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1.
Let (R, m) be a complete Noetherian local ring, I an ideal of R and M a nonzero Artinian R-module. In this paper it is shown that if p is a prime ideal of R such that dim R/p = 1 and (0:M p) is not finitely generated and for each i ? 2 the R-module Ext R i (M,R/p) is of finite length, then the R-module Ext R 1 (M, R/p) is not of finite length. Using this result, it is shown that for all finitely generated R-modules N with Supp(N) ? V (I) and for all integers i ? 0, the R-modules Ext R i (N,M) are of finite length, if and only if, for all finitely generated R-modules N with Supp(N) ? V (I) and for all integers i ? 0, the R-modules Ext R i (M,N) are of finite length. 相似文献
2.
Sarah Scherotzke 《Journal of Pure and Applied Algebra》2011,215(5):829-838
We construct rank varieties for the Drinfeld double of the Taft algebra Λn and for uq(sl2). For the Drinfeld double when n=2 this uses a result which identifies a family of subalgebras that control projectivity of Λ-modules whenever Λ is a Hopf algebra satisfying a certain homological condition. In this case we show that our rank variety is homeomorphic to the cohomological support variety. We also show that Ext∗(M,M) is finitely generated over the cohomology ring of the Drinfeld double for any finitely generated module M. 相似文献
3.
Let R be any ring. A right R-module M is called n-copure projective if Ext1(M, N) = 0 for any right R-module N with fd(N) ≤ n, and M is said to be strongly copure projective if Ext i (M, F) = 0 for all flat right R-modules F and all i ≥ 1. In this article, firstly, we present some general properties of n-copure projective modules and strongly copure projective modules. Then we define and investigate copure projective dimensions of modules and rings. Finally, more properties and applications of n-copure projective modules, strongly copure projective modules and copure projective dimensions are given over coherent rings with finite self-FP-injective dimension. 相似文献
4.
We show that the conditions defining total reflexivity for modules are independent. In particular, we construct a commutative
Noetherian local ring R and a reflexive R-module M such that ExtRi(M,R)=0 for all i>0, but ExtRi(M*,R)≠0 for all i>0.
Presented by Juergen Herzog
Mathematics Subject Classification (2000) 13D07. 相似文献
5.
We consider two finitely generated graded modules over a homogeneous Noetherian ring $R = \oplus _{n \in \mathbb{N}_0 } R_n$ with a local base ring (R 0, m0) and irrelevant ideal R + of R. We study the generalized local cohomology modules H b i (M,N) with respect to the ideal b = b0 + R +, where b0 is an ideal of R 0. We prove that if dimR 0/b0 ≤ 1, then the following cases hold: for all i ≥ 0, the R-module H b i (M,N)/a0 H b i (M,N) is Artinian, where $\sqrt {\mathfrak{a}_0 + \mathfrak{b}_0 } = \mathfrak{m}_0$ ; for all i ≥ 0, the set $Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)$ is asymptotically stable as n→?∞. Moreover, if H b i (M,N) n is a finitely generated R 0-module for all n ≤ n 0 and all j < i, where n 0 ∈ ? and i ∈ ?0, then for all n ≤ n 0, the set $Ass_{R_0 } \left( {H_\mathfrak{b}^i \left( {M,N} \right)_n } \right)$ is finite. 相似文献
6.
Let M be a finitely generated module over a local ring R of characteristic p > 0. If depth(R) = s, then the property that M has finite projective dimension can be characterized by the vanishing of the functor ExtiR(M, fnR){{\rm Ext}^i_R(M, ^{f^n}R)} for s + 1 consecutive values i > 0 and for infinitely many n. In addition, if R is a d-dimensional complete intersection, then M has finite projective dimension can be characterized by the vanishing of the functor ExtiR(M, fnR){{\rm Ext}^i_R(M, ^{f^n}R)} for some i ≥ d and some n > 0. 相似文献
7.
8.
