Let $\mathbb K$ be a field of characteristic 0, let $S$ be a complete local ring with coefficient field $\mathbb K$, let $\mathbb K[[x_1,\dots,x_n]]$ be the ring of formal power series in variables $x_1,\dots, x_n$ with coefficients from $\mathbb K$, let $\mathbb K[[x_1,\dots,x_n]]\to S$ be a $\mathbb K$-algebra surjection and let $E_\bullet^{\bullet,\bullet} $ be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of $S$. Nicholas Switala proved that this spectral sequence is independent of the surjection beginning with the $E_2$ page, and the groups $E^{p,q}_2$ are all finite-dimensional over $\mathbb K$.
In this paper we extend this result to affine varieties. Namely, let $Y$ be an affine variety over $\mathbb K$, let $X$ be a non-singular affine variety over $\mathbb K$, let $Y\subset X$ be an embedding over $\mathbb K$ and let $E_\bullet^{\bullet,\bullet} $ be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of $Y$. Then this spectral sequence is independent of the embedding beginning with the $E_2$ page, and the groups $E^{p,q}_2$ are all finite-dimensional over $\mathbb K$.
Over any smooth algebraic variety over a $p$-adic local field $k$, we construct the de Rham comparison isomorphisms for the étale cohomology with partial compact support of de Rham $\mathbb Z_p$-local systems, and show that they are compatible with Poincaré duality and with the canonical morphisms among such cohomology. We deduce these results from their analogues for rigid analytic varieties that are Zariski open in some proper smooth rigid analytic varieties over $k$. In particular, we prove finiteness of étale cohomology with partial compact support of any $\mathbb Z_p$-local systems, and e…
We define an overconvergent version of the Hyodo-Kato complex for semistable varieties $Y$ over perfect fields of positive characteristic, and prove that its hypercohomology tensored with $\mathbb{Q}$ recovers the log-rigid cohomology when $Y$ is quasi-projective. We then describe the monodromy operator using the overconvergent Hyodo-Kato complex. Finally, we show that overconvergent Hyodo-Kato cohomology agrees with log-crystalline cohomology in the projective semistable case.
We compute the rational Borel-Moore homology groups for affine determinantal varieties in the spaces of general, symmetric, and skew-symmetric matrices, solving a problem suggested by the work of Pragacz and Ratajski. The main ingredient is the relation with Hartshorne's algebraic de Rham homology theory, and the calculation of the singular cohomology of matrix orbits, using the methods of Cartan and Borel. We also establish the degeneration of the Čech-de Rham spectral sequence for determinantal varieties, and compute explicitly the dimensions of de Rham cohomology groups of local cohomology …
根据Bernhard Keller的一个定理，光滑簇的de Rham上同调同构于该簇上完全复形的微分分次范畴的周期循环同调. de Rham上同调和循环同调都可以被簇上的正则函数的指数扭曲。我们解释了这种同构在扭曲情形下成立，并得出了与正则函数相关的代数Gauss-Manin系统的导出不变性的一些结论。
Institute of Mathematics, Vietnam Academy of Science and Technology（越南科学技术院数学研究所）；Camille Jordan Institute, Claude Bernard University Lyon 1（里昂第一大学卡米耶·乔丹研究所）
For a Nash submersion $ϕ\colon X\to Y$, we study the complex $\mathcal{SDR}(ϕ)$ of Schwartz sections of the relative de Rham complex of $ϕ$. We define the notion of Schwartz sections of constructible sheaves on Nash manifolds and prove that $\mathcal{SDR}(ϕ)$ is homotopy equivalent to the Schwartz sections of the proper push-forward $ϕ_!\mathbb{R}_X$ of the constant sheaf $\mathbb{R}_X$. Using this equivalence, we show that $\mathcal{SDR}(ϕ)$ depends (up to homotopy equivalence) only on the homology type of the map $ϕ$. We also deduce that $\mathcal{SDR}(ϕ)$ has Hausdorff homology spaces.
In this paper, we introduce the notion of relation type of analytic and formal algebras and prove that it is well-defined and invariant by describing this notion in terms of the André-Quillen homology and using the Jacobi-Zariski long exact sequence of homology. In particular, the relation type is an invariant of schemes of finite type over a field, analytic varieties, and algebroid varieties.
Differential forms on singular varieties and cyclic homology
A. Connes断言光滑紧致流形X上光滑函数的Frechet代数通过一个纯代数过程提供了X的de Rham上同调。即利用该代数的Hochschild和循环同调的方法。在一个M-Mather分层簇的情形下，我们构造了一个正则部分上的函数的Frechet代数和一个沿奇异部分的极点模。我们与这些对象相关联的复杂的微分形式和Hochschild复杂的，在正常的一部分，都与极点沿奇异部分。第一个复形的de Rham上同调和第二个复形的圈同调与簇的交同调有关，对应的反常性由极点的阶数决定。
We introduce the categories of geometric mixed Hodge modules on algebraic varieties over a subfield $k\subset\mathbb C$, and for a prime number $p$, the categories of geometric $p$-adic mixed Hodge modules on algebraic varieties over a subfield $k\subset K\subset\mathbb C_p$ of a p-adic field $K$. We then give an Hodge realization functor on the de derived category of relative motives over $k\subset\mathbb C$, and for a prime number $p$ a $p$-adic Hodge realization functor on the derived category of relative motives over $k\subset K\subset\mathbb C_p$.
We give an explicit formula for the zeroth $\mathbb{A}^1$-homology sheaf of a smooth proper variety. We also provide a simple proof of a theorem of Kahn-Sujatha which describes hom sets in the birational localization of the category of smooth varieties.
我们介绍了“尖锐”（普遍）的扩展1动机（添加剂的因素和扭转）的领域的特征为零。我们通过传递到李代数来定义“夏普德拉姆实现”。在复数上，我们证明了形式Hodge结构范畴中的一个（sharp de Rham）比较定理.对于一个自由的1-动机及其Cartier对偶，我们得到了一个关于它们的尖锐扩张的正则联系，从而产生了一个关于尖锐实现的完美配对。因此，我们提供了“一维sharp de Rham上同调”的代数簇。
We compute the rational homology of the moduli stack $\mathcal{M}$ of objects in the derived category of certain smooth complex projective varieties $X$ including toric varieties, flag varieties, curves, surfaces, and some 3- and 4-folds. We identify Joyce's vertex algebra construction on $H_\ast(\mathcal{M},\mathbb{Q})$ with a generalized super-lattice vertex algebra associated to $K^0_{\rm top}(X^{\rm an}) \oplus K^1_{\rm top}(X^{\rm an})$.
