Non-commutative desingularization of determinantal varieties I

We show that determinantal varieties defined by maximal minors of a generic matrix have a non-commutative desingularization, in that we construct a maximal Cohen-Macaulay module over such a variety whose endomorphism ring is Cohen-Macaulay and has finite global dimension. In the case of the determinant of a square matrix, this gives a non-commutative crepant resolution.     

The G-biliaison class of symmetric determinantal schemes

We consider a family of schemes, that are defined by minors of a homogeneous symmetric matrix with polynomial entries. We assume that they have maximal possible codimension, given the size of the matrix and of the minors that define them. We show that these schemes are G-bilinked to a linear variety of the same dimension. In particular, they can be obtained from a linear variety by a finite sequence of ascending G-biliaisons on some determinantal schemes. We describe the biliaisons explicitly in the proof of Theorem 2.3. In particular, it follows that these schemes are glicci.     

Spanning-tree extensions of the Hadamard-Fischer inequalities

All possible graph-theoretic generalizations of a certain sort for the Hadamard-Fischer determinantal inequalities are determined. These involve ratios of products of principal minors which dominate the determinant. Furthermore, the cases of equality in these inequalities are characterized, and equality is possible for every set of values which can occur for the relevant minors. This relates recent work of the authors on positive definite completions and determinantal identities. When applied to the same collections of principal minors, earlier generalizations give poorer, more difficult to compute bounds than the present inequalities. Thus, this work extends, and in a certain sense completes, a series of generalizations of Hadamard-Fischer begun in the 1960s.     

Criteria for Componentwise Linearity

We establish characteristic-free criteria for the componentwise linearity of graded ideals. As applications, we classify the componentwise linear ideals among the Gorenstein ideals, the standard determinantal ideals, and the ideals generated by the submaximal minors of a symmetric matrix.     

On the Symmetric and Rees algebras of an ideal

Some necessary and sufficient conditions are given for the Rees and Symmetric Algebra of an ideal being canonically isomorphic. Some applications are obtained to the study of the relations between the generators of ideals which are maximal minors of a generic t by (t+1) matrix, prime ideals of finite projective dimension, almost complete intersections or d-sequences.This paper was supported by C. N. R. (Consiglio Nazionale delle Ricerche)     

Arithmetic-Geometric Mean determinantal identity

In this paper, we give a generalization of a determinantal identity posed by Charles R. Johnson about minors of a Toeplitz matrix satisfying a specific matrix identity. These minors are those appear in the Dodgson’s condensation formula.     

Ideals Generated by Diagonal 2-Minors

With a simple graph G on [n], we associate a binomial ideal P_G generated by diagonal minors of an n × n matrix X = (x_ij) of variables. We show that for any graph G, P_G is a prime complete intersection ideal and determine the divisor class group of K[X]/P_G. By using these ideals, one may find a normal domain with free divisor class group of any given rank.     

Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus-invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non-empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size. As a corollary, we obtain explicit generating sets of quantum minors for the torus-invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over ${\mathbb{Q}}$ .     

The Gale transform and multi-graded determinantal schemes

Eisenbud and Popescu showed that certain finite determinantal subschemes of projective spaces defined by maximal minors of adjoint matrices of homogeneous linear forms are related by Veronese embeddings and a Gale transform. We extend this result to adjoint matrices of multihomogeneous multilinear forms. The subschemes now lie in products of projective spaces and the Veronese embeddings are replaced with Segre embeddings.     

