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

Gersten-Witt Complex of Hirzebruch Surfaces

Gersten-Witt cohomologies of Hirzebruch surfaces are computed.

An $8$-Periodic Exact Sequence of Witt Groups of Azumaya Algebras with Involution

Given an Azumaya algebra with involution $(A,σ)$ over a commutative ring $R$ and some auxiliary data, we construct an $8$-periodic chain complex involving the Witt groups of $(A,σ)$ and other algebras with involution, and prove it is exact when $R$ is semilocal. When $R$ is a field, this recovers an $8$-periodic exact sequence of Witt groups of Grenier-Boley and Mahmoudi, which in turn generalizes exact sequences of Parimala--Sridharan--Suresh and Lewis. We apply this result in several ways: We establish the Grothendieck--Serre conjecture on principal homogeneous bundles and the local purity c…

实概形上的Gersten复形

A Gersten complex on real schemes

本文通过实概形上的Gersten型层复形建立相干对偶与Verdier对偶的联系，给出导出范畴中的对偶对象，其超上同调与Gersten-Witt复形一致，并关联到Borel-Moore同调。

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…

On the Grothendieck-Serre Conjecture for Classical Groups

We prove some new cases of the Grothendieck-Serre conjecture for classical groups. This is based on a new construction of the Gersten-Witt complex for Witt groups of Azumaya algebras with involution on regular semilocal rings, with explicit second residue maps; the complex is shown to be exact when the ring is of dimension $\le 2$ (or $\le 4$, with additional hypotheses on the algebra with involution). Note that we do not assume that the ring contains a field.

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。