scholarly journals Examples of Stability of Tensor Products in Positive Characteristic

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
A. S. Argáez
Keyword(s):  
Tensor Product ◽  
Positive Characteristic ◽  
Tensor Products ◽  
Necessary Condition ◽  
Closed Field ◽  
Base Field ◽  
Smooth Variety ◽  
Algebraically Closed Field ◽  
Analogous Theorem

Let X be projective smooth variety over an algebraically closed field k and let ℰ, ℱ be μ-semistable locally free sheaves on X. When the base field is ℂ, using transcendental methods, one can prove that the tensor product is always a μ-semistable sheaf. However, this theorem is no longer true over positive characteristic; for an analogous theorem one needs the hypothesis of strong μ-semistability; nevertheless, this hypothesis is not a necessary condition. The objective of this paper is to construct, without the strongly μ-semistability hypothesis, a family of locally free sheaves with μ-stable tensor product.

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

scholarly journals Fibrations with moving cuspidal singularities

1991 ◽  
Vol 122 ◽  
pp. 161-179 ◽  
Author(s):  
Yoshifumi Takeda
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Surface ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Smooth Projective Surface ◽  
Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

scholarly journals Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristic p

2019 ◽  
Author(s):  
Dmitry Kubrak ◽  
Roman Travkin
Keyword(s):  
Differential Operators ◽  
Closed Field ◽  
Brauer Group ◽  
Affine Variety ◽  
Smooth Variety ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Quiver Varieties ◽  
Hypertoric Varieties ◽  
General Reduction

Abstract “Even more so is the word ‘crystalline’, a glacial and impersonal concept of his which disdains viewing existence from a single portion of time and space” Eileen Myles, “The Importance of Being Iceland” For a smooth variety $X$ over an algebraically closed field of characteristic $p$ to a differential 1-form $\alpha $ on the Frobenius twist $X^{\textrm{(1)}}$ one can associate an Azumaya algebra ${{\mathcal{D}}}_{X,\alpha }$, defined as a certain central reduction of the algebra ${{\mathcal{D}}}_X$ of “crystalline differential operators” on $X$. For a resolution of singularities $\pi :X\to Y$ of an affine variety $Y$, we study for which $\alpha $ the class $[{{\mathcal{D}}}_{X,\alpha }]$ in the Brauer group $\textrm{Br}(X^{\textrm{(1)}})$ descends to $Y^{\textrm{(1)}}$. In the case when $X$ is symplectic, this question is related to Fedosov quantizations in characteristic $p$ and the construction of noncommutative resolutions of $Y$. We prove that the classes $[{{\mathcal{D}}}_{X,\alpha }]$ descend étale locally for all $\alpha $ if ${{\mathcal{O}}}_Y\widetilde{\rightarrow }\pi _\ast{{\mathcal{O}}}_X$ and $R^{1}\pi _*\mathcal O_X = R^2\pi _*\mathcal O_X =0$. We also define a certain class of resolutions, which we call resolutions with conical slices, and prove that for a general reduction of a resolution with conical slices in characteristic $0$ to an algebraically closed field of characteristic $p$ classes $[{{\mathcal{D}}}_{X,\alpha }]$ descend to $Y^{\textrm{(1)}}$ globally for all $\alpha $. Finally we give some examples; in particular, we show that Slodowy slices, Nakajima quiver varieties, and hypertoric varieties are resolutions with conical slices.

scholarly journals THE 2-RANKS OF HYPERELLIPTIC CURVES WITH EXTRA AUTOMORPHISMS

2009 ◽  
Vol 05 (05) ◽  
pp. 897-910 ◽  
Author(s):  
DARREN GLASS
Keyword(s):  
Automorphism Group ◽  
Hyperelliptic Curve ◽  
Hyperelliptic Curves ◽  
Closed Field ◽  
Base Field ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Characteristic Two ◽  
Carry Over ◽  
The Relationship

This paper examines the relationship between the automorphism group of a hyperelliptic curve defined over an algebraically closed field of characteristic two and the 2-rank of the curve. In particular, we exploit the wild ramification to use the Deuring–Shafarevich formula in order to analyze the ramification of hyperelliptic curves that admit extra automorphisms and use this data to impose restrictions on the genera and 2-ranks of such curves. We also show how some of the techniques and results carry over to the case where our base field is of characteristic p > 2.

