embedding codimension three. In the third paper, "The structure of Gorenstein-linear resolutions of Artinian algebras", the embedding codimension is arbitrary.