Openness of various loci over Noetherian rings

Open loci results for commutative DG-rings

Given a commutative noetherian non-positive DG-ring with bounded cohomology which has a dualizing DG-module, we study its regular, Gorenstein and Cohen-Macaulay loci. We give a sufficient condition for the regular locus to be open, and show that the Gorenstein locus is always open. However, both of these loci are often empty: we show that no matter how nice $\mathrm{H}^0(A)$ is, there are examples where the Gorenstein locus of $A$ is empty. We then show that the Cohen-Macaulay locus of a commutative noetherian DG-ring with bounded cohomology which has a dualizing DG-module always contains a de…

有界余维的完全交：Nagata准则的失效与开性的恢复

Complete Intersections of Bounded Codimension: Failure of the Nagata Criterion and Recovery of Openness

本文证明余维至多为$c$的完全交性质不满足Nagata准则，构造反例并给出NC的刻画条件，恢复其轨迹开性，还证明相关子集在拟紧优概形上可构造。

valuation环上的正则环

Regular rings over valuation rings

本文探讨了估值环上本质有限生成局部环的正则性性质，研究了开放位、完美大Cohen-Macaulay代数和切线复形等经典与现代主题，并给出了诺特环上的柯代瓦消失定理的应用。

Tame Loci of Certain Local Cohomology Modules

Let $M$ be a finitely generated graded module over a Noetherian homogeneous ring $R = \bigoplus_{n \in \mathbb{N}_0}R_n$. For each $i \in \mathbb{N}_0$ let $H^i_{R_{+}}(M)$ denote the $i$-th local cohomology module of $M$ with respect to the irrelevant ideal $R_+ = \bigoplus_{n > 0} R_n$ of $R$, furnished with its natural grading. We study the tame loci $\ft^i(M)^{\leq 3}$ at level $i \in \mathbb{N}_0$ in codimension $\leq 3$ of $M$, that is the sets of all primes $\fp_0 \subset R_0$ of height $\leq 3$ such that the graded $R_{\fp_0}$-modules $H^i_{R_{+}}(M)_{\fp_0}$ are tame.

Open loci of graded modules

Let $A=\oplus_{i\in \nn}A_i$ be an excellent homogeneous Noetherian graded ring and let $M=\oplus_{n\in \zz}M_n$ be a finitely generated graded $A$-module. We consider $M$ as a module over $A_0$ and show that the $(S_k)$-loci of $M$ are open in $\Spec(A_0)$. In particular, the Cohen-Macaulay locus $U^0_{CM}=\{\p\in \Spec(A_0) \mid M_\p {is Cohen-Macaulay}\}$ is an open subset of $\Spec(A_0)$. We also show that the $(S_k)$-loci on the homogeneous parts $M_n$ of $M$ are eventually stable. As an application we obtain that for a finitely generated Cohen-Macaulay module $M$ over an excellent ring $…

拟优良诺特概形上的正则性有界

Regularity is bounded on a quasi-excellent Noetherian scheme

本文证明拟优良诺特概形上切锥的梯度Betti表只有有限种可能，从而正则性有界。

On liftable and weakly liftable modules

Let $T$ be a Noetherian ring and $f$ a nonzerodivisor on $T$. We study concrete necessary and sufficient conditions for a module over $R=T/(f)$ to be weakly liftable to $T$, in the sense of Auslander, Ding and Solberg. We focus on cyclic modules and get various positive and negative results on the lifting and weak lifting problems. For a module over $T$ we define the loci for certain properties: liftable, weakly liftable, having finite projective dimension and study their relationships.

Jordan块、微分迹与Noether商环上的余理想非包含关系

Jordan Blocks, Differential Traces, and Colon-Ideal Noncontainment over Noetherian Quotients

本文在Noether商环上证明幂零微分为零元素的余理想非包含定理，推广至孤立及非孤立临界轨迹，并构造反例否证积分闭包加强。

$F$-幂零环与持久性质

$F$-nilpotent rings and permanence properties

本文研究$F$-幂零、弱$F$-幂零及广义弱$F$-幂零奇点类在忠实平坦局部环映射下的持久性，并证明在$F$-有限或优局部环上本质上有限型的诺特环中，相应素理想轨迹关于扎里斯基拓扑是开集。

DG奇异等价与奇异轨迹

DG singular equivalence and singular locus

本文从DG奇异范畴构造仿射概形，证明其到奇异轨迹的有限满射，并得出等价DG奇异范畴的环具有相同维数的奇异轨迹。