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This paper studies local properties of Jacobson-Witt algebras over fields of prime characteristic, i.e., initiates the study on 2-local derivations of Lie algebras of prime characteristic. Let W n be a simple Jacobson-Witt algebra over a field F of prime characteristic p with | F | ≥ p n . In this paper, it is shown that every 2-local derivation on W n is a derivation.

In this paper, a finite-dimensional Lie superalgebra K n , m over a field of prime characteristic is constructed. Then, we study some properties of K n , m . Moreover, we prove that K n , m is an extension of a simple Lie superalgebra, and if m = n − 1 , then it is isomorphic to a subalgebra of a restricted Lie superalgebra.

In this paper, we study the $F$-rationality of the Rees algebra and the extended Rees algebra of $\mathfrak{m}$-primary ideals in excellent local rings $(R, \mathfrak{m})$ of prime characteristic. We partially answer some conjectures and questions raised by N. Hara, K.-i. Watanabe and K.-i. Yoshida (J. Algebra, pp.153--190, vol 247, 2002)., 13 pages; comments welcome

In this note we compute the centers of the categories of tilting modules for G=SL2 in prime characteristic, of tilting modules for the corresponding quantum group at a complex root of unity, and of projective GgT-modules when g=1,2., Comment: 18 pages, some figures, revised version, to appear in Glasg. Math. J., comments welcome

Let $(R,\frak m)$ be a Noetherian local ring of prime characteristic $p>0$, and $t$ an integer such that $H_{\frak m}^j(R)/0^F_{H^j_{\frak m}(R)}$ has finite length for all $j, Comment: To appear in Acta Mathematica Vietnamica

Musta\c{t}\u{a} defined Bernstein-Sato polynomials in prime characteristic for principal ideals and proved that the roots of these polynomials are related to the $F$-jumping numbers of the ideal. This approach was later refined by Bitoun. Here we generalize these techniques to develop analogous notions for the case of arbitrary ideals and prove that these have similar connections to $F$-jumping numbers., Comment: v2: added Subsections 6.1 (some remarks about compatibility with char. 0) 6.5 (other characterizations of Bernstein-Sato roots) and 6.6 (examples). Relaxed assumptions on R. 33 pages

The concept of frame multiresolution analysis (FMRA) on local fields of positive characteristic was given by Shah in his paper, Frame Multiresolution Analysis on Local Fields published by Journal of Operators. The author has studied the concept of minimum-energy wavelet frames on these prime characteristic fields. We continued the studies based on frame multiresolution analysis and minimum-energy wavelet frames on local fields of positive characteristic. In this paper, we introduce the notion of the construction of minimum-energy wavelet frames based on FMRA on local fields of positive characteristic. We provide a constructive algorithm for the existence of the minimum-energy wavelet frame on the local field of positive characteristic. An explicit construction of the frames and bases is given. In the end, we exhibit an example to illustrate our algorithm.

Let g be the Witt algebra over an infinite field of prime characteristic p. In this paper, we discuss the properties of local derivations on g , and prove that every local derivation on g is a deri...

Let g be a restricted Lie superalgebra of Cartan type W(n), S(n)or H(n) over an algebraically closed field k of prime characteristic p > 3, in the sense of modular version of Kac’s definition in 1977. In this note, we show that the restricted representation category over g has only one block (reckoning parities in). This phenomenon is very different from the case of characteristic zero.

