Commutative Noetherian Ring Research Articles

Subrings of polynomial rings and the conjectures of Eisenbud and Evans

Let R be a commutative Noetherian ring of dimension d. In 1973, Eisenbud and Evans proposed three conjectures on the polynomial ring R[T]. These conjectures were settled in the affirmative by Sathaye, Mohan Kumar and Plumstead. One of the primary objectives of this article is to investigate the validity of these conjectures over Noetherian subrings of R[T] of dimension d+1, containing R. We formulate a class of such rings, which includes polynomial rings, Rees algebras, Rees-like algebras and Noetherian symbolic Rees algebras, and exhibit that all three conjectures hold for rings belonging to this class.

Categorical properties of reduction functors over non-positive DG-rings

Given a non-positive DG-ring A A , associated to it are the reduction and coreduction functors F ( − ) = H 0 ( A ) ⊗ A L − F(-) = \mathrm {H}^0(A)\otimes ^{\mathrm {L}}_A - and G ( − ) = R Hom A ⁡ ( H 0 ( A ) , − ) G(-) = \mathrm {R}\operatorname {Hom}_A(\mathrm {H}^0(A),-) , considered as functors D ⁡ ( A ) → D ⁡ ( H 0 ( A ) ) \operatorname {\mathsf {D}}(A) \to \operatorname {\mathsf {D}}(\mathrm {H}^0(A)) , as well as the forgetful functor S : D ⁡ ( H 0 ( A ) ) → D ⁡ ( A ) S:\operatorname {\mathsf {D}}(\mathrm {H}^0(A)) \to \operatorname {\mathsf {D}}(A) . In this paper we carry a systematic study of the categorical properties of these functors. As an application, a new descent result for vanishing of Ext \operatorname {Ext} and Tor \operatorname {Tor} over ordinary commutative noetherian rings is deduced.

Rigidity of Ext and Tor via flat–cotorsion theory

AbstractLet $\mathfrak{p}$ be a prime ideal in a commutative noetherian ring R and denote by $k(\mathfrak{p})$ the residue field of the local ring $R_\mathfrak{p}$. We prove that if an R-module M satisfies $\operatorname{Ext}_R^{n}(k(\mathfrak{p}),M)=0$ for some $n\geqslant\dim R$, then $\operatorname{Ext}_R^i(k(\mathfrak{p}),M)=0$ holds for all $i \geqslant n$. This improves a result of Christensen, Iyengar and Marley by lowering the bound on n. We also improve existing results on Tor-rigidity. This progress is driven by the existence of minimal semi-flat-cotorsion replacements in the derived category as recently proved by Nakamura and Thompson.

Duality between injective envelopes and flat covers

We establish a duality between injective envelopes and flat covers over a commutative Noetherian ring. One case of this duality states that a morphism is an injective envelope, if and only if its Matlis dual is a flat cover. We also show that if we swap injective envelopes and flat covers in this duality, neither implication is true in general.

Solid generators in module categories and applications

Let R be a commutative noetherian ring. Denote by mod R the category of finitely generated R -modules. In the present paper, we introduce the notion of solid subcategories of mod R and investigate it. The main result of this paper not only recovers results of Schoutens, Krause and Stevenson, and Takahashi on thick subcategories, but also unifies and extends them to solid subcategories. Moreover, it provides some contributions to the study of the question asking when a thick subcategory is Serre.

A Friedlander-Suslin Theorem over a Noetherian Base Ring

AbstractLet k be a noetherian commutative ring and let G be a finite flat group scheme over k. Let G act rationally on a finitely generated commutative k-algebra A. We show that the cohomology algebra $$H^*(G,A)$$ H ∗ ( G , A ) is a finitely generated k-algebra. This unifies some earlier results: If G is a constant group scheme, then it is a theorem of Evens (Trans. Amer. Math. Soc. 101, 224–239, 1961, Theorem 8.1), and if k is a field of finite characteristic, then it is a theorem of Friedlander and Suslin (Invent. Math. 127, 209–270, 1997). If k is a field of characteristic zero, then there is no higher cohomology, so then it is a theorem of invariant theory.

