实概形上的Gersten复形

De Rham complex of a Gerstenhaber algebra

We introduce a notion of the De Rham complex of a Gerstenhaber algebra which produces a notion of a "quasi-BV structure", and allows to classify these structures, generalizing the classical results for polyvector fields.

On The Gersten-Witt Complex of an Azumaya Algebra with Involution

Let $(A,σ)$ be an Azumaya algebra with involution over a regular ring $R$. We prove that the Gersten-Witt complex of $(A,σ)$ defined by Gille is isomorphic to the Gersten-Witt complex of $(A,σ)$ defined by Bayer-Fluckiger, Parimala and the author. Advantages of both constructions are used to show that the Gersten-Witt complex is exact when $\dim R\leq 3$, $\mathrm{ind}\, A\leq 2$ and $σ$ is orthogonal or symplectic. This means that the Grothendieck-Serre conjecture holds for the group $R$-scheme of $σ$-unitary elements in $A$ under the same hypotheses; $R$ is not required to contain a field.

光滑情形下关于 Milnor K-理论的相对 Gersten 猜想

On the relative Gersten conjecture for Milnor K-theory in the smooth case

本文证明优秀离散赋值环上光滑概形的改进 Milnor K-层 Gersten 复形除首项外正合，并将首项正合性归结到特殊纤维一般点对应的离散赋值环。

混合特征下光滑与半稳定概形的Bloch-Ogus理论

Bloch-Ogus theory for smooth and semi-stable schemes in mixed characteristic

本文研究了混合特征下光滑与半稳定概形上对偶同调理论的Bloch-Ogus理论与Gersten猜想，证明了光滑情形下的猜想及半稳定情形的特例，并由此导出优秀离散赋值环上局部环的Galois符号映射的满射性。

Milnor-Witt cycle modules over an excellent DVR

The definition of Milnor-Witt cycle modules in [Feld, N., Milnor-Witt cycle modules, Journal of Pure and Applied Algebra 224 (2020) 106298] can easily be adapted over general regular base schemes. However, there are simple examples to show that Gersten complex fails to be exact for cycle modules in general if the base is not a field. The goal of this article is to show that, for a restricted class of Milnor-Witt cycle modules over an excellent DVR satisfying an extra axiom, called here as R5, the expected properties of exactness of Gersten complex and $\mathbb{A}^1$-invariance hold. Moreover R…

Variations on the Bloch-Ogus Theorem

In the present paper we discuss questions concerning the arithmetic resolution for etale cohomology. Namely, consider a smooth quasi-projective variety X over a field k together with the local scheme U at a point x. Let Y be a smooth proper scheme over U. We prove there is the Gersten-type exact sequence for etale cohomology with coefficients in a locally constant etale sheaf F of Z/nZ-modules on Y which has finite stalks and (n,char(k))=1.

henselian局部环的正常交叉簇的gersten型猜想

Gersten-type conjecture for henselian local rings of normal crossing varieties

本文证明了关于henselian局部环的正常交叉簇的gersten型猜想，并推广了artin关于brauer群的定理。

Étale duality for constructible sheaves on arithmetic schemes

In this note we relate three topics for arithmetic schemes: a general duality for étale constructible torsion sheaves, an étale homology theory, and a Gersten-Bloch-Ogus-Kato complex. The results in this paper have been used in other papers of the authors ([JS], [Sa], [SaH] in the list of references).

A Nisnevich Local Bloch-Ogus Theorem over a General Base

We prove the exactness of the Nisnevich Gersten complex over a base under some conditions. We also obtain, as a consequence, a Nisnevich analogue of the Bloch-Ogus theorem for étale cohomology in this setting.

关于多面体锥和环面多对数的一个注记

A note on polyhedral cones and toric polylogarithms

本文通过建立单纯锥链复形与Milnor K-理论Gersten复形之间的同构，推广了先前关于代数环面上Milnor K-理论特殊元素的方法，改进了Charlton-Radchenko-Rudenko的结果。

Jet schemes of complex plane branches and equisingularity

For $m \in \mathbb{N}$, we determine the irreducible components of the $m$-th Jet Scheme of a complex branch $C$ and give formulas for their number $N(m)$ and for their codimensions, in terms of $m$ and the generators of the semigroup of $C$. This structure of the Jet Schemes determines and is determined by the topological type of $C$.

On the decomposition of the De Rham complex on formal schemes

We show that, for a pseudo-proper smooth noetherian formal scheme $\mathfrak{X}$ over a positive characteristic $p$ field, its truncated De Rham complex up to the characteristic $p$ is decomposable. Moreover, if the dimension of $\mathfrak{X}$ is exactly $p$, then the full De Rham complex is decomposable. Along the way we establish the Cartier isomorphism associated to a smooth morphism of positive characteristic noetherian formal schemes.

The units-Picard complex of a reductive group scheme

Let S be a locally noetherian regular scheme. We compute the units-Picard complex of a reductive S-group scheme G in terms of the dual algebraic fundamental complex of G. To do so, we establish a units-Picard-Brauer exact sequence for a torsor under a smooth S-group scheme.

Dualizing Complexes and Perverse Sheaves on Noncommutative Ringed Schemes

A quasi-coherent ringed scheme is a pair (X,A), where X is a scheme, and A is a noncommutative quasi-coherent O_X-ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.

Topological properties of punctual Hilbert schemes of almost-complex fourfolds (I)

In this article, we study topological properties of Voisin's punctual Hilbert schemes of an almost-complex fourfold $X$. In this setting, we compute their Betti numbers and construct Nakajima operators. We also define tautological bundles associated with any complex bundle on $X$, which are shown to be canonical in $K$-theory.

Initial Complex Associated to a Jet Scheme of a Determinantal Variety

We show in this paper that the principal component of the first order jet scheme over the classical determinantal variety of m x n matrices of rank at most 1 is arithmetically Cohen-Macaulay, by showing that an associated Stanley-Reisner simplicial complex is shellable.