D. Bennis, N. Mahdou
A ring is called $n$-perfect ($n\geq 0$), if every flat module has projective dimension less or equal than $n$. In this paper, we show that the $n$-perfectness relate, via homological approach, some homological dimension of rings. We study $n$-perfectness in some known ring constructions. Finally, several examples of $n$-perfect rings satisfying special conditions are given.
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Cotorsion pairs and Tor-pairs over commutative noetherian rings
本文对交换诺特环上的遗传余挠对进行分类，利用环谱上的整值函数和局部深度条件描述左类，并通过对偶对应到遗传 Tor-对。
Let R be a ring (not necessarily commutative). A left R-module is said to be cotorsion if Ext 1 R (G, M) = 0 for any flat R-module G. It is well known that each pure-injective left R-module is cotorsion, but the converse does not hold: for instance, if R is left perfect but not left pure-semisimple then each left R-module is cotorsion but there exist non-pure-injective left modules. The aim of this paper is to describe the class C of commutative rings R for which each cotorsion R-module is pure-injective. It is easy to see that C contains the class of von Neumann regular rings and the one of p…
In this paper, we establish, as a generalization of a result on the classical homological dimensions of commutative rings, an upper bound on the Gorenstein global dimension of commutative rings using the global cotorsion dimension of rings. We use this result to compute the Gorenstein global dimension of some particular cases of trivial extensions of rings and of group rings.
Let R be a ring and n,k be two non-negative integers. As an extension of several known notions, we introduce and study (n,k)-weak cotorsion modules using the class of right R-modules with n-weak flat dimensions at most k. Various examples and applications are also given.
Balanced pairs, virtually Gorenstein rings, and cotorsion torsion triples
本文研究了环R中平衡对与余 torsion 三元组的关系，证明了右几乎格罗蒙德环的条件，并通过无限维翻转理论分类了所有余 torsion 三元组。
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…
We are interested in characterising the commutative rings for which a $1$-tilting cotorsion pair $(\mathcal{A}, \mathcal{T})$ provides for covers, that is when the class $\mathcal{A}$ is a covering class. We use Hrbek's bijective correspondence between the $1$-tilting cotorsion pairs over a commutative ring $R$ and the faithful finitely generated Gabriel topologies on $R$. Moreover, we use results of Bazzoni-Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for $1$-tilting cotorsion pairs arising from flat injective ring epimorphis…
For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a consequence, we show that pointwise Matlis duality gives a bijective correspondence between the isoclasses of indecomposable injective left modules and the isoclasses of indecomposable flat cotorsion right modules. This correspondence is an explicit realization of Herzog's homeomorphism induced from elementary duality of Ziegler spectra.
Cotorsion modules relative to a hereditary torsion theory
本文针对遗传挠理论τ_w，引入Tor对、余挠对及强平坦模，以此刻画完全环等多种已知环类。
We describe a general correspondence between injective (resp. projective) recollements of triangulated categories and injective (resp. projective) cotorsion pairs. This provides a model category description of these recollement situations. Our applications focus on displaying several recollements that glue together various full subcategories of K(R), the homotopy category of chain complexes of modules over a general ring R. When R is (left) Noetherian ring, these recollements involve complexes built from the Gorenstein injective modules. When R is a (left) coherent ring for which all flat modu…
S-flat cotorsion pair
本文研究了交换环乘法子集下的S-余挠模，证明了S-平坦模与S-余挠模构成遗传完美余挠对，并以此刻画S-完美环及S-凝聚环的预包络性质。
Cotorsion pairs, thick subcategories, and finitely generated Gorenstein projective modules
在Cohen-Macaulay环上的诺特代数上，证明有限生成Gorenstein投射模构成遗传cotorsion对的左半部分，并回答Takahashi的问题，刻画左弱Gorenstein性质。
Rigidity of Ext and Tor via flat-cotorsion theory
