scholarly journals Uniform bounds in F-finite rings and their applications

2017 ◽  
Author(s):  
◽  
Thomas Marion Polstra
Keyword(s):  
Finite Type ◽  
Local Ring ◽  
Lower Semicontinuous ◽  
Multiplicity Function ◽  
Characteristic P ◽  
Finite Rings ◽  
Uniform Bounds ◽  
Frobenius Splitting ◽  
Lower Semi ◽  
Splitting Numbers

This dissertation establishes uniform bounds in characteristic p rings which are either F-finite or essentially of finite type over an excellent local ring. These uniform bounds are then used to show that the Hilbert-Kunz length functions and the normalized Frobenius splitting numbers defined on the spectrum of a ring converge uniformly to their limits, namely the Hilbert-Kunz multiplicity function and the Fsignature function. From this we establish that the F-signature function is lower semicontinuous. Lower semi-continuity of the F-signature of a pair is also established. We also give a new proof of the upper semi-continuity of Hilbert-Kunz multiplicity, a result originally proven by Ilya Smirnov.

scholarly journals The lower semicontinuity of the Frobenius splitting numbers

2010 ◽  
Vol 150 (1) ◽  
pp. 35-46 ◽  
Author(s):  
FLORIAN ENESCU ◽  
YONGWEI YAO
Keyword(s):  
Local Ring ◽  
Lower Semicontinuity ◽  
Lower Semicontinuous ◽  
Prime Characteristic ◽  
Frobenius Splitting ◽  
Mild Conditions ◽  
Splitting Numbers

AbstractWe show that, under mild conditions, the (normalized) Frobenius splitting numbers of a local ring of prime characteristic are lower semicontinuous.

scholarly journals Upper semi-continuity of the Hilbert–Kunz multiplicity

2016 ◽  
Vol 152 (3) ◽  
pp. 477-488 ◽  
Author(s):  
Ilya Smirnov
Keyword(s):  
Finite Type ◽  
Local Ring ◽  
Finite Rings ◽  
Rings And Algebras

We prove that the Hilbert–Kunz multiplicity is upper semi-continuous in F-finite rings and algebras of essentially finite type over an excellent local ring.

scholarly journals On the Standard Poisson Structure and a Frobenius Splitting of the Basic Affine Space

2019 ◽  
Author(s):  
Jun Peng ◽  
Shizhuo Yu
Keyword(s):  
General Theory ◽  
Poisson Structure ◽  
Borel Subgroup ◽  
Closed Field ◽  
Poisson Geometry ◽  
Characteristic P ◽  
Frobenius Splitting ◽  
Algebraic Torus ◽  
Poisson Varieties ◽  
Simply Connected

Abstract The goal of this paper is to construct a Frobenius splitting on $G/U$ via the Poisson geometry of $(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$, where $G$ is a simply connected semi-simple algebraic group defined over an algebraically closed field of characteristic $p> 3$, $U$ is the uniradical of a Borel subgroup of $G$, and $\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}}$ is the standard Poisson structure on $G/U$. We first study the Poisson geometry of $(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$. Then we develop a general theory for Frobenius splittings on $\mathbb{T}$-Poisson varieties, where $\mathbb{T}$ is an algebraic torus. In particular, we prove that compatibly split subvarieties of Frobenius splittings constructed in this way must be $\mathbb{T}$-Poisson subvarieties. Lastly, we apply our general theory to construct a Frobenius splitting on $G/U$.

The maximal dimension of formal fibers of local rings of an algebraic scheme of finite type

2019 ◽  
Vol 18 (06) ◽  
pp. 1950120
Author(s):  
Đoàn Trung Cu’ò’ng
Keyword(s):  
Finite Type ◽  
Local Ring ◽  
Local Algebra ◽  
Local Rings ◽  
Maximal Dimension ◽  
Algebraic Varieties ◽  
Noetherian Local Ring ◽  
Closed Point ◽  
Dimension One ◽  
Algebraic Scheme

For a scheme [Formula: see text] of finite type over a Noetherian local ring [Formula: see text] with a closed point [Formula: see text] of the special fiber, we show that the maximal dimension of the formal fibers of the local algebra [Formula: see text] equals to [Formula: see text] provided that either [Formula: see text] is complete of dimension one or the dimensions of the formal fibers of [Formula: see text] are less than [Formula: see text]. This extends Matsumura’s theorem for algebraic varieties.

Units of commutative completely primary finite rings of characteristic p^n

2013 ◽  
Vol 7 ◽  
pp. 259-266
Author(s):  
Owino Maurice Oduor ◽  
Ojiema Michael Onyango ◽  
Mmasi Eliud
Keyword(s):  
Characteristic P ◽  
Finite Rings

On a Class of Finite Rings of Characteristic p 2

2000 ◽  
Vol 38 (3-4) ◽  
pp. 377-390 ◽  
Author(s):  
Graham D. Williams
Keyword(s):  
Characteristic P ◽  
Finite Rings

scholarly journals On the relationship between depth and cohomological dimension

2015 ◽  
Vol 152 (4) ◽  
pp. 876-888 ◽  
Author(s):  
Hailong Dao ◽  
Shunsuke Takagi
Keyword(s):  
Finite Type ◽  
Local Ring ◽  
Characteristic Zero ◽  
Residue Field ◽  
Cohomological Dimension ◽  
Picard Group ◽  
Regular Local Ring ◽  
Sharp Bounds ◽  
The Relationship

Let $(S,\mathfrak{m})$ be an $n$-dimensional regular local ring essentially of finite type over a field and let $\mathfrak{a}$ be an ideal of $S$. We prove that if $\text{depth}\,S/\mathfrak{a}\geqslant 3$, then the cohomological dimension $\text{cd}(S,\mathfrak{a})$ of $\mathfrak{a}$ is less than or equal to $n-3$. This settles a conjecture of Varbaro for such an $S$. We also show, under the assumption that $S$ has an algebraically closed residue field of characteristic zero, that if $\text{depth}\,S/\mathfrak{a}\geqslant 4$, then $\text{cd}(S,\mathfrak{a})\leqslant n-4$ if and only if the local Picard group of the completion $\widehat{S/\mathfrak{a}}$ is torsion. We give a number of applications, including a vanishing result on Lyubeznik’s numbers, and sharp bounds on the cohomological dimension of ideals whose quotients satisfy good depth conditions such as Serre’s conditions $(S_{i})$.

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