1,720,965 research outputs found
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
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
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
Gröbner Bases, Initial Ideals and Initial Algebras
No full textThe first chapter gives a compact, but quite complete introduction to Gröbner bases and Sagbi bases in general. The focus is on the structural aspects, namely, the use of Gröbner and Sagbi degenerations in the transfer of homological and enumerative information from Stanley-Reisner and/or toric rings to those objects that degenerate to them
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
Cohomology and Regularity in Characteristic Zero
No full textThe goal of this chapter is to extend and prove sharper versions of the results of the preceding chapter, when working over a field of characteristic zero. Two crucial advantages are the linear reductivity of the general linear group and the Borel-Weil-Bott theorem. The class of ideals defined by shape can be enlarged to that of GL-invariant ideals, and the formulas for Castelnuovo-Mumford regularity can be significantly sharpened. As an application of the calculation of Ext modules, we explain how to describe the GL-structure for the local cohomology with support in determinantal ideals. Finally, we conclude the book with a quick survey of the important topic of free resolutions of determinantal ideals
Gröbner Bases of Determinantal Ideals
No full textChapter 4 presents the computation of Gröbner bases, based on standard bitableaux and the Robinson-Schensted-Knuth correspondence to which it gives a short introduction. It exploits the information on the initial ideal for structural results with a homological flavor, as well as in the computation of enumerative data such as multiplicities and Hilbert series. We give also a more recent approach to Gröbner bases of determinantal ideals via secant varieties
Grassmannians, Flag Varieties, Schur Functors and Cohomology
No full textIn this chapter we introduce the basic theory of flag varieties, and describe general results regarding cohomology groups of line bundles, with an emphasis on vanishing statements. We develop the theory in the relative setting, which offers significant flexibility for the inductive arguments that we employ. We also introduce Schur functors and explain their relationship to direct images of line bundles associated to dominant weights. We end with a brief discussion of Grothendieck duality and some applications, that will be instrumental in our calculation of Ext modules in subsequent chapters
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