We determine which faithful irreducible representations $V$ of a simple linear algebraic group $G$ are generically free for Lie($G$), i.e., which $V$ have an open subset consisting of vectors whose stabilizer in Lie($G$) is zero. This relies on bounds on $\dim V$ obtained in prior work (part I), which reduce the problem to a finite number of possibilities for $G$ and highest weights for $V$, but still infinitely many characteristics. The remaining cases are handled individually, some by computer calculation. These results were previously known for fields of characteristic zero, although new phenomena appear in prime characteristic; we provide a shorter proof that gives the result with very mild hypotheses on the characteristic. (The few characteristics not treated here are settled in part III.) These results are related to questions about invariants and the existence of a stabilizer in general position., Part I is arxiv preprint 1711.05502. Part III is arxiv preprint 1801.06915. v2: minor text changes to align with part III; v3: updated to align with v3 of Part I. Supporting Magma code available at http://garibaldibros.com

Given a polynomial $f$ with coefficients in a field of prime characteristic $p$, it is known that there exists a differential operator that raises $1/f$ to its $p$th power. We first discuss a relation between the `level' of this differential operator and the notion of `stratification' in the case of hyperelliptic curves. Next we extend the notion of level to that of a pair of polynomials. We prove some basic properties and we compute this level in certain special cases. In particular we present examples of polynomials $g$ and $f$ such that there is no differential operator raising $g/f$ to its $p$th power., 14 pages, comments are welcome

We study the conditions under which the highest nonvanishing local cohomology module of a domain $R$ with support in an ideal $I$ is faithful over $R$, i.e., which guarantee that $H^c_I(R)$ is faithful, where $c$ is the cohomological dimension of $I$. In particular, we prove that this is true for the case of positive prime characteristic when $c$ is the number of generators of $I$.

The concepts of multiresolution analysis (MRA) and wavelet have been generalized to a local field K of positive characteristic by using a prime element p of such a field. A MRA is a sequence of closed subspaces of L 2 (K) satisfying certain properties. In this paper, we are interested in a nonstationary MRA and related wavelets on local fields of positive characteristic.

Cholangiocarcinoma (CCA) is a heterogeneous group of malignancies that primarily originate from the bile duct. Tumor heterogeneity is a prime characteristic of CCA and considering the scarcity of approved targeted therapy drugs, this makes precision oncology impractical in CCA. Stratifying patients based on their molecular signature and biomarker-guided therapy may offer a conducive solution. Receptors tyrosine kinases (RTK) are potential targets for novel therapeutic strategies in CCA as RTK signaling is dysregulated in CCA. This study aims to identify targetable RTK profile in CCA using a bioinformatic approach. We discovered that CCA samples could be grouped into molecular subtypes based on the gene expression profile of selected RTKs (RTK25). Using the RTK25 gene list, we discovered five distinct molecular subtypes of CCA in this cohort. Tyrosine kinase inhibitors that target each RTK profile and their subsequent molecular signatures were also discovered. These results suggest that certain RTKs correlate with each other, indicating that tailored dual inhibition of RTKs may be more favorable than monotherapy. The results from this study can direct future investigative attention towards validating this concept in in vivo and in vitro systems. Ultimately, this will facilitate biomarker-guided clinical trials for the successful approval of novel therapeutic options in CCA.

Companion matrices over fields of prime characteristic, p, that are sums of two idempotents and a nilpotent are characterized in terms of dimension and trace of such a matrix and of p. Companion matrices over fields of positive characteristic, p, that are sums of m idempotents, m ≥ 2, and a nilpotent are characterized in terms of dimension and trace of such a matrix and of p.

Let F be an algebraically closed field of prime characteristic, and W(m, n, 1) be the simple restricted Lie superalgebra of Witt type over F, which is the Lie superalgebra of superderivations of the superalgebra $$\mathfrak{A}(m;1)\otimes\wedge(n)$$, where $$\mathfrak{A}(m;1)$$ is the truncated polynomial algebra with m indeterminants and ∧(n) is the Grassmann algebra with n indeterminants. In this paper, the author determines the character formulas for a class of simple restricted modules of W(m, n, 1) with atypical weights of type I.

