Francois Couchot
It is proven that each commutative arithmetical ring $R$ has a finitistic weak dimension $\leq 2$. More precisely, this dimension is 0 if $R$ is locally IF, 1 if $R$ is locally semicoherent and not IF, and 2 in the other cases.
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Finitistic injective dimension exceeding finitistic projective dimension for a commutative ring
构造一个交换环，其大有限射影维数为1而内射维数为2，反驳了Bass关于交换环维数比较的猜想。
The small finitistic dimensions of commutative rings, II
本文研究多项式环、形式幂级数环等四类环构造的小有限维数，证明环的小有限维数不超过其Krull维数，并给出一个具有无限小有限维数的全商环实例。
Let $R$ be a commutative ring with identity. The small finitistic dimension $\fPD(R)$ of $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we characterize a ring $R$ with $\fPD(R)\leq n$ using finitely generated semi-regular ideals, tilting modules, cotilting modules of cofinite type or vaguely associated prime ideals. As an application, we obtain that if $R$ is a Noetherian ring, then $\fPD(R)= \sup\{\grade(\m,R)|\m\in \Max(R)\}$ where $\grade(\m,R)$ is the grade of $\m$ on $R$ . We also show that a ring $R$ satisfies …
The small finitistic dimensions of commutative rings, III
本文研究了交换环的小有限维数，证明了其与有限生成理想和Ext的零性之间的关系，并应用于多种环类。
We introduce the finitistic extension degree of a ring and investigate rings for which it is finite. The Auslander-Reiten Conjecture is proved for rings of finite finitistic extension degree and these rings are also shown to have finite finitistic dimension. We apply these results to better understand a gen- eralized version of the Auslander-Reiten Condition for Gorenstein rings. We also record how the finitistic extension degree behaves with respect to many change of ring procedures that arise frequently in the commutative setting.
Finitistic dimensions over commutative DG-rings
本文研究有限振幅交换诺特非正DG-环的有限维数，证明有限平坦维数DG-模的投射维数上界，构造指定投射维数的DG-模并给出大有限投射维数的上下界及可达性，还得到同调光滑代数导出Hochschild（上）同调的新零化结果。
Change-of-Rings Theorems for the Small Finitistic Dimension
本文从有限制平坦同调代数角度研究交换环的小有限维数，探讨了有限制平坦维数的基本性质，并建立了环变换定理，包括商扩张和多项式扩展结果，以及局部不等式。
The finitistic weak dimensions of Gaussian rings
该论文证明每个交换高斯环的有限弱维数至多为二，通过构造矛盾论证，不假设弱整体维数有限，主要贡献是给出了这一维数上界的完整证明。
Finitistic dimensions of formal power series rings
该研究探讨交换环及其形式幂级数环的大有限投射与平坦维数的关系，明确诺特环时维数增1，给出各类维数对的实现情况及完全系数环的二分结论。
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…
Transfer Krull monoids are a recent concept including all commutative Krull domains and also, for example, wide classes of non-commutative Dedekind domains. We show that transfer Krull monoids are fully elastic (i.e., every rational number between $1$ and the elasticity of the monoid can be realized as the elasticity of an element). In commutative Krull monoids which have sufficiently many prime divisors in all classes of their class group, the set of catenary degrees and the set of tame degrees are intervals. Without the assumption on the distribution of prime divisors, arbitrary finite sets …
Goldman primes and higher cyclic presentations for the small finitistic dimension
本文研究交换环的小有限维数，通过 Koszul 级数逃脱同调检验的两种方式，给出多项式扩张的公式，并构造例子说明循环条件仅在呈现层级一上控制该维数。
A definition of quasi-flat left module is proposed and it is shown that any left module which is either quasi-projective or flat is quasi-flat. A characterization of local commutative rings for which each ideal is quasi-flat (resp. quasi-projective) is given. It is also proven that each commutative ring R whose finitely generated ideals are quasi-flat is of $λ$-dimension $\le$ 3, and this dimension $\le$ 2 if R is local. This extends a former result about the class of arithmetical rings. Moreover, if R has a unique minimal prime ideal then its finitely generated ideals are quasi-projective if …
We compute the Gorenstein weak dimension of a coherent power series rings over a commutative rings and we show that, in general, $\gwd(R) \leq 1$ does not imply that $R$ is an arithmetical ring.
This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. Section 2 provides some preliminaries on quasi-projective modules over commutative rings. Section 3 investigates the correlation with well-known Prufer conditions; namely, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky's theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni-Glaz's related conjecture on Gaussian rings. We also e…
Formalization in Lean of faithfully flat descent of projectivity
本文在Lean中形式化了项目性忠实平坦降级的理论，修正了Raynaud和Gruson的经典结果，并为研究交换诺特环的有限制维度提供了关键工具。
G-dimensions for DG-modules over commutative DG-rings
本文在非正分次交换诺特DG环上定义并研究了DG模的G-维数，通过Minamoto引入的投射分解的DG版本给出了G-维数有限性的判据，并证明了G-维数的有限性刻画了环的局部Gorenstein性质，进而应用于建立与little finitistic维数的联系、刻画Cohen-Macaulay和Gorenstein DG环以及推广经典的Buchweitz-Happel定理及其逆定理。
We introduce in this work, the class of commutative rings whose lattice of ideals forms an MTL-algebra which is not necessary a BL-algebra. The so-called class of rings will be named MTL-rings. We prove that a local commutative ring with identity is an MTL-ring if and only if it is an arithmetical ring. It is shown that a noetherian commutative ring R with an identity is an MTL-ring if and only if ideals of the localization RM at a maximal ideal M are totally ordered by the set inclusion. Remarking that noetherian MTL-rings are again BL-rings, we work outside of the noetherian case by consider…
Over a commutative noetherian ring $R$ of finite Krull dimension, we show that every complex of flat cotorsion $R$-modules decomposes as a direct sum of a minimal complex and a contractible complex. Moreover, we define the notion of a semi-flat-cotorsion complex as a special type of semi-flat complex, and provide functorial ways to construct a quasi-isomorphism from a semi-flat complex to a semi-flat-cotorsion complex. Consequently, every $R$-complex can be replaced by a minimal semi-flat-cotorsion complex in the derived category over $R$. Furthermore, we describe structure of semi-flat-cotors…
It is shown that a ring is left semihereditary if and only each homomorphic image of its injective hull as left module is FP-injective. It is also proven that a commutative ring R is reduced and arithmetical if and only if E/U if FP-injective for any FP-injective R-module E and for any submodule U of finite Goldie dimension. A characterization of commutative rings for which each module of finite Goldie dimension is of injective dimension at most one is given. Let R be a chain ring and Z its subset of zerodivisors. It is proven that E/U is FP-injective for each FP-injective R-module E and each …
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