A Question about Pic(X) as a G-module

Extended equivariant Picard complexes and homogeneous spaces

Let k be a field of characteristic 0 and let X be a smooth geometrically integral k-variety. In our previous paper we defined the extended Picard complex UPic(X) as a certain complex of Galois modules in degrees 0 and 1. We computed the isomorphism class of UPic(G) in the derived category of Galois modules for a connected linear k-group G. In this paper we assume that X is a homogeneous space of a connected linear k-group G with geometric stabilizer H. We compute the isomorphism class of UPic(X) in the derived category of Galois modules in terms of the character groups of G and H. The proof is…

Finite dual of a cocommutative Hopf algebroid. Application to linear differential matrix equations and Picard-Vessiot theory

A fundamental tool of Differential Galois Theory is the assignment of an algebraic group to each finite-dimensional differential module over differential field in such a way that the category of differential modules it generates is equivalent, as a symmetric monoidal category, to the category of representations of the group. Its underlying set is then recognized as the group of differential automorphisms of the Picard-Vessiot field extension of the base field for this differential module. These results can be obtained by means of a Tannaka reconstruction process, applied to the abelian categor…

Derived equivalences of gerbey curves

We study derived equivalences of certain stacks over genus $1$ curves, which arise as connected components of the Picard stack of a genus $1$ curve. To this end, we develop a theory of integral transforms for these algebraic stacks. We use this theory to answer the question of when two stacky genus $1$ curves are derived equivalent. We use integral transforms and intersection theory on stacks to answer the following questions: if $C'=Pic^d(C)$, is $C=Pic^f(C')$ for some integer $f$? If $C'=Pic^d(C)$ and $C''=Pic^f(C')$, then is $C''=Pic^g(C)$ for some integer $g$?

Lines generate the Picard groups of certain Fermat surfaces

We answer a question of T.Shioda and show that, for any positive integer $m$ prime to 6, the Picard group of the Fermat surface $Φ_m$ is generated by the classes of lines contained in $Φ_m$.

半环上模理论的各个侧面

Facets of module theory over semirings

本文建立了半环上的基本模理论，证明了线丛的通常定义在半环上保持一致，并指出数域的窄类群可由其全非负代数整数子半环的反身Picard群恢复。

Cohen-Macaulay modules over the algebra of planar quasi-invariants and Calogero-Moser systems

In this paper, we study properties of the algebras of planar quasi-invariants. These algebras are Cohen-Macaulay and Gorenstein in codimension one. Using the technique of matrix problems, we classify all Cohen-Macaulay modules of rank one over them and determine their Picard groups. In terms of this classification, we describe the spectral modules of the planar rational Calogero-Moser systems. Finally, we elaborate the theory of the algebraic inverse scattering method, computing a new unexpected explicit example of a deformed Calogero-Moser system.