本文利用 Nakamura 和 Thompson 建立的极小半平坦-余挠替换，将 Ext 刚性结果中 n 的下界降至 dim R，并改进 Tor-刚性结论。
The strongly flat dimension of modules and rings
本文引入模与环的S-强平坦维数，度量模偏离S-强平坦及环偏离S-几乎半单的程度，并探讨其与其他维数的关系。
Restricted injective dimensions over Cohen-Macaulay rings
本文证明有限 Krull 维数 Cohen-Macaulay 环上大小受限内射维数一致，并借助倾斜理论提出 Cohen-Macaulay Hom 内射维数新定义，证明其构成完美余挠对的右半部分。
Let $R$ be a ring, and $n$ a fixed nonnegative integer. An $R$-module $W$ is called $L_{n}$-injective if ${\rm Ext}_{R}^{1}(M,W)=0$ for any $R$-module $M$ with flat dimension at most $n$. In this paper, we prove first that ($\mathcal{F}_{n},\mathcal{L}_{n}$) is a complete hereditary cotorsion theory, where $\mathcal{F}_n$ (resp. $\mathcal{L}_n$) denotes the class of all $R$-modules with flat dimension at most $n$ (resp. $L_{n}$-injective $R$-modules). Then we introduce the $L_{n}$-injective dimension of a module and $L_n$-global dimension of a ring. Finally, over rings with weak global dimensi…
Given a multiplicative subset $S$ in a commutative ring $R$, we consider $S$-weakly cotorsion and $S$-strongly flat $R$-modules, and show that all $R$-modules have $S$-strongly flat covers if and only if all flat $R$-modules are $S$-strongly flat. These equivalent conditions hold if and only if the localization $R_S$ is a perfect ring and, for every element $s\in S$, the quotient ring $R/sR$ is a perfect ring, too. The multiplicative subset $S\subset R$ is allowed to contain zero-divisors.
A note on Matlis localizations
本文研究了交换环上的$S$-Matlis环，通过$S$-强平坦模、$S$-弱余挠模和$S$-$h$-可除模给出了该类环的若干新刻画。
When every S-flat module is (flat) projective
本文研究含单位元交换环中S-平坦模均为平坦或投射模的环类，给出两类环的刻画，定义S-完全环，举例区分三类环并研究S-完全性的环构造传递性。
In this paper we introduce the notion of "strong $n$-perfect rings" which is in some way a generalization of the notion of "$n$-perfect rings". We are mainly concerned with those class of rings in the context of pullbacks. Also we exhibit a class of $n$-perfect rings that are not strong $n$-perfect rings. Finally, we establish the transfer of this notion to the direct product notions.
Given a $1$-tilting cotorsion pair over a commutative ring, we characterise the rings over which the $1$-tilting class is an enveloping class. To do so, we consider the faithful finitely generated Gabriel topology $\mathcal{G}$ associated to the $1$-tilting class $\mathcal{T}$ over a commutative ring as illustrated by Hrbek. We prove that a $1$-tilting class $\mathcal{T}$ is enveloping if and only if $ \mathcal{G}$ is a perfect Gabriel topology (that is, it arises from a perfect localisation) and $R/J$ is a perfect ring for each $J \in \mathcal{G}$, or equivalently $\mathcal{G}$ is a perfect G…
In this paper, we introduce the notion of "(n,d)-perfect rings" which is in some way a generalization of the notion of "S-rings". After we give some basic results of this rings and we survey the relationship between "A(n) property" and "(n,d)-perfect property". Finally, we investigate the "(n,d)-perfect property" in pullback rings.
Let $R$ be a commutative ring with unity not equal to zero and let $Γ(R)$ be a zero-divisor graph realized by $R$. For a simple, undirected, connected graph $G = (V, E)$, a {\it total perfect code} denoted by $C(G)$ in $G$ is a subset $C(G) \subseteq V(G)$ such that $|N(v) \cap C(G)| = 1$ for all $v \in V(G)$, where $N(v)$ denotes the open neighbourhood of a vertex $v$ in $G$. In this paper, we study total perfect codes in graphs which are realized as zero-divisor graphs. We show a zero-divisor graph realized by a local commutative ring with unity admits a total perfect code if and only if the…
We prove a $p$-adic analog of Kunz's theorem: a $p$-adically complete noetherian ring is regular exactly when it admits a faithfully flat map to a perfectoid ring. This result is deduced from a more precise statement on detecting finiteness of projective dimension of finitely generated modules over noetherian rings via maps to perfectoid rings. We also establish a version of the $p$-adic Kunz's theorem where the flatness hypothesis is relaxed to almost flatness.
