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1.
Let k(x) be the field of fractions of the polynomial algebra k[x] over the field k. We prove that, for an arbitrary finite dimensional k-algebra Λ, any finitely generated Λ ⊗k k(x)-module M such that its minimal projective presentation admits no non-trivial selfextension is of the form M ≅ Nk(x), for some finitely generated Λ-module N. Some consequences are derived for tilting modules over the rational algebra Λ ⊗k k(x) and for some generic modules for Λ.
Received: 24 November 2003; revised: 11 February 2005 相似文献
2.
Let R be a commutative ring with identity, let M be an R-module, and let K
1, . . . ,K
n
be submodules of M: We construct an algebraic object called the product of K
1, . . . ,K
n
: This structure is equipped with appropriate operations to get an R(M)-module. It is shown that the R(M)-module M
n
= M . . .M and the R-module M inherit some of the most important properties of each other. Thus, it is shown that M is a projective (flat) R-module if and only if M
n
is a projective (flat) R(M)-module. 相似文献
3.
Daniel Maycock 《代数通讯》2013,41(7):2367-2387
This paper generalises a result for upper triangular matrix rings to the situation of upper triangular matrix differential graded algebras. An upper triangular matrix DGA has the form (R, S, M) where R and S are differential graded algebras and M is a DG-left-R-right-S-bimodule. We show that under certain conditions on the DG-module M and with the existance of a DG-R-module X, from which we can build the derived category D(R), that there exists a derived equivalence between the upper triangular matrix DGAs (R, S, M) and (S, M′, R′), where the DG-bimodule M′ is obtained from M and X and R′ is the endomorphism differential graded algebra of a K-projective resolution of X. 相似文献
4.
Ivo Herzog 《Algebras and Representation Theory》2007,10(2):135-155
Given an R-T-bimodule
R
K
T
and R-S-bimodule
R
M
S
, we study how properties of
R
K
T
affect the K-double dual M** = Hom
T
[Hom
R
(M, K), K] considered as a right S-module. If
R
K is a cogenerator, then for every R-S-bimodule, the natural morphism Φ
M
: M → M** is a pure-monomorphism of right S-modules. If
R
K is the minimal (injective) cogenerator and K
T
is quasi-injective, then M
** is a pure-injective right S-module. If
R
K is the minimal (injective) cogenerator, and T = End
R
K it is shown that K
T
is quasi-injective if and only if the K-topology on R is linearly compact. If the
R
K-topology on R is of finite type, then the natural morphism Φ
R
: R → R** is the pure-injective envelope of R
R
as a right module over itself.
The author is partially supported by NSF Grant DMS-02-00698. 相似文献
5.
David E. Dobbs 《Rendiconti del Circolo Matematico di Palermo》2009,58(3):327-336
If T is a (commutative unital) ring extension of a ring R, then Λ(T /R) is defined to be the supremum of the lengths of chains of intermediate fields between R
P
/P R
P
and T
Q
/QT
Q
, where Q varies over Spec(T) and P:= Q ∩ R. The invariant σ(R):= sup Λ(T/R), where T varies over all the overrings of R. It is proved that if Λ(S/R)< ∞ for all rings S between R and T, then (R, T) is an INC-pair; and that if (R, T) is an INC-pair such that T is a finite-type R-algebra, then Λ(T/R)< ∞. Consequently, if R is a domain with σ(R) < ∞, then the integral closure of R is a Prüfer domain; and if R is a Noetherian G-domain, then σ(R) < ∞, with examples showing that σ(R) can be any given non-negative integer. Other examples include that of a onedimensional Noetherian locally pseudo-valuation
domain R with σ(R)=∞. 相似文献
6.
Mitchel T. Keller Yi-Huang Shen Noah Streib Stephen J. Young 《Journal of Algebraic Combinatorics》2011,33(2):313-324
Let K be a field and S=K[x
1,…,x
n
]. In 1982, Stanley defined what is now called the Stanley depth of an S-module M, denoted sdepth (M), and conjectured that depth (M)≤sdepth (M) for all finitely generated S-modules M. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in
the case when M=I/J with J⊂I being monomial S-ideals. Specifically, their method associates M with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze
squarefree Veronese ideals in S. In particular, if I
n,d
is the squarefree Veronese ideal generated by all squarefree monomials of degree d, we show that if 1≤d≤n<5d+4, then sdepth (I
n,d
)=⌊(n−d)/(d+1)⌋+d, and if d≥1 and n≥5d+4, then d+3≤sdepth (I
n,d
)≤⌊(n−d)/(d+1)⌋+d. 相似文献
7.
