scholarly journals F-THRESHOLDS OF GRADED RINGS

2016 ◽  
Vol 229 ◽  
pp. 141-168 ◽  
Author(s):  
ALESSANDRO DE STEFANI ◽  
LUIS NÚÑEZ-BETANCOURT
Keyword(s):  
Characteristic Zero ◽  
Projective Dimension ◽  
Graded Rings ◽  
Projective Varieties ◽  
Regular Sequences ◽  
Serre’S Condition ◽  
Regular Rings

The $a$-invariant, the $F$-pure threshold, and the diagonal $F$-threshold are three important invariants of a graded $K$-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly $F$-regular rings. In this article, we prove that these relations hold only assuming that the algebra is $F$-pure. In addition, we present an interpretation of the $a$-invariant for $F$-pure Gorenstein graded $K$-algebras in terms of regular sequences that preserve $F$-purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo–Mumford regularity, and Serre’s condition $S_{k}$. We also present analogous results and questions in characteristic zero.

REGULAR LOCAL ALGEBRAS OVER VALUATION DOMAINS: WEAK DIMENSION AND REGULAR SEQUENCES

2008 ◽  
Vol 07 (05) ◽  
pp. 575-591
Author(s):  
HAGEN KNAF
Keyword(s):  
Projective Dimension ◽  
Regular Sequence ◽  
Homological Dimension ◽  
Finitely Generated ◽  
Prüfer Domain ◽  
Regular Sequences ◽  
Local Algebras ◽  
Weak Dimension ◽  
Valuation Domains ◽  
Finitely Presented

A local ring O is called regular if every finitely generated ideal I ◃ O possesses finite projective dimension. In the article localizations O = Aq, q ∈ Spec A, of a finitely presented, flat algebra A over a Prüfer domain R are investigated with respect to regularity: this property of O is shown to be equivalent to the finiteness of the weak homological dimension wdim O. A formula to compute wdim O is provided. Furthermore regular sequences within the maximal ideal M ◃ O are studied: it is shown that regularity of O implies the existence of a maximal regular sequence of length wdim O. If q ∩ R has finite height, then this sequence can be chosen such that the radical of the ideal generated by its members equals M. As a consequence it is proved that if O is regular, then the factor ring O/(q ∩ R)O, which is noetherian, is Cohen–Macaulay. If in addition (q ∩ R)Rq ∩ R is not finitely generated, then O/(q ∩ R)O itself is regular.

scholarly journals The Bogomolov-Beauville-Yau decomposition for klt projective varieties with trivial first Chern class - without tears

2021 ◽  
Vol 149 (1) ◽  
pp. 1-13
Author(s):  
Frédéric Campana
Keyword(s):  
Chern Class ◽  
Characteristic Zero ◽  
Vector Fields ◽  
Positive Characteristic ◽  
Decomposition Theorem ◽  
Canonical Bundle ◽  
Projective Varieties ◽  
Holomorphic Vector Fields ◽  
First Chern Class ◽  
Complex Projective

We give a simplified proof (in characteristic zero) of the decomposition theorem for connected complex projective varieties with klt singularities and a numerically trivial canonical bundle. The proof mainly consists in reorganizing some of the partial results obtained by many authors and used in the previous proof but avoids those in positive characteristic by S. Druel. The single, to some extent new, contribution is an algebraicity and bimeromorphic splitting result for generically locally trivial fibrations with fibers without holomorphic vector fields. We first give the proof in the easier smooth case, following the same steps as in the general case, treated next. The last two words of the title are plagiarized from [4].

A generalisation of the Abhyankar Jung theorem to associated graded rings of valuations

2015 ◽  
Vol 160 (2) ◽  
pp. 233-255 ◽  
Author(s):  
STEVEN DALE CUTKOSKY
Keyword(s):  
Characteristic Zero ◽  
Positive Characteristic ◽  
Natural Extension ◽  
Graded Rings ◽  
Blowing Up ◽  
Associated Graded Rings

AbstractSuppose thatR→Sis an extension of local domains andν* is a valuation dominatingS. We consider the natural extension of associated graded rings along the valuation grν*(R) → grν*(S). We give examples showing that in general, this extension does not share good properties of the extensionR→S, but after enough blow ups above the valuations, good properties of the extensionR→Sare reflected in the extension of associated graded rings. Stable properties of this extension (after blowing up) are much better in characteristic zero than in positive characteristic. Our main result is a generalisation of the Abhyankar–Jung theorem which holds for extensions of associated graded rings along the valuation, after enough blowing up.

scholarly journals Which abelian tensor categories are geometric?