Cofiniteness with respect to Extension of Serre Subcategories at Small Dimensions

Let A denote a commutative Noetherian ring, a be an ideal of A , X be an arbitrary dense subcategory of A -modules, and N be the full subcategory of finite A -modules. In this paper, we study N ⋆ X - c -cofinite modules with respect to the extension subcategory N ⋆ X when dim A / c ≤ 2 . We also study c -cofiniteness with respect to a new dimension.

Openness of various loci over Noetherian rings

In this paper, we consider the openness of the P-locus of a finitely generated module over a commutative noetherian ring in the case where P is each of the properties FID, Gor, Can, CM, MCM, (Sn), and (Tn). One of the main results asserts that FID-loci over an acceptable ring are open. We give a module version of the Nagata criterion, and prove that it holds for all of the aforementioned properties.

Annihilator of top local cohomology and Lynch’s conjecture

Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] a proper ideal of [Formula: see text] and [Formula: see text] a nonzero finitely generated [Formula: see text]-module with [Formula: see text]. Let [Formula: see text] (respectively, [Formula: see text]) be the smallest (respectively, greatest) non-negative integer [Formula: see text] such that the local cohomology [Formula: see text] is nonzero. In this paper, we provide sharp bounds under inclusion for the annihilators of the local cohomology modules [Formula: see text], [Formula: see text] and these annihilators are computed in certain cases. Also, we construct a counterexample to Lynch’s conjecture.

Singular equivalences to locally coherent hearts of commutative noetherian rings

We show that Krause's recollement exists for any locally coherent Grothendieck category whose derived category is compactly generated. As a source of such categories, we consider the hearts of intermediate and restrictable t-structures in the derived category of a commutative noetherian ring. We show that the induced tilting object over such a heart gives rise to an equivalence between the two Krause's recollements, and in particular, to a singular equivalence.

General local cohomology modules in view of low points and high points

UDC 512.5 Let R be a commutative Noetherian ring, let Φ be a system of ideals of R , let M be a finitely generated R -module, and let t be a nonnegative integer. We first show that a general local cohomology module H Φ i ( M ) is a finitely generated R -module for all i < t if and only if A s s R ( H Φ i ( M ) ) is a finite set and H Φ 𝔭 i ( M 𝔭 ) is a finitely generated R 𝔭 -module for all i < t and all 𝔭 ∈ S p e c ( R ) . Then, as a consequence, we prove that if ( R , 𝔪 ) is a complete local ring, Φ is countable, and n ∈ ℕ 0 is such that ( A s s R ( H Φ h Φ n ( M ) ( M ) ) ) ≥ n is a finite set, then f Φ n ( M ) = h Φ n ( M ) . In addition, we show that the properties of vanishing and finiteness of general local cohomology modules are equivalent on high points over an arbitrary Noetherian (not necessary local) ring. For each covariant R -linear functor T from M o d ( R ) into itself, which has the global vanishing property on M o d ( R ) and for an arbitrary Serre subcategory 𝒮 and t ∈ ℕ , we prove that ℛ i T ( R ) ∈ 𝒮 for all i ≥ t if and only if ℛ i T ( M ) ∈ 𝒮 for any finitely generated R -module M and all i ≥ t . Then we obtain some results on general local cohomology modules.

Cellular Noetherian algebras with finite global dimension are split quasi-hereditary

We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. In the process, we establish that a split quasi-hereditary algebra is semi-perfect if and only if the ground ring is a local commutative Noetherian ring. We give a formula to determine the global dimension of a split quasi-hereditary algebra over a commutative regular Noetherian ring (with finite Krull dimension) in terms of the ground ring and finite-dimensional split quasi-hereditary algebras. For the general case, we give upper bounds for the finitistic dimension of split quasi-hereditary algebras over arbitrary commutative Noetherian rings. We apply these results to Schur algebras over regular Noetherian rings and to Schur algebras over quotients rings of the integers.

Theory of linkage, Gorenstein dimension and two semidualizing modules

Let [Formula: see text] be a semiperfect commutative Noetherian ring with identity and [Formula: see text] be semidualizing [Formula: see text]-modules. We study the theory of linkage with respect to [Formula: see text] for modules of finite [Formula: see text]-dimension. For a module which is horizontally linked with respect to [Formula: see text], the connections of its invariants reduced grade with respect to [Formula: see text], [Formula: see text]-dimension and depth are discussed. Along the way, we provide a characterization of horizontally linked modules with respect to [Formula: see text] of [Formula: see text]-dimension zero.

