Dedekind方案上的稳定$\mathbb{A}^1$ -连通性

$\mathbb{A}^1$-connectivity of moduli of vector bundles on a curve

In this note we prove that the moduli stack of vector bundles on a curve, with a fixed determinant is $\mathbb{A}^1$-connected. We obtain this result by classifying vector bundles on a curve upto $\mathbb{A}^1$-concordance. Consequently we classify$\mathbb{P}^n$- bundles on a curve upto $\mathbb{A}^1$-weak equivalence, extending a result of Asok-Morel. We also give an explicit example of a variety which is $\mathbb{A}^1$-h-cobordant to a projective bundle over $\mathbb{P}^2$ but does not have the structure of a projective bundle over $\mathbb{P}^2$, thus answering a question of Asok-Kebekus-We…

Conjectures on L-functions for flag bundles on Dedekind domains

Let $\mathcal{O}_K$ be the ring of integers in an algebraic number field $K$ and let $S:=\operatorname{Spec}(\mathcal{O}_K)$. Let $T_0,\ldots,T_n$ be regular schemes of finite type over $S$ and let $X$ be a scheme of finite type over $T_n$ with a stratification of closed subschemes (a generalized cellular decomposition) \[ \emptyset=X_{-1} \subseteq X_0 \subseteq \cdots \subseteq X_{n-1} \subseteq X_n:=X \] with $X_i-X_{i-1}=E_i$ where $E_i$ is a vector bundle of rank $d_i$ on $T_i$. We prove that if the Beilinson-Soule vanishing conjecture and Soule conjecture holds for $T_i$ it follows the s…

Stable connectivity over a base

Morel's stable connectivity theorems state that for any connective $S^1$-spectrum $F$ of motivic spaces (Nisnevich simplicial sheaves) over an arbitrary field, the spectrum $L_{\mathbb A^1}(F)$ is connective, and the same property for $\mathbb P^1$-spectra of motivic spaces. Here $L_{\mathbb A^1}$ denotes the $\mathbb A^1$-localisation in the category of motivic spectra over a field $k$. Originally the same property was conjectured for the case of motivic $S^1$-spectra over a base scheme $S$.In view of Ayoub's conterexamples the modified version of conjecture states that $L_{\mathbb A^1}(F)$ i…

连通性与一个形式几何问题

Connectivity and a Problem of Formal Geometry

设$P=\mathbb P^m(e)\times\mathbb P^n(h)$是加权射影空间的乘积，$Δ_P$是$P\times P$的对角线.证明了形式有理函数在$P\times P$的某些闭子簇$X$上沿交$X\capΔ_P$的代数化结果.

$\mathbb A^1$ -Chow么半群上的连通性与代数圈的有理等价

$\mathbb A^1$-connectivity on Chow monoids v.s. rational equivalence of algebraic cycles

设$k$是一个特征为零的域，$X$是一个嵌入到$k$上的射影空间中的射影簇。对于两个自然数$r$和$d$，令$C_{r,d}(X)$是Chow方案，其参数化了变量$X$上维度$r$和度$d$的有效圈。$X$上的一个有效的最小度$r$ -圈产生一个$C_{r,d}(X)$嵌入$C_{r,d+1}(X)$的链，其上极限是$X$上的$r$ -圈的联结Chow幺半群$C_r^{\infty }(X)$。设$BC_r^{\infty }(X)$是这个幺半群的动机分类空间。本文建立了$X$上的度$0$维$r$代数圈模有理等价的Chow群$CH_r(X)_0$与$Spec(k)$上的$BC_r^{\infty }(X)$的圈空间的$\mathbb A^1$ -路连通分支的层的截面群之间的同构.等价地，$CH_r(X)_0$同构于$S^1\wedge \mathbb A^1$ -基本群$Π_1^{S^1\wedge \mathbb A^1}(BC_r^{\infty }(X))$在$Spec(k)$的截面群。

Cohomological connectivity of perturbations of map-germs

Let $f\colon (\mathbb C^n,S)\to (\mathbb C^p,0)$ be a finite map-germ with $n<p$ and $Y_δ$ the image of a small perturbation $f_δ$. We show that the reduced cohomology of $Y_δ$ is concentrated in a range of degrees determined by the dimension of the instability locus of $f$. In the case $n\geq p$ we obtain an analogous result, replacing finiteness by $\mathcal K$-finiteness and $Y_δ$ by the discriminant $Δ(f_δ)$. We also study the monodromy associated to the perturbation $f_δ$.

$\u2248$ 概形的 $\mathbb{P}^1$ 同伦理论

$\mathbb{P}^1$ Homotopy Theory of Schemes

该论文研究概形的 $\mathbb{P}^1$ 局部 motivic 同伦理论，证明了非稳定与稳定的 $\mathbb{P}^1$-连通性定理，并建立了同伦 $t$-结构。

A Bloch-Ogus Theorem for henselian local rings in mixed characteristic

We show a conditional exactness statement for the Nisnevich Gersten complex associated to an $\mathbb{A}^1$-invariant cohomology theory with Nisnevich descent for smooth schemes over a Dedekind ring with only infinite residue fields. As an application we derive a Nisnevich analogue of the Bloch-Ogus theorem for étale cohomology over a henselian discrete valuation ring with infinite residue field.

切片滤过的连通性

Connectivity of the slice filtration

本文证明Tate截断函子保持motivic谱的连通性，并建立相应的Hurewicz定理和重构定理的切片类比。

Polyhedral divisors and torus actions of complexity one over arbitrary fields

We show that the presentation of affine $\mathbb{T}$-varieties of complexity one in terms of polyhedral divisors holds over an arbitrary field. We also describe a class of multigraded algebras over Dedekind domains. We study how the algebra associated to a polyhedral divisor changes when we extend the scalars. As another application, we provide a combinatorial description of affine $\mathbf{G}$-varieties of complexity one over a field, where $\mathbf{G}$ is a (not-nescessary split) torus, by using elementary facts on Galois descent. This class of affine $\mathbf{G}$-varieties is described via …

Connectivity of joins, cohomological quantifier elimination, and an algebraic Toda's theorem

Let $X \subset \mathbb{P}^{n}$ be a non-empty closed subscheme over an algebraically closed field $k$, and $\mathrm{J}^{[p]}(X) = \mathrm{J}(X,\mathrm{J}(X,\cdots,\mathrm{J}(X,X)\cdots)$ denote the $p$-fold iterated join of $X$ with itself. In this article, we prove that the restriction homomorphism on cohomology $\mathrm{H}^{i}(\mathbb{P}^{N}) \rightarrow \mathrm{H}^{i}(\mathrm{J}^{[p]}(X))$, with $N = (p+1)(n+1)-1$, is an isomorphism for $0 \leq i < p$, and injective for $i=p$, for any good cohomology theory. We also prove this result in the more general setting of relative joins for $X$ ove…

超概形的平坦族中局部性质的开性

Openness of local properties in flat families of superschemes

本文证明在平坦真态射中，超概形纤维的正规性、GFRR、正则性和 Cohen-Macaulay 性质构成开集，从而可从特殊纤维提升到一般纤维。