Relative Inverse Limit Perfection of Derived Commutative Rings
该文研究正特征域上导出交换环的相对Frobenius映射，构造相对逆极限完备化，探究动化环的半完备性、F-有限性相对版本，证明F-有限动化环映射分解及Noetherian F-有限环态射余切复形零化与相对完备性的关系。
In this paper, we introduce and study the notion of linkage by perfect modules, which we call perfect linkage, for Cohen-Macaulay modules over Cohen--Macaulay local rings. We explore perfect linkage in connection with syzygies, maximal Cohen-Macaulay approximations and Yoshino-Isogawa linkage. We recover a theorem of Yoshino and Isogawa, and analyze the structure of double perfect linkage. Moreover, we establish a criterion for two Cohen-Macaulay modules of codimension one to be perfectly linked, and apply it to the classical linkage theory for ideals. We also construct various examples of lin…
Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, by using the classes $ \mathcal{P}_C $ and $ \mathcal{I}_C $, we extend the notions of perfect and coperfect modules introduced by D.Rees \cite{R} and O.Jenda \cite{J1}. First, we study the basic properties of these modules and relations between them. Next, we characterize local rings in terms of the existence of special perfect (resp. coperfect) modules.
Let S be a polynomial algebra over a field. If I is the edge ideal of a perfect semiregular tree, then we give precise formulas for values of depth, Stanley depth, projective dimension, regularity and Krull dimension of S/I.
On the Witt vectors of perfect rings in positive characteristic
本文研究了正特征下完美环的Witt向量性质，证明在特定条件下，完美环的Witt向量环具有积分闭包性质。
We investigate homological properties of perfect algebras of prime characteristic. The principle is as follows: perfect algebras resolve the singularities. For example, we show any module over the ring of absolute integral closure has finite flat dimension. Under some mild conditions, we show any module over that ring has finite projective dimension. We compute weak dimension and global dimension of perfect rings in a series of nontrivial cases. Some interesting applications are given. In particular, we answer some questions asked by Shimomoto.
We give necessary and sufficient conditions for strong Koszulness of toric rings associated with the stable set polytope of graphs.
Let $k$ be a perfect field of characteristic $p>0$, $k(t)_{per}$ the perfect closure of $k(t)$ and $A$ a $k$-algebra. We characterize whether the ring $A\otimes_k k(t)_{per}$ is noetherian or not. As a consequence, we prove that the ring $A\otimes_k k(t)_{per}$ is noetherian when $A$ is the ring of formal power series in $n$ indeterminates over $k$.
We consider the tensor product of modules over the polynomial algebra corresponding to the usual tensor product of linear operators. We present a general description of the representation ring in case the ground field k is perfect. It is made explicit in the special cases when k is real closed respectively algebraically closed. Furthermore, we discuss the generalisation of this problem to representations of quivers. In particular the representation ring of quivers of extended Dynkin type A is provided.
