The Rings of Witt Vectors for Noncommutative Rings with Some Rich Structure

In this article, for a noncommutative ring A with some rich structure, we define a ring of Witt vectors with coefficients in A, which is noncommutative unless A is commutative.     

关于Witt向量之序列

对出现于公式Π_ｎ≥１１/(１＋ｄ_ｎｔ^ｎ／ｎ！)＝（１－ｔ）ｅ^ｔ中的整数ｄ_ｎ给出了精确估计     

On the Restricted Lie Algebra Structure of the Witt Lie Algebra in Finite Characteristic

The article contains an explicit formula for the restricted Lie algebra structure in the Witt Lie algebra over a field of finite characteristic. Some combinatorial lemmas can be of independent interest.     

On the cohomology of Witt vectors of p-adic integers and a conjecture of Hesselholt

Let K be a complete discrete valued field of characteristic zero with residue field kK of characteristic p>0. Let L/K be a finite Galois extension with Galois group G such that the induced extension of residue fields kL/kK is separable. Hesselholt (2004) [2] conjectured that the pro-abelian group {H¹(G,Wn(OL))}_n∈N is zero, where OL is the ring of integers of L and W(OL) is the ring of Witt vectors in OL w.r.t. the prime p. He partially proved this conjecture for a large class of extensions. In this paper, we prove Hesselholt?s conjecture for all Galois extensions.     

Remarks on Semistability of G-Bundles in Positive Characteristic

We analyzes a notion of strong semistability of principal G-bundles by including reduction to nonreduced parabolic subgroup schemes. It turns out that strong semistability is equivalent to the Frobenius semistability of Ramanan and Rananathan. We also give a bound for nonstrongly semsitability of a semistable GL(n)-bundle improving a previous result of Shepherd-Barron.     

Group completion in the K-theory and Grothendieck–Witt theory of proto-exact categories

We study the algebraic K-theory and Grothendieck–Witt theory of proto-exact categories, with a particular focus on classes of examples of ${\mathbb{F}}_1$-linear nature. Our main results are analogues of theorems of Quillen and Schlichting, relating the K-theory or Grothendieck–Witt theory spaces of proto-exact categories defined using the (hermitian) Q-construction and group completion.     

Emergence of the Witt group in the cellular lattice of rational spaces

We give an embedding of a quotient of the Witt semigroup into the lattice of rational cellular classes represented by formal -cones between and the two-cell complex ().

     

Ramification of Valuations and Local Rings in Positive Characteristic

In characteristic zero, local monomialization is true along any valuation. However, we have recently shown that local monomialization is not always true in positive characteristic, even in two dimensional algebraic function fields. In this paper we show that local monomialization is true for defectless extensions of two dimensional excellent local rings, extending an earlier result of Piltant and the author for two dimensional algebraic function fields over an algebraically closed field. We also give theorems showing that in many cases there are good stable forms of the extension of associated graded rings in a finite separable field extension.     

On the commutation of the test ideal with localization and completion

Let be a reduced ring that is essentially of finite type over an excellent regular local ring of prime characteristic. Then it is shown that the test ideal of commutes with localization and, if is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull of the residue field of every local ring of is equal to the finitistic tight closure of zero in . It is conjectured that this latter condition holds for all local rings of prime characteristic; it is proved here for all Cohen-Macaulay singularities with at most isolated non-Gorenstein singularities, and in general for all isolated singularities. In order to prove the result on the commutation of the test ideal with localization and completion, a ring of Frobenius operators associated to each -module is introduced and studied. This theory gives rise to an ideal of which defines the non-strongly F-regular locus, and which commutes with localization and completion. This ideal is conjectured to be the test ideal of in general, and shown to equal the test ideal under the hypothesis that in every local ring of .

     

Projective push-forwards in the Witt theory of algebraic varieties

We define push-forwards along projective morphisms in the Witt theory of smooth quasi-projective varieties over a field. We prove that they have standard properties such as functoriality, compatibility with pull-backs and projection formulas.     

关于n维欧氏空间R^n中闭集的教学思考

结合教材《工科数学分析基础》,对n维欧氏空间R^n中闭集的教学进行探讨和设计,同时分别给出了导集、闭包这两类特殊闭集的特征和性质．     

环的代数封闭性

证明了任一环有代数封闭的扩张环,且实封闭域上的四元数体是代数封闭的,给出了代数封闭环的若干性质.     

Integro-Local and Integral Limit Theorems on the Large Deviations of Sums of Random Vectors: Regular Distributions

We study the asymptotics of the probability that the sum of random vectors belongs to a relatively small cube in the range of large deviations.     

Realization of numbers as the degrees of maps between manifolds

We determine the set of degrees between some classes of oriented closed （n - 1）-connected 2n-manifolds by using the arithmetic theory of quadratic forms.     

随机向量自正则和的重对数律

本文给出了独立随机向量序列自正则和的重对数律成立的一个充分条件.     

On Singular Curves in the Case of Positive Characteristic

Nuclear convergence spaces are studied. It is shown that an L_e-embedded convergence vector space E is L_eL_M-embedded if it is Schwartz and satisfies a certain countability condition which expresses that the set of filters converging to zero is essentially countable. Further it is shown that if E is L_eL_M-embedded and nuclear, then the identity E → E can be approximated with finite operators in the equable continuous convergence structure on L(E, E). This result is used in the study of the spectrum Hom_cH_e(U) of the convergence algebra H_e(U) of holomorphic functions on a circled convex open set to prove sufficient conditions for the validity of the formula Hom_cH_e(U) ～ U.     

On the complexity of the integral closure

The computation of the integral closure of an affine ring has been the focus of several modern algorithms. We will treat here one related problem: the number of generators the integral closure of an affine ring may require. This number, and the degrees of the generators in the graded case, are major measures of cost of the computation. We prove several polynomial type bounds for various kinds of algebras, and establish in characteristic zero an exponential type bound for homogeneous algebras with a small singular locus.

     

Gauss整数环诸类问题探

本文主要探讨：（１） Ｇａｕｓｓ 整数环的定义及性质;（２） 其商环及性质;（３） Ｚ（ｘ）ｘ２＋ １同构于 Ｚ（ｉ）;（４） 两点猜想