2018 ◽  
Vol 2018 (734) ◽  
pp. 145-186 ◽  
Author(s):  
Daniel Schäppi
Keyword(s):  
Characteristic Zero ◽  
Intrinsic Properties ◽  
Tensor Categories ◽  
Projective Varieties ◽  
Coherent Sheaves ◽  
Geometric Objects ◽  
Necessary And Sufficient ◽  
Tannaka Duality ◽  
Tannakian Categories

AbstractFor a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding.However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties—expressible entirely within the language of tensor categories—which are necessary and sufficient for the existence of a fiber functor. In this paper we generalize Deligne’s result to weakly Tannakian categories in characteristic zero. The class of geometric objects whose tensor categories of quasi-coherent sheaves can be recognized in this manner includes both the gerbes arising in classical Tannaka duality and more classical geometric objects such as projective varieties over a field of characteristic zero.Our proof uses a different perspective on fiber functors, which we formalize through the notion of geometric tensor categories. A second application of this perspective allows us to describe categories of quasi-coherent sheaves on fiber products.

scholarly journals Seminormal graded rings and weakly normal projective varieties

1985 ◽  
Vol 8 (2) ◽  
pp. 231-240 ◽  
Author(s):  
John V. Leahy ◽  
Marie A. Vitulli
Keyword(s):  
Graded Rings ◽  
Projective Varieties ◽  
Normal Variety ◽  
Blowing Up ◽  
Weak Normality

This paper is concerned with the seminormality of reduced graded rings and the weak normality of projective varieties. One motivation for this investigation is the study of the procedure of blowing up a non-weakly normal variety along its conductor ideal.

On 2-dimensional local rings with Artin's approximation property

1983 ◽  
Vol 94 (1) ◽  
pp. 35-52
Author(s):  
M. L. Brown
Keyword(s):  
Characteristic Zero ◽  
Approximation Property ◽  
Local Rings ◽  
Projective Varieties ◽  
Blowing Up

AbstractExtending results of Popescu and Brown, the main result of this paper is that excellent henselian R1 and S1 2-dimensional local rings, at least in characteristic zero, have the approximation property of M. Artin.Most of the paper consists of an extension of Néron's desingularization to rings which are R1 and S1; such a theorem was previously known for factorial domains. The main theorem is then deduced from this desingularization theorem using a theorem of Elkik.Because of cohomological obstructions, the desingularization theorem is proved only for quasi-projective varieties. In the previously known case for factorial domains, these obstructions are always zero and the desingularization can be obtained by blowing up subschemes. The more general desingularization of this paper is obtained by blowing up locally free sheaves instead, the obstructions being zero for this case.

On the Curves Associated to Certain Rings of Automorphic Forms

2001 ◽  
Vol 53 (1) ◽  
pp. 98-121 ◽  
Author(s):  
Kamal Khuri-Makdisi
Keyword(s):  
Ad Hoc ◽  
Automorphic Forms ◽  
Complex Multiplication ◽  
Graded Rings ◽  
Projective Varieties ◽  
Hecke Operator ◽  
Complex Tori ◽  
Hecke Correspondences ◽  
Cm Points ◽  
Projective Lines

AbstractIn a 1987 paper, Gross introduced certain curves associated to a definite quaternion algebra B over Q; he then proved an analog of his result with Zagier for these curves. In Gross’ paper, the curves were defined in a somewhat ad hoc manner. In this article, we present an interpretation of these curves as projective varieties arising from graded rings of automorphic forms on B×, analogously to the construction in the Satake compactification. To define such graded rings, one needs to introduce a “multiplication” of automorphic forms that arises from the representation ring of B×. The resulting curves are unions of projective lines equipped with a collection of Hecke correspondences. They parametrize two-dimensional complex tori with quaternionic multiplication. In general, these complex tori are not abelian varieties; they are algebraic precisely when they correspond to CM points on these curves, and are thus isogenous to a product E × E, where E is an elliptic curve with complex multiplication. For these CM points one can make a relation between the action of the p-th Hecke operator and Frobenius at p, similar to the well-known congruence relation of Eichler and Shimura.

scholarly journals On Additive invariants of actions of additive and multiplicative groups

2013 ◽  
Vol 12 (3) ◽  
pp. 551-568 ◽  
Author(s):  
Wenchuan Hu
Keyword(s):  
Fixed Point ◽  
Algebraic Group ◽  
Finite Fields ◽  
Algebraic Variety ◽  
Characteristic Zero ◽  
Multiplicative Group ◽  
Fixed Point Set ◽  
Projective Varieties ◽  
Point Set ◽  
Multiplicative Groups

AbstractLet X be an algebraic variety with an action of either the additive or multiplicative group. We calculate the additive invariants of X in terms of the additive invariants of the fixed point set, using a formula of Białynicki-Birula. The method is also generalized to calculate certain additive invariants for Chow varieties. As applications, we obtain results on the Hodge polynomial of Chow varieties in characteristic zero and the number of points for Chow varieties over finite fields. As applications, we obtain the l-adic Euler-Poincaré characteristic for the Chow varieties of certain projective varieties over a field of arbitrary characteristic. Moreover, we show that the virtual Hodge (p,0) and (0,q)-numbers of the Chow varieties and affine algebraic group varieties are zero for all p,q positive.

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