On the stability by tensor products of complexes of arithmetic D-modules

Theory of weights in p-adic cohomology

Let k be a finite field of characteristic p>0. We construct a theory of weights for overholonomic complexes of arithmetic D-modules with Frobenius structure on varieties over k. The notion of weight behave like Deligne's one in the l-adic framework: first, the six operations preserve weights, and secondly, the intermediate extension of an immersion preserves pure complexes and weights.

Tensor product of dualizing complexes over a field

Let $k$ be a field, and let $X,Y$ be two locally noetherian $k$-schemes (respectively $k$-formal schemes) with dualizing complexes $R_X$ and $R_Y$ respectively. We show that $R_X \boxtimes_{k} R_Y$ (respectively its derived completion) is a dualizing complex over $X\times_{k} Y$ if and only if $X\times_{k} Y$ is locally noetherian of finite Krull dimension.

Obstructions to weak approximation for reductive groups over $p$-adic function fields

We establish arithmetic duality theorems for short complexes associated to reductive groups over $p$-adic function fields. Using dualities, we deduce obstructions to weak approximation for certain reductive groups (especially quasi-split ones) and relate this obstruction to an unramified Galois cohomology group.

On Canonical Homomorphisms of Tensor Sheaves

In this paper we define tensor modules(sheaves) of Schur type,or of generalized Schur type associated with the give module(sheaf), using the so-called Schur functors. Then using global method we construct canonical homomorphisms between these modules(sheaves). We will get canonical isomorphisms if the original sheaf is locally free using idea of algebraic geometry, which is in fact a generalization of result in linear algebra. In the final section, we give canonical complexes using homomorphisms constructed before, and these complexes will become split exact sequence if further condition holds…

Local-global principles for homogeneous spaces of reductive groups over global function fields

Let $K$ be a global field of positive characteristic. We prove that the Brauer-Manin obstructions to the Hasse principle, to weak approximation and to strong approximation are the only ones for homogeneous spaces of reductive groups with reductive stabilizers. The methods involve abelianization techniques and arithmetic duality theorems for complexes of tori over K.

p进GKZ超几何D-模与复形的比较定理

A Comparison Theorem for $p$-adic GKZ Hypergeometric $\mathcal D$-Modules and Complexes

本文比较p进GKZ超几何复形与超几何算术D-模，在特定条件下证明比较态射为拟同构，还建立复形的次数集中性、全纯性，构造Frobenius结构并证明其限制为特定过收敛F-等晶体。

T-model structures on chain complexes of presheaves

A tensor model structure is constructed on the category of chain complexes of presheaves of R-modules for a sheaf of rings R in a Grothendieck topos. If the topos has enough points, then the homotopy category is equivalent to the derived category. Some t-structures, generalizing the perverse t-structures, are constructed on the derived category.

有限域上的算术傅里叶变换：一般性消没、卷积与等分布

Arithmetic Fourier transforms over finite fields: generic vanishing, convolution, and equidistribution

本文研究一般连通交换代数群上迹函数的算术傅里叶变换，通过证明通用消没定理并构造Tannakian范畴，利用Deligne的黎曼假设导出等分布定理，推广了Katz的工作，并给出应用实例与开放问题。

相对 Du Bois 复形的 Hodge 模构造

A Hodge module construction of relative Du Bois complexes

本文通过混合 Hodge 模和 Grothendieck 对偶构造相对 Du Bois 复形，并建立有限滤过及纤维限制与符号 m-Du Bois 奇点的联系。

Stability of holonomicity over quasi-projective varieties

Let $\V$ be a mixed characteristic complete discrete valuation ring with perfect residue field $k$. We solve Berthelot's conjectures on the stability of the holonomicity over smooth projective formal $\V$-schemes. Then we build a category of complexes of arithmetic $\D$-modules over quasi-projective $k$-varieties with bounded, $F$-holonomic cohomology. We get its stability under Grothendieck's six operations.

The homotopy category of flat functors

Let C be a small category and G be a tensor Grothendieck category. We define a notion of atness in the category Fun(C; G) of all covariant functors from C to G and show that the inclusion K(FlatA) ---> K(A) has a right adjoint where K(A) is the homotopy category of A and K(FlatA) its subcategory consisting of complexes of at functors. In addition, we find a replacement for the quotient Dpac(FlatA) = K(FlatA)Kp(FlatA) of triangulated categories where Kp(FlatA) is the homotopy category of all pure acyclic complexes of at functors.

具有大（反）典范丛的代数簇的导出范畴中的张量三角范畴结构

Tensor triangulated category structures in the derived category of a variety with big (anti-)canonical bundle

针对具有大（反）典范丛的复光滑射影代数簇$X$，研究其完美复形范畴$Perf(X)$上的所有可能张量三角范畴结构，并证明了Bondal-Orlov重构定理的幺半版本。

On the Structure of Double Complexes

We study consequences and applications of the folklore statement that every double complex over a field decomposes into so-called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences easy to understand. We describe a notion of `universal' quasi-isomorphism, investigate the behaviour of the decomposition under tensor product and compute the Grothendieck ring of the category of bounded double complexes over a field with finite cohomologies up to such quasi-isomorphism (and some variants). Applying the theory to the double complexes of smo…