Massoud Tousi 《代数通讯》2013,41(11):3977-3987
ABSTRACT Assume that ?:(R, ± 𝔪) → (S, ± 𝔫) is a local flat homomorphism between commutative Noetherian local rings R and S. Let M be a finitely generated R-module. We investigate the ascent and descent of sequentially Cohen-Macaulay properties between the R-module M and the S-module M ? R S. 相似文献
8.
BASUDEB DHARA 《Proceedings Mathematical Sciences》2012,122(1):121-128
Let R be a prime ring with its Utumi ring of quotient U, H and G be two generalized derivations of R and L a noncentral Lie ideal of R. Suppose that there exists 0 ≠ a ∈ R such that a(H(u)u − uG(u))
n
= 0 for all u ∈ L, where n ≥ 1 is a fixed integer. Then there exist b′,c′ ∈ U such that H(x) = b′x + xc′, G(x) = c′x for all x ∈ R with ab′ = 0, unless R satisfies s
4, the standard identity in four variables. 相似文献
9.
Xiaofei Qi 《Linear and Multilinear Algebra》2013,61(4):391-397
Let 𝒜 and ? be unital algebras over a commutative ring ?, and ? be a (𝒜,??)-bimodule, which is faithful as a left 𝒜-module and also as a right ?-module. Let 𝒰?=?Tri(𝒜,??,??) be the triangular algebra and 𝒱 any algebra over ?. Assume that Φ?:?𝒰?→?𝒱 is a Lie multiplicative isomorphism, that is, Φ satisfies Φ(ST???TS)?=?Φ(S)Φ(T)???Φ(T)Φ(S) for all S, T?∈?𝒰. Then Φ(S?+?T)?=?Φ(S)?+?Φ(T)?+?Z S,T for all S, T?∈?𝒰, where Z S,T is an element in the centre 𝒵(𝒱) of 𝒱 depending on S and T. 相似文献
10.
Let R be a left Noetherian ring, S a right Noetherian ring and R ω a Wakamatsu tilting module with S = End( R ω). We introduce the notion of the ω-torsionfree dimension of finitely generated R-modules and give some criteria for computing it. For any n ? 0, we prove that l.id R (ω) = r.id S (ω) ? n if and only if every finitely generated left R-module and every finitely generated right S-module have ω-torsionfree dimension at most n, if and only if every finitely generated left R-module (or right S-module) has generalized Gorenstein dimension at most n. Then some examples and applications are given. 相似文献
11.
We establish an order-preserving bijective correspondence between the sets of coclosed elements of some bounded lattices related by suitable Galois connections. As an application, we deduce that if M is a finitely generated quasi-projective left R-module with S = End R (M) and N is an M-generated left R-module, then there exists an order-preserving bijective correspondence between the sets of coclosed left R-submodules of N and coclosed left S-submodules of Hom R (M, N). 相似文献
12.
Thomas Marley 《manuscripta mathematica》2001,104(4):519-525
Let R be a commutative Noetherian local ring of dimension d, I an ideal of R, and M a finitely generated R-module. We prove that the set of associated primes of the local cohomology module H
i
I
(M) is finite for all i≥ 0 in the following cases: (1) d≤ 3; (2) d= 4 and $R$ is regular on the punctured spectrum; (3) d= 5, R is an unramified regular local ring, and M is torsion-free. In addition, if $d>0$ then H
d
− 1
I
(M) has finite support for arbitrary R, I, and M.
Received: 31 October 2000 / Revised version: 8 January 2001 相似文献
13.
Let R be a Noetherian ring and M be a finitely generated R-module. Let I(M) be the first nonzero Fitting ideal of M. The main result of this paper asserts that when I(M) = Q is a regular maximal ideal of R, then M?R∕Q⊕P, for some projective R-module P of constant rank if and only if T(M)?QM. As a consequence, it is shown that if M is an Artinian R-module and I(M) = Q is a regular maximal ideal of R, then M?R∕Q. 相似文献
14.
For a ring R and a right R-module M, a submodule N of M is said to be δ-small in M if, whenever N+X=M with M/X singular, we have X=M. Let ℘ be the class of all singular simple modules. Then δ(M)=Σ{ L≤ M| L is a δ-small submodule of M} = Re
jm(℘)=∩{ N⊂ M: M/N∈℘. We call M δ-coatomic module whenever N≤ M and M/N=δ(M/N) then M/N=0. And R is called right (left) δ-coatomic ring if the right (left) R-module R
R(RR) is δ-coatomic. In this note, we study δ-coatomic modules and ring. We prove M=⊕
i=1
n
Mi is δ-coatomic if and only if each M
i (i=1,…, n) is δ-coatomic. 相似文献
15.