Suppose $\mathbb{F}$ is a field of prime characteristic $p$ and $E$ is a finite subgroup of the additive group $(\mathbb{F},+)$. Then $E$ is an elementary abelian $p$-group. We consider two such subgroups, say $E$ and $E'$, to be equivalent if there is an $\alpha\in\mathbb{F}^*:=\mathbb{F}\setminus\{0\}$ such that $E=\alpha E'$. In this paper we show that rational functions can be used to distinguish equivalence classes of subgroups and, for subgroups of prime rank or rank less than twelve, we give explicit finite sets of separating invariants., Comment: There has been a minor change of title from the previous version. We have expanded the introduction, simplified some of the proofs and corrected a number of minor errors. The document is now 25 pages

In this paper, we study A/k-bialgebras in prime characteristic. Firstly, we prove a Cartier type structure theorem for cocomplete A/k-bialgebras. Secondly, we generalize Michaelis’s theorem...

In this paper we show that the Frobenius test exponent for parameter ideals of a local ring of prime characteristic is always bigger than or equal to its Hartshorne-Speiser-Lyubeznik number. Our argument is based on an isomorphism of Nagel and Schenzel on local cohomology that we will provide an elementary proof., Comment: final version, major change in Section 2, to appear in Communication in Algebra

We classify connected graphs $G$ whose binomial edge ideal is Gorenstein. The proof uses methods in prime characteristic., Comment: 11 pages. Comments welcome

We extend the notion of Frobenius Betti numbers to large classes of finitely generated modules over rings of prime characteristic, which are not assumed to be local. To do so, we introduce new invariants, which we call Frobenius Euler characteristics. We prove uniform convergence and upper semicontinuity results for Frobenius Betti numbers and Euler characteristics. These invariants detect the singularities of a ring, extending two results from the local to the global setting.

In prime characteristic there are important invariants that allow us to measure singularities. For certain cases, it is known that they are rational numbers. In this article, we show this property for Stanley-Reisner rings in several cases., 19 pages. Minor corrections from previous version. Accepted in Journal of Pure and Applied Algebra

We initiate the study of Nash blowups in prime characteristic. First, we show that a normal variety is non-singular if and only if its Nash blowup is an isomorphism, extending a theorem by A. Nobile. We also study higher Nash blowups, as defined by T. Yasuda. Specifically, we give a characteristic-free proof of a higher version of Nobile's Theorem for quotient varieties and hypersurfaces. We also prove a weaker version for $F$-pure varieties., Comment: 10 pages, comments welcome

We prove that the higher Nash blowup of a normal toric variety defined over a field of positive characteristic is an isomorphism if and only if it is non-singular. We also extend a result by R. Toh-Yama which shows that higher Nash blowups do not give a one-step resolution of the $A_3$-singularity. These results were previously known only in characteristic zero., Comment: 12 pages, comments welcome

Let f(x) be a formal power series with coefficients in the field k and let n ? 1. We define the notion of n-transcendence of f(x) over k and, more generally, the stable transcendence function dk(f(x), n). It is shown that, if k has prime characteristic p, this function determines the minimal Krull dimension dk(G) of the universal modular Galois-algebras of an elementary Abelian p-group G, introduced in [2, 3, 4, 5]. Since the concept of n-transcendence is of independent interest in all characteristics, a number of fundamental theorems are proved where the generalized Artin-Schreier polynomials surprisingly play a central role. We make a plausible conjecture in the case when k = Fp, the truth of which would imply a conjectural result concerning dFp (G) previously investigated by the authors.

We focus on the classification of pointed p3-dimensional Hopf algebras H over any algebraically closed field of prime characteristic p > 0. In particular, we consider certain cases when the group of grouplike elements is of order p or p2, that is, when H is pointed but is not connected nor a group algebra. The structures of the associated graded algebra gr H are completely described as bosonizations of graded braided Hopf algebras over group algebras, and most of the lifting structures of H are given. This work provides many new examples of (parametrized) non-commutative, non-cocommutative finite-dimensional Hopf algebras in positive characteristic.

We classify the simple modules of the exceptional algebraic supergroups over an algebraically closed field of prime characteristic., v3, 24 pages, minor edits, Math. Z. (to appear)