1,720,974 research outputs found
Invariant rings of the special orthogonal group have nonunimodal h -vectors
No full textFor K an infinite field of characteristic other than two, consider the action of the special orthogonal group SOt(K) on a polynomial ring via copies of the regular representation. When K has characteristic zero, Boutot's theorem implies that the invariant ring has rational singularities; when K has positive characteristic, the invariant ring is F-regular, as proven by Hashimoto using good filtrations. We give a new proof of this, viewing the invariant ring for SOt(K) as a cyclic cover of the invariant ring for the corresponding orthogonal group; this point of view has a number of useful consequences, for example, it readily yields the a-invariant and information on the Hilbert series. Indeed, we use this to show that the h-vector of the invariant ring for SOt(K) need not be unimodal
Castelnuovo-Mumford Regularity and Powers
No full textThis note has two goals. The first is to give a short and self contained introduction to the Castelnuovo-Mumford regularity for standard graded rings R= iEN Ri over general base rings R0. The second is to present a simple and concise proof of a classical result due to Cutkosky, Herzog and Trung and, independently, to Kodiyalam asserting that the regularity of powers Iv of an homogeneous ideal I of R is eventually a linear function in v. Finally we show how the flexibility of the definition of the Castelnuovo-Mumford regularity over general base rings can be used to give a simple proof of a result proved by the authors in “Maximal minors and linear powers”
Regularity of prime ideals
No full textWe answer several natural questions which arise from a recent paper of McCullough and Peeva providing counterexamples to the Eisenbud–Goto Regularity Conjecture. We give counterexamples using Rees algebras, and also construct counterexamples that do not rely on the Mayr–Meyer construction. Furthermore, examples of prime ideals for which the difference between the maximal degree of a minimal generator and the maximal degree of a minimal first syzygy can be made arbitrarily large are given. Using a result of Ananyan-Hochster we show that there exists an upper bound on regularity of prime ideals in terms of the multiplicity alone
Hankel determinantal rings have rational singularities
No full textHankel determinantal rings, i.e., determinantal rings defined by minors of
Hankel matrices of indeterminates, arise as homogeneous coordinate rings of higher order
secant varieties of rational normal curves; they may also be viewed as linear specializations
of generic determinantal rings. We prove that, over fields of characteristic zero, Hankel
determinantal rings have rational singularities; in the case of positive prime characteristic,
we prove that they are F-pure. Independent of the characteristic, we give a complete
description of the divisor class groups of these rings, and show that each divisor class
group element is the class of a maximal Cohen-Macaulay module
Determinantal Ideals and the Straightening Law
No full textIn this chapter gives a short introduction to standard bitableaux and the straightening law. This powerful technique is the key to structural properties of determinantal rings. But it is also of central importance for the computation of Gröbner and Sagbi bases on the one hand and for the representation theoretic approach on the other
Asymptotic Regularity for Symbolic Powers of Determinantal Ideals
No full textThis chapter applies the methods of Chap. 9 to the study the ideals “defined by shape”. They allow natural filtrations that lead to cohomology computations in a characteristic-free fashion. The filtrations take a particularly nice form for symbolic powers of determinantal ideals, where the vanishing theorems from Chap. 9, combined with the characterization of Castelnuovo-Mumford regularity in Chap. 8, allow us to determine an explicit formula for the asymptotic regularity. We end Chapter 10 with a brief survey of several other homological and arithmetic properties of determinantal ideals that can be derived in a compact way via geometric arguments
F-singularities of Determinantal Rings
No full textThis chapter treats ring theoretic properties derived from the Frobenius functor in positive characteristics, related to tight closure theory. We develop them far enough to prove that determinantal rings are strongly F-regular. For deformation arguments one needs F-rationality. It is closely related to the rationality of singularities in characteristic 0, so that we can at least briefly discuss this property for determinantal rings. F-rationality is a very handy tool for the exploitation of toric and equivariant deformations
Universal Gröbner Bases
No full textChapter 5 covers the existence of universal Gröbner bases of determinantal ideals as far as they are known, namely maximal minors and 2-minors. The approach to the case of maximal minors is particularly simple
Algebras Defined by Minors
No full textIn Chap. 4 we have studied the Gröbner deformations of determinantal ideals defined by their initial ideals. We now turn to the study of algebras generated by minors through their initial algebras. Since the initial algebras are normal monoid domains, toric algebra can be applied to them. Since normal monoid domains are very well understood, we can draw strong consequences for the algebras defined by minors
Castelnuovo–Mumford Regularity
No full textFor standard graded algebras over fields, Castelnuovo-Mumford regularity has become an indispensable invariant. Chapter 8 develops this notion from scratch, but in a more general version for standard graded algebras over Noetherian base rings. As in the classical case, regularity can be computed from local cohomology, minimal free resolutions and Koszul homology. In the given generality we prove the theorems on the regularity of powers and products of ideals. In the context of determinantal rings we are mainly interested in linear free resolutions of powers of ideals of maximal minors in the non-generic case, exemplified by ideals of rational normal scrolls
- …