Zhanmin Zhu 《Czechoslovak Mathematical Journal》2018,68(2):455-474
Let R be a ring. A subclass T of left R-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let T be a weak torsion class of left R-modules and n a positive integer. Then a left R-module M is called T-finitely generated if there exists a finitely generated submodule N such that M/N ∈ T; a left R-module A is called (T,n)-presented if there exists an exact sequence of left R-modules such that F0,..., Fn?1 are finitely generated free and Kn?1 is T-finitely generated; a left R-module M is called (T,n)-injective, if Ext n R (A,M) = 0 for each (T, n+1)-presented left R-module A; a right R-module M is called (T,n)-flat, if Tor R n (M,A) = 0 for each (T, n+1)-presented left R-module A. A ring R is called (T,n)-coherent, if every (T, n+1)-presented module is (n + 1)-presented. Some characterizations and properties of these modules and rings are given.
相似文献
$$0 \to {K_{n - 1}} \to {F_{n - 1}} \to \cdots \to {F_1} \to {F_0} \to M \to 0$$
9.
Let Λ be an Artin algebra over a commutative Artinian ring, k. If M is a finitely generated left Λ -module, we denote by Ω (M) the kernel of η M : P M → M a minimal projective cover. We prove that if M and N are finitely generated left Λ -modules and Ext Λ 1 (M, M) = 0, Ext Λ 1 (N, N) = 0, then M? N if and only if M/rad M? N/rad N and Ω (M)? Ω (N). Now if k is an algebraically closed field and (d i ) i?? is a sequence of nonnegative integers almost all of them zero, then we prove that the family of objects X ? b (Λ), the bounded derived category of Λ, with Hom b (Λ)(X,X[1]) = 0 and dim k H i (X) = d i for all i ? ?, has only a finite number of isomorphism classes (see Huisgen-Zimmermann and Saorín, 2001). 相似文献
10.
Amir Mafi 《Proceedings Mathematical Sciences》2009,119(2):159-164
Let a be an ideal of a commutative Noetherian ring R with non-zero identity and let N be a weakly Laskerian R-module and M be a finitely generated R-module. Let t be a non-negative integer. It is shown that if H
a
i
(N) is a weakly Laskerian R-module for all i < t, then Hom
R
(R/a, H
a
t
(M, N)) is weakly Laskerian R-module. Also, we prove that Ext
R
i
(R/a, H
a
t
)) is weakly Laskerian R-module for all i = 0, 1. In particular, if Supp
R
(H
a
i
(N)) is a finite set for all i < t, then Ext
R
i
(R/a, H
a
t
(N)) is weakly Laskerian R-module for all i = 0, 1. 相似文献
11.
Ahmad Abbasi Hajar Roshan-Shekalgourabi Dawood Hassanzadeh-Lelekaami 《Czechoslovak Mathematical Journal》2014,64(2):327-333
Let R be a commutative Noetherian ring with identity and I an ideal of R. It is shown that, if M is a non-zero minimax R-module such that dim Supp H I i (M) ? 1 for all i, then the R-module H I i (M) is I-cominimax for all i. In fact, H I i (M) is I-cofinite for all i ? 1. Also, we prove that for a weakly Laskerian R-module M, if R is local and t is a non-negative integer such that dim Supp H I i (M) ? 2 for all i < t, then Ext R j (R/I,H I i (M)) and Hom R (R/I,H I t (M)) are weakly Laskerian for all i < t and all j ? 0. As a consequence, the set of associated primes of H I i (M) is finite for all i ? 0, whenever dim R/I ? 2 and M is weakly Laskerian. 相似文献
12.
We show that a right artinian ring R is right self-injective if and only if ψ(M)?=?0 (or equivalently ?(M)?=?0) for all finitely generated right R-modules M, where ψ, $\phi :\!\!\!\! \mod R \to \mathbb N$ are functions defined by Igusa and Todorov. In particular, an artin algebra Λ is self-injective if and only if ?(M)?=?0 for all finitely generated right Λ-modules M. 相似文献
13.
14.
