Finiteness results concerning algebraic power series

Finiteness properties of affine difference algebraic groups

We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the matrix entries can indeed be defined by finitely many such equations. As an application, we show that the difference ideal of all difference algebraic relations among the solutions of a linear differential equation is finitely generated.

Finiteness in derived categories of local rings

New homotopy invariant finiteness conditions on modules over commutative rings are introduced, and their properties are studied systematically. A number of finiteness results for classical homological invariants like flat dimension, injective dimension, and Gorenstein dimension, are established. It is proved that these specialize to give results concerning modules over complete intersection local rings. A noteworthy feature is the use of techniques based on thick subcategories of derived categories.

有限类型代数可积导出模的跃迁有限性

Finiteness of leaps of modules of integrable derivations of algebras of finite type

本文证明了有限类型代数可积导出模的跃迁有限性，并建立了无限可积导出模的紧致性。

Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals

We study chains of lattice ideals that are invariant under a symmetric group action. In our setting, the ambient rings for these ideals are polynomial rings which are increasing in (Krull) dimension. Thus, these chains will fail to stabilize in the traditional commutative algebra sense. However, we prove a theorem which says that "up to the action of the group", these chains locally stabilize. We also give an algorithm, which we have implemented in software, for explicitly constructing these stabilization generators for a family of Laurent toric ideals involved in applications to algebraic sta…

论无限对应的有限轨道

On finite orbits of infinite correspondences

本文整理代数对应的相关结果并推广到射影直线上的对应，重点研究有限轨道，并给出有限轨道数目无穷性的域论刻画。

On Gorenstein Projective, Injective and Flat Dimensions - A Functorial Description with Applications

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions do not immediately come with practical and robust criteria for finiteness, not even over commutative noetherian local rings. In this paper we enlarge the class of rings known to admit good criteria for finiteness of Gorenstein dimensions: It now includes, for instance, the rings encountered in commutative algebraic geometry and, in the…