On the characterization of almost strictly totally positive matrices

A nonsingular matrix is called almost strictly totally positive when all its minors are nonnegative and, furthermore, these minors are strictly positive if and only if their diagonal entries are strictly positive. Almost strictly totally positive matrices are useful in Approximation Theory and Computer Aided Geometric Design to generate bases of functions with good shape preserving properties. In this paper we give an algorithmic characterization of these matrices. Moreover, we provide a determinantal characterization of them in terms of the positivity of a very reduced number of their minors and also in terms of their factorizations. Both authors were partially supported by the DGICYT Spain Research Grant PB93-0310     

On the characterization of almost strictly totally positive matrices

A nonsingular matrix is called almost strictly totally positive when all its minors are nonnegative and, furthermore, these minors are strictly positive if and only if their diagonal entries are strictly positive. Almost strictly totally positive matrices are useful in Approximation Theory and Computer Aided Geometric Design to generate bases of functions with good shape preserving properties. In this paper we give an algorithmic characterization of these matrices. Moreover, we provide a determinantal characterization of them in terms of the positivity of a very reduced number of their minors and also in terms of their factorizations.     

Universal and comprehensive Gröbner bases of the classical determinantal ideal

Let A =(x _ij ), i =1,2,... ,k, j =1,2,... ,l, 1 ≤ k ≤ l, be a matrix of independent variables, G be the set of maximal minors of A, and I = (G) be the classical determinantal ideal. We show that G is a universal Gr?bner basis of I. Also, a sufficient condition for G to be a universal comprehensive Gr?bner basis is proved. Bibliography: 12 titles.     

The initial algebra of maximal minors of a generalized hankel matrix

One deals with a certain generalization of the (generic) Hankel matrix. Fixing a field K, for a suitable diagonal term order on the polynomial ring B over K generated by the entries of this matrix, one considers the K-subalgebra A ? B generated by the set of initial terms of the the maximal minors of the matrix. The algebra A detects the underlying combinatorics in terms of certain generalized standard tableaux and these give a way of showing that A is normal. A more involved technique also yields that the Rees algebra of the ideal of B generated by the generators of A is normal. In particular, both algebras are Cohen-Macaulay. We also present a Gröbner basis for the presentation ideal of A and compute some numerical invariants using the simplicial complex associated to A.     

A Gorenstein simplicial complex for symmetric minors

We show that the ideal generated by the (n - 2) minors of a general symmetric n by n matrix has an initial ideal that is the Stanley–Reisner ideal of the boundary complex of a simplicial polytope and has the same graded Betti numbers.     

Remarks on pseudo-valuation rings

A prime ideal P of a ring A is said to be a strongly prime ideal if aP and bA are comparable for all a,b ε A. We shall say that a ring A is a pseudo-valuation ring (PVR) if each prime ideal of A is a strongly prime ideal. We show that if A is a PVR with maximal ideal M, then every overring of A is a PVR if and only if M is a maximal ideal of every overring of M that does not contain the reciprocal’of any element of M.We show that if R is an atomic domain and a PVD, then dim(R) ≤ 1. We show that if R is a PVD and a prime ideal of R is finitely generated, then every overring of R is a PVD. We give a characterization of an atomic PVD in terms of the concept of half-factorial domain.     

Gorenstein Ladder Determinantal Rings

Ladder determinantal rings are rings associated with idealsof minors of certain subsets of a generic matrix of indeterminates.By results of Abhyankar, Narasimhan, Herzog and Trung, and Conca,they are known to be Cohen-Macaulay normal domains. In thispaper we characterize the Gorenstein property of ladder determinantalrings in terms of the shape of the ladder.     

Decompositions of Ideals of Minors Meeting a Submatrix

We compute the primary decomposition of certain ideals generated by subsets of minors in a generic matrix or in a generic symmetric matrix, or subsets of Pfaffians in a generic skew-symmetric matrix. Specifically, the ideals we consider are generated by minors that have at least some given number of rows and columns in certain submatrices.     

On divided commutative rings

Let R be a commutative ring with identity having total quotient ring T. A prime ideal P of R is called divided if P is comparable to every principal ideal of R. If every prime ideal of R is divided, then R is called a divided ring. If P is a nonprincipal divided prime, then P^-1 = { x ? T : xP ? P} is a ring. We show that if R is an atomic domain and divided, then the Krull dimension of R ≤ 1. Also, we show that if a finitely generated prime ideal containing a nonzerodivisor of a ring R is divided, then it is maximal and R is quasilocal.