Let R be a left coherent ring. We first prove that a right R-module M is strongly copure flat if and only if Ext i (M, C) = 0 for all flat cotorsion right R-modules C and i ≥ 1. Then we define and investigate copure flat dimensions of left coherent rings. Finally, we give some new characterizations of n-FC rings. 相似文献
15.
Sohrab Sohrabi Laleh Mir Yousef Sadeghi Mahdi Hanifi Mostaghim 《Czechoslovak Mathematical Journal》2012,62(1):105-110
Let R be a commutative Noetherian ring, a an ideal of R, M an R-module and t a non-negative integer. In this paper we show that the class of minimax modules includes the class of AF modules. The main result is that if the R-module Ext
R
t
(R/a,M) is finite (finitely generated), H
a
i
(M) is a-cofinite for all i < t and H
a
t
(M) is minimax then H
a
t
(M) is a-cofinite. As a consequence we show that if M and N are finite R-modules and H
a
i
(N) is minimax for all i < t then the set of associated prime ideals of the generalized local cohomology module H
a
t
(M,N) is finite. 相似文献
16.
José L. Bueso J. Gómez-Torrecillas F. J. Lobillo 《Algebras and Representation Theory》2001,4(3):201-218
In this paper the Poincaré–Birkhoff–Witt (PBW) rings are characterized. Gröbner bases techniques are also developed for these rings. An explicit presentation of Ext
i
(M,N) is provided when N is a centralizing bimodule. 相似文献
17.
A right module M over a ring R is said to be retractable if Hom R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ? R RG is a retractable RG-module if and only if M R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings. 相似文献
18.
Claus Scheiderer 《Mathematische Zeitschrift》2010,266(1):1-19
Let A be an excellent local ring of real dimension ≤2, let T be a finitely generated preordering in A, and let ${\widehat{T}}We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if ${{\rm Ext}^{1}_{R}\,(M, T)\,=\,0}We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules,
and the Mittag-Leffler ones. A right R-module M is called Baer if Ext1R (M, T) = 0{{\rm Ext}^{1}_{R}\,(M, T)\,=\,0} for all torsion modules T, and M is Mittag-Leffler in case the canonical map M?R ?i ? IQi? ?i ? I(M?RQi){M\otimes_R \prod _{i\in I}Q_i\to \prod _{i\in I}(M\otimes_RQ_i)} is injective where {Qi}i ? I{\{Q_i\}_{i\in I}} are arbitrary left R-modules. We show that a module M is Baer iff M is p-filtered where p is the preprojective component of the tame hereditary algebra R. We apply this to prove that the universal localization of a Baer module is projective in case we localize with respect to
a complete tube. Using infinite dimensional tilting theory we then obtain a structure result showing that Baer modules are
more complex then the (infinite dimensional) preprojective modules. In the final section, we give a complete classification
of the Mittag-Leffler modules. 相似文献
19.
Reza Sazeedeh 《Proceedings Mathematical Sciences》2007,117(4):429-441
Let M be a finitely generated graded module over a Noetherian homogeneous ring R with local base ring (R
0, m0). If R
0 is of dimension one, then we show that reg
i+1(M) and coreg
i+1(M) are bounded for all i ∈ ℕ0. We improve these bounds, if in addition, R
0 is either regular or analytically irreducible of unequal characteristic. 相似文献
20.
Let R be a complete semi-local ring with respect to the topology defined by its Jacobson radical, a an ideal of R, and M a finitely generated R-module. Let D
R
(−) := Hom
R
(−, E), where E is the injective hull of the direct sum of all simple R-modules. If n is a positive integer such that Ext
R
j
(R/a, D
R
(H
a
t
(M))) is finitely generated for all t > n and all j ⩾ 0, then we show that Hom
R
(R/a, D
R
(H
a
n
(M))) is also finitely generated. Specially, the set of prime ideals in Coass
R
(H
a
n
(M)) which contains a is finite.
Next, assume that (R, m) is a complete local ring. We study the finiteness properties of D
R
(H
a
r
(R)) where r is the least integer i such that H
a
r
(R) is not Artinian. 相似文献