The Krull and global dimension of the tensor product of quantum tori

2016 ◽  
Vol 15 (09) ◽  
pp. 1650174
Author(s):  
Ashish Gupta
Keyword(s):  
Tensor Product ◽  
Upper Bound ◽  
Tensor Products ◽  
Global Dimension ◽  
Base Field ◽  
Quantum Torus ◽  
Common Value ◽  
Quantum Tori ◽  
Special Cases ◽  
The Common

An [Formula: see text]-dimensional quantum torus is defined as the [Formula: see text]-algebra generated by variables [Formula: see text] together with their inverses satisfying the relations [Formula: see text], where [Formula: see text]. The Krull and global dimensions of this algebra are known to coincide and the common value is equal to the supremum of the rank of certain subgroups of [Formula: see text] that can be associated with this algebra. In this paper we study how these dimensions behave with respect to taking tensor products of quantum tori over the base field. We derive a best possible upper bound for the dimension of such a tensor product and from this special cases in which the dimension is additive with respect to tensoring.

scholarly journals Multiplicities in the Tensor Product of Finite-Dimensional Representations of Discrete Groups

1978 ◽  
Vol 21 (1) ◽  
pp. 17-19
Author(s):  
Dragomir Ž. Djoković
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Unitary Representations ◽  
Vector Spaces ◽  
Closed Field ◽  
Discrete Groups ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations

Let G be a group and ρ and σ two irreducible unitary representations of G in complex Hilbert spaces and assume that dimp ρ= n < ∞. D. Poguntke [2] proved that is a sum of at most n2 irreducible subrepresentations. The case when dim a is also finite he attributed to R. Howe.We shall prove analogous results for arbitrary finite-dimensional representations, not necessarily unitary. Thus let F be an algebraically closed field of characteristic 0. We shall use the language of modules and we postulate that allour modules are finite-dimensional as F-vector spaces. The field F itself will be considered as a trivial G-module.

THE LOEWY STRUCTURE OF -VERMA MODULES OF SINGULAR HIGHEST WEIGHTS

2015 ◽  
Vol 16 (4) ◽  
pp. 887-898
Author(s):  
Noriyuki Abe ◽  
Masaharu Kaneda
Keyword(s):  
Algebraic Group ◽  
Maximal Torus ◽  
Positive Characteristic ◽  
Closed Field ◽  
Verma Modules ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Loewy Length ◽  
Frobenius Kernel ◽  
Field Of Positive Characteristic

Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_{1}$ the Frobenius kernel of $G$, and $T$ a maximal torus of $G$. We show that the parabolically induced $G_{1}T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan–Lusztig $P$- and $Q$-polynomials. We assume that the characteristic of the field is sufficiently large that, in particular, Lusztig’s conjecture for the irreducible $G_{1}T$-characters holds.

Centers and Azumaya loci for finite W-algebras in positive characteristic

2021 ◽  
pp. 1-32
Author(s):  
BIN SHU ◽  
YANG ZENG
Keyword(s):  
Lie Algebra ◽  
Nilpotent Element ◽  
Positive Characteristic ◽  
Irreducible Representations ◽  
Maximal Dimension ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Simple Lie Algebra ◽  
Maximal Spectrum ◽  
Smooth Locus

Abstract In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

scholarly journals The construction problem for Hodge numbers modulo an integer in positive characteristic

2020 ◽  
Vol 8 ◽  
Author(s):  
Remy van Dobben de Bruyn ◽  
Matthias Paulsen
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Construction Problem ◽  
Algebraically Closed Field ◽  
Serre Duality ◽  
Hodge Numbers ◽  
Field Of Positive Characteristic

Abstract Let k be an algebraically closed field of positive characteristic. For any integer $m\ge 2$ , we show that the Hodge numbers of a smooth projective k-variety can take on any combination of values modulo m, subject only to Serre duality. In particular, there are no non-trivial polynomial relations between the Hodge numbers.

ENRIQUES’ CLASSIFICATION IN CHARACTERISTIC 0$" height="12pt">: THE -THEOREM

2018 ◽  
Vol 235 ◽  
pp. 201-226
Author(s):  
FABRIZIO CATANESE ◽  
BINRU LI
Keyword(s):  
Positive Characteristic ◽  
Kodaira Dimension ◽  
Algebraic Surfaces ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Precise Version

The main goal of this paper is to show that Castelnuovo–Enriques’ $P_{12}$ - theorem (a precise version of the rough classification of algebraic surfaces) also holds for algebraic surfaces $S$ defined over an algebraically closed field $k$ of positive characteristic ( $\text{char}(k)=p>0$ ). The result relies on a main theorem describing the growth of the plurigenera for properly elliptic or properly quasielliptic surfaces (surfaces with Kodaira dimension equal to 1). We also discuss the limit cases, i.e., the families of surfaces which show that the result of the main theorem is sharp.

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