A mixed characteristic analogue of the perfection of rings and its almost Cohen-Macaulay property
本文在混合特征完备局部整环上构造了一个类似正特征完美闭包的 perfectoid 环，并利用 André 平坦性引理和 Riemann 延拓定理证明了其几乎 Cohen-Macaulay 性质。
The Rouquier dimension of the category of perfect complexes over a regular ring
本文证明了正则环上完美复形范畴的 Rouquier 维数恰好等于环的 Krull 维数，并指出正则序列的最大长度为其提供下界，解决了正则局部环情形下下界未知的问题。
In this paper, we give a criterion of the nearly Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ with connected components $G^{(1)}, \ldots, G^{(\ell)}$ is nearly Gorenstein if and only if (1) for each $i$, the Ehrhart ring of the stable set polytope of $G^{(i)}$ is Gorenstein and (2) $|ω(G^{(i)})-ω(G^{(j)})|\leq 1$ for any $i$ and $j$, where $ω(G^{(i)})$ is the clique number of $G^{(i)}$. We also show that the Segre product of Cohen-Macaulay graded rings with linear non-zerodivisor …
Valuation rings and perfectoid rings are examples of (usually non-noetherian) rings that behave in some sense like regular rings. We give and study an extension of the concept of regular local rings to non-noetherian rings so that it includes valuation and perfectoid rings and it is related to Grothendieck's definition of formal smoothness as in the noetherian case. For that, we have to take into account the topologies. We prove a descent theorem for regularity along flat homomorphisms (in fact for homomorphisms of finite flat dimension), extending some known results from the noetherian to the…
δ-rings, perfectoid towers, and lim Cohen-Macaulay sequences
本文研究正特征或混合特征Noetherian环上δ-环、完美塔和极限Cohen-Macaulay序列的基本结构及其相互关系，并讨论完美塔的形变及构造方法。
Let $R$ be a polynomial ring and $I \subset R$ be a perfect ideal of height two minimally generated by forms of the same degree. We provide a formula for the multiplicity of the saturated special fiber ring of $I$. Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of $I$. Applying ideas introduced in arXiv:1805.05180, we obtain the value of the j-multiplicity of $I$ and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of $I$.
In this paper, we give a criterion of the Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ is Gorenstein if and only if (1) sizes of maximal cliques are constant (say $n$) and (2) (a) $n=1$, (b) $n=2$ and there is no odd cycle without chord and length at least 7 or (c) $n\geq 3$ and there is no odd cycle without chord and length at least 5.
Let $K$ be a Gorenstein noetherian ring of finite Krull dimension, and consider the category of cohomologically noetherian commutative differential graded rings $A$ over $K$, such that $H^0(A)$ is essentially of finite type over $K$, and $A$ has finite flat dimension over $K$. We extend Grothendieck's twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup. We prove functoriality results with respect to cohomologically finite and cohomologically essentially smooth maps, and prove a perfect base change result for $f^{!}$ in this…
While every grade 2 perfect ideal in a regular local ring is linked to a complete intersection ideal, it is known not to be the case for ideals of grade 3. We soften the blow by proving that every grade 3 perfect ideal in a regular local ring is linked to a complete intersection or a Golod ideal. Our proof is indebted to a homological classification of Cohen-Macaulay local rings of codimension 3. That debt is swiftly repaid, as we use linkage to reveal some of the finer structures of this classification.
A characterization of Cohen-Macaulay rings in terms of levels of perfect complexes
本文研究了以Gorenstein C-投射模为子类的复叠层水平，通过完美复叠层的有限性表征Cohen-Macaulay环，恢复了Christensen等人关于R的Gorenstein性的最新定理。
F-coherent rings with applications to tight closure theory
本文引入正特征诺特环的新类F-凝聚环（完全闭包为凝聚的诺特环），研究其性质，并结合紧闭包理论给出应用，讨论了F-凝聚环与几类F-环的关系及凝聚性对一般完全环紧闭包行为的影响。
The non-Gorenstein locus of stable set rings of finite simple perfect graphs is studied. We describe combinatorially those perfect graphs whose stable set rings are Gorenstein on the punctured spectrum. In addition, we show that, in general, for Cohen--Macaulay graded algebras, their Cohen--Macaulay type and residue are largely independent.
Local and category-theoretical entropies associated with an endomorphism of finite length (i.e., with zero-dimensional closed fiber) of a commutative Noetherian local ring are compared. Local entropy is shown to be less than or equal to category-theoretical entropy. The two entropies are shown to be equal when the ring is regular, and also for the Frobenius endomorphism of a complete local ring of positive characteristic. Furthermore, given a flat morphism of Cohen-Macaulay local rings endowed with compatible endomorphisms of finite length, it is shown that local entropy is "additive". Finally…
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