Characters of the group [formula omitted] for a commutative Noetherian ring [formula omitted

Let R be a commutative Noetherian ring with unit. We classify the characters of the group ELd(R) provided that d is greater than the stable range of the ring R. It follows that every character of ELd(R) is induced from a finite dimensional representation. Towards our main result we classify ELd(R)-invariant probability measures on the Pontryagin dual group of Rd.

Tilting and Silting Theory of Noetherian Algebras

Abstract We develop silting theory of a Noetherian algebra $\Lambda $ over a commutative Noetherian ring $R$. We study mutation theory of $2$-term silting complexes of $\Lambda $, and as a consequence, we see that mutation exists. As in the case of finite-dimensional algebras, functorially finite torsion classes of $\Lambda $ bijectively correspond to silting $\Lambda $-modules, if $R$ is complete local. We show a reduction theorem of $2$-term silting complexes of $\Lambda $, and by using this theorem, we study torsion classes of the module category of $\Lambda $. When $R$ has Krull dimension one, we describe the set of torsion classes of $\Lambda $ explicitly by using the set of torsion classes of finite-dimensional algebras.

The $$\mathfrak a$$-filter Grade of an Ideal $$\mathfrak b$$ and $$(\mathfrak a,\mathfrak b)$$-f-modules

Let $$\mathfrak a,\mathfrak b$$ be two ideals of a commutative noetherian ring R and M a finitely generated R-module. We continue to study f-gradR( $$\mathfrak a,\mathfrak b$$ ,M) which was introduced in [Bull. Malays. Math. Sci. Soc., 2015, 38(2): 467–482]. Some computations and bounds of f-gradR( $$\mathfrak a,\mathfrak b$$ ,M) are provided. We also give the structure of ( $$\mathfrak a,\mathfrak b$$ )-f-modules. Some properties which are analogous to those of Cohen-Macaulay modules are discovered.

Melkersson condition for extension of Serre subcategories

Let R be a commutative noetherian ring and let $$\mathfrak {a}$$ be an ideal of R. In this paper, we study a certain condition, namely $$C_{\mathfrak {a}}$$ , introduced by Aghapournahr and Melkersson, on the extension of two subcategories of R-modules. We develop some of the main results of Yoshizawa (Proc Am Math Soc 140(7):2293–2305, 2012; J Commut Algebra 13:137–155, 2021). As an example of extension of subcategories, we study the weakly Laskerian modules and we find some conditions under which the local cohomology modules of a weakly Laskerian module lie in an arbitrary Serre subcategory. Eventually, we investigate the cofiniteness of the local cohomology modules of weakly Laskerian modules.

Some Properties of n-semidualizing Modules

Let R be a commutative noetherian ring. The n-semidualizing modules of R are generalizations of its semidualizing modules. We will prove some basic properties of n-semidualizing modules. Our main result and example shows that the divisor class group of a Gorenstein determinantal ring over a field is the set of isomorphism classes of its 1-semidualizing modules. Finally, we pose some questions about n-semidualizing modules.

A partial converse ghost lemma for the derived category of a commutative Noetherian ring

In this article a condition is given to detect the containment among thick subcategories of the bounded derived category of a commutative noetherian ring. More precisely, for a commutative noetherian ring R R and complexes of R R -modules with finitely generated homology M M and N N , we show N N is in the thick subcategory generated by M M if and only if the ghost index of N p N_\mathfrak {p} with respect to M p M_\mathfrak {p} is finite for each prime p \mathfrak {p} of R R . To do so, we establish a “converse coghost lemma” for the bounded derived category of a non-negatively graded DG algebra with noetherian homology.

Cofiniteness with respect to extension of Serre subcategories

Let R be a commutative Noetherian ring, be an ideal of R, be a Serre subcategory of R-modules satisfying the condition , N be a finitely generated R-module and M be an arbitrary R-module with . We show that if M is an --cofinite R-module with , then is --cofinite for each . If with , then M is --cofinite if and only if Moreover, we prove that the subcategory and M is --cofinite of the category of R-modules is abelian. Furthermore, for a nonnegative integer n, we prove that for all if and only if is --cofinite for all and .