Modules whose finiteness dimensions coincide with their cohomological dimensions

局部上同调维数的拓扑计算

Topological calculation of local cohomological dimension

本文证明了复流形中闭解析子空间的局部上同调维数与环境流形维数之和等于环境流形维数，并通过链接同调计算局部上同调维数，并推论出拟射影簇的局部上同调维数由一般超平面截面及0维分层链接同调决定。

A uniform proof of the finiteness of the class group of a global field

We give a definition of a class of Dedekind domains which includes the rings of integers of global fields and give a proof that all rings in this class have finite ideal class group. We also prove that this class coincides with the class of rings of integers of global fields.

Cartier Crystals

Building on our previous work "Cartier modules: finiteness results" we start in this manuscript an in depth study of the derived category of Cartier modules and the cohomological operations which are defined on them. After localizing at the sub-category of locally nilpotent objects we show that for a morphism essentially of finite type $f$ the operations $Rf_*$ and $f^!$ are defined for Cartier crystals. We show that, if $f$ is of finite type (but not necessarily proper) $Rf_*$ preserves coherent cohomology (up to nilpotence) and that $f^!$ has bounded cohomological dimension. In a sequel we w…

On the generalized multiplicities of maximal minors and sub-maximal pfaffians

Let $S=\mathbb{C}[x_{ij}]$ be a polynomial ring of $m\times n$ generic variables (resp. a polynomial ring of $(2n+1) \times (2n+1)$ skew-symmetric variables) over $\mathbb{C}$ and let $I$ (resp. Pf) be the determinantal ideal of maximal minors (resp. sub-maximal pfaffians) of $S$. Using the representation theoretic techniques introduced in the work of Raicu et al, we study the asymptotic behavior of the length of the local cohomology module of determinantal and pfaffian thickenings for suitable choices of cohomological degrees. This asymptotic behavior is also defined as a notion of multiplict…

通过消除上同调来生成奇点范畴和无限射影维数定位

Generation of singularity categories and infinite injective dimension locus via annihilation of cohomologies

本文研究了奇点范畴的强生成与上同调消 annihilator 的关系，证明了在 Krull 维数至多为一的 Noetherian 域中，奇点范畴具有强生成当且仅当其上同调消 annihilator 非零，并引入了 co-cohomological annihilator 的概念。

约化模与余约化模的应用 III：函子的同调性质与凝聚性

Applications of reduced and coreduced modules III: homological properties and coherence of functors

本文研究 I-约化与 I-余约化 R-模在局部上同调、局部同调及相关函子凝聚性中的应用，并建立理想维数与 R/I 投射或平坦维数的等式。

On determinantal ideals and algebraic dependence

Let $X$ be a matrix with entries in a polynomial ring over an algebraically closed field $K$. We prove that, if the entries of $X$ outside some $(t \times t)$-submatrix are algebraically dependent over $K$, the arithmetical rank of the ideal $I_t(X)$ of $t$-minors of $X$ drops at least by one with respect to the generic case; under suitable assumptions, it drops at least by $k$ if $X$ has $k$ zero entries. This upper bound turns out to be sharp if $\mathrm{char}\, K=0$, since it then coincides with the lower bound provided by the local cohomological dimension.

Connectedness of Hilbert scheme strata defined by bounding cohomology

Let Hilb^p be the Hilbert scheme parametrizing the closed subschemes of P^n with Hilbert polynomial p \in Q[t] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilb^p we define locally closed subspaces of the Hilbert scheme. The aim of this thesis is to show that some of these subspaces are connected. For this we exploit the ideals constructed by D. Mall. It turns out that these ideals are sequentially Cohen-Macaulay and that their initial ideals and their generic initial ideals coincide for any admissible term order.

The depth, the delooping level and the finitistic dimension

We investigate two invariants of Noetherian semiperfect rings, namely the depth and a new invariant we call the "delooping level". These give lower and upper bounds for the finitistic dimension, respectively. As first theorems, we give a necessary and sufficient criterion for the delooping level to be finite in terms of the splitting of the unit map of a related adjunction, and use this to give a sufficient torsionfreeness criterion for finiteness. We further relate these invariants to the Auslander-Bridger grade conditions for modules, which are vanishing conditions on double Ext duals. As ma…