Helmut Zöschinger 《代数通讯》2013,41(6):1977-1994
Let (R, 𝔪) be a commutative, noetherian, local ring, E the injective hull of the residue field R/𝔪, and M ○○ = Hom R (Hom R (M, E), E) the bidual of an R-module M. We investigate the elements of Ass(M ○○) as well as those of Coatt(M) = {𝔭 ∈ Spec(R)|𝔭 = Ann R (Ann M (𝔭))} and provide criteria for equality in one of the two inclusions Ass(M) ? Ass(M ○○) ? Coatt(M). If R is a Nagata ring and M a minimax module, i.e., an extension of a finitely generated R-module by an artinian R-module, we show that Ass(M ○○) = Ass(M) ∪ {𝔭 ∈ Coatt(M)| R/𝔭 is incomplete}. 相似文献
16.
KARIM SAMEI 《Proceedings Mathematical Sciences》2011,121(2):121-132
An R-module M is called a multiplication module if for each submodule N of M, N = IM for some ideal I of R. As defined for a commutative ring R, an R-module M is said to be reduced if the intersection of prime submodules of M is zero. The prime spectrum and minimal prime submodules of the reduced module M are studied. Essential submodules of M are characterized via a topological property. It is shown that the Goldie dimension of M is equal to the Souslin number of Spec(M)\mbox{\rm Spec}(M). Also a finitely generated module M is a Baer module if and only if Spec(M)\mbox{\rm Spec}(M) is an extremally disconnected space; if and only if it is a CS-module. It is proved that a prime submodule N is minimal in M if and only if for each x ∈ N,
Ann(x) \not í (N:M).\mbox{\rm Ann}(x) \not \subseteq (N:M). When M is finitely generated; it is shown that every prime submodule of M is maximal if and only if M is a von Neumann regular module (VNM); i.e., every principal submodule of M is a summand submodule. Also if M is an injective R-module, then M is a VNM. 相似文献
17.
Let G = GL
N
or SL
N
as reductive linear algebraic group over a field k of characteristic p > 0. We prove several results that were previously established only when N ⩽ 5 or p > 2
N
: Let G act rationally on a finitely generated commutative k-algebra A and let grA be the Grosshans graded ring. We show that the cohomology algebra H
*(G, grA) is finitely generated over k. If moreover A has a good filtration and M is a Noetherian A-module with compatible G action, then M has finite good filtration dimension and the H
i
(G, M) are Noetherian A
G
-modules. To obtain results in this generality, we employ functorial resolution of the ideal of the diagonal in a product
of Grassmannians. 相似文献
18.
A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext1(G, M) = 0 (resp. Tor1(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given. 相似文献
19.
Let G{{\mathcal G}} be a group, Λ a G{{\mathcal G}}-graded Artin algebra and gr(Λ) denote the category of finitely generated G{{\mathcal G}}-graded Λ-modules. This paper provides a framework that allows an extension of tilting theory to Db(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)) and to study connections between the tilting theories of Db(L){{\mathcal D}}^b(\Lambda) and Db(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)). In particular, using that if T is a gradable Λ-module, then a grading of T induces a G{{\mathcal G}}-grading on EndΛ(T), we obtain conditions under which a derived equivalence induced by a gradable Λ-tilting module T can be lifted to a derived equivalence between the derived categories Db(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)) and Db(gr(EndL(T))){{\mathcal D}}^b(\rm gr(\rm End_{\Lambda}(\textit T))). 相似文献
20.
We prove real Paley-Wiener type theorems for the Dunkl transform ℱ
D
on the space
of tempered distributions. Let T∈S′(ℝ
d
) and Δ
κ
the Dunkl Laplacian operator. First, we establish that the support of ℱ
D
(T) is included in the Euclidean ball
, M>0, if and only if for all R>M we have lim
n→+∞
R
−2n
Δ
κ
n
T=0 in S′(ℝ
d
). Second, we prove that the support of ℱ
D
(T) is included in ℝ
d
∖B(0,M), M>0, if and only if for all R<M, we have lim
n→+∞
R
2n
ℱ
D
−1(‖y‖−2n
ℱ
D
(T))=0 in S′(ℝ
d
). Finally, we study real Paley-Wiener theorems associated with
-slowly increasing function.
相似文献