Noetherianity and Ext

Let R=R₀⊕R₁⊕R₂⊕? be a graded algebra over a field K such that R₀ is a finite product of copies of K and each R_i is finite dimensional over K. Set J=R₁⊕R₂⊕? and . We study the properties of the categories of graded R-modules and S-modules that relate to the noetherianity of R. We pay particular attention to the case when R is a Koszul algebra and S is the Koszul dual to R.     

Murley Groups and the Torsion-Freeness of Ext

This article has been retracted.     

Filtrations for the Roots of Ext

We survey the set–theoretic methods of module theory that make it possible to equip roots of the contravariant Ext functor with filtrations built from the small roots. The power of these methods is illustrated by several applications: a solution to the Kaplansky problem on Baer modules and some of the related problems for relative Baer modules, the structure of tilting modules and classes, the structure of Matlis localizations of commutative rings, and in particular cases, proofs of the finitistic dimension conjectures, and of the telescope conjecture for module categories. Received: January 2007     

On thep-rank of Ext

AssumeV =L andλ is regular smaller than the first weakly compact cardinal. Under those circumstances and with arbitrary requirements on the structure of Ext (G, ℤ) (under well known limitations), we construct an abelian groupG of cardinalityλ such that for noG′ ⊆G, |G′| <λ isG/G′ free and Ext (G, ℤ) realizes our requirements. Deceased. Partially supported by NSERC. Partially supported by the United States — Israel Binational Science Foundation. Publication No. 314.     

Ext vanishing and infinite Auslander-Buchsbaum

A vanishing theorem is proved for Ext groups over non-commutative graded algebras. Along the way, an ``infinite' version is proved of the non-commutative Auslander-Buchsbaum theorem.

     

Isomorphism of one-place functors Ext

Let be an associative ring with identity. One considers the category of left (unitary) -modules m and also the contravariant and the covariant functors Ext ¹ ( ,A) and Ext ¹ (A, ): M_z M. One proves the following results: (1) If the homomorphism of -modules A B induces an isomorphism Ext ¹ ( ,A)Ext ¹ ( ,B), then there exist injective -modules J₁ and J₂ such that AJ₁BJ₂. (2) Every functorial morphism Ext ¹ ( ,A)Ext ¹ ( ,B) induces a certain homomorphism of -modules AB. One also obtains a dual result.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 71–74, 1981.     

Polynomially bounded operators and Ext groups

In this paper, we consider the Ext functor in the category
of Hilbert modules over the disk algebra. We characterize the group
as a quotient of operators and explicitly calculate
, where is a weighted Hardy space. We then use our results to give a simple proof of a result due to Bourgain.

     

Homological behavior of idempotent subalgebras and Ext algebras

Let A be a(left and right) Noetherian ring that is semiperfect. Let e be an idempotent of A and consider the ring Γ :=(1-e)A(1-e) and the semi-simple right A-module Se := e A/e rad A. In this paper, we investigate the relationship between the global dimensions of A and Γ, by using the homological properties of Se. More precisely, we consider the Yoneda ring Y(e) := Ext_A~*(Se, Se) of e. We prove that if Y(e) is Artinian of finite global dimension, then A has finite global dimension if and only if so does Γ. We also investigate the situation where both A and Γ have finite global dimension. When A is Koszul and finite dimensional, this implies that Y(e) has finite global dimension. We end the paper with a reduction technique to compute the Cartan determinant of Artin algebras. We prove that if Y(e) has finite global dimension, then the Cartan determinants of A and Γ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture.     

Generalized local cohomology and regularity of Ext modules

Let R be a Noetherian standard graded ring, and M and N two finitely generated graded R-modules. We introduce ${\mathrm{reg}}_R(M,N)$ by using the notion of generalized local cohomology instead of local cohomology, in the definition of regularity. We prove that ${\mathrm{reg}}_R(M,N)$ is finite in several cases. In the case that the base ring is a field, we show that${\mathrm{reg}}_R(M,N)=\mathrm{reg}(N)?\mathrm{indeg}(M).$ This formula, together with a graded version of duality for generalized local cohomology, gives a formula for the minimum of the initial degrees of some Ext modules (in the case R is Cohen–Macaulay), of which the three usual definitions of regularity are special cases. Bounds for regularity of certain Ext modules are obtained, using the same circle of ideas.     

Some bounds for the annihilators of local cohomology and Ext modules

Czechoslovak Mathematical Journal - Let $$\mathfrak{a}$$ be an ideal of a commutative Noetherian ring R and t be a nonnegative integer. Let M and N be two finitely generated R-modules. In certain...     

Local Limitations of the Ext Functor Do Not Exist

In this note it is shown that for k a field, and for the four-dimensionalalgebra = kx, y/x², y², xy + qyx when qⁿ 1, 0 for all n, thereexist a two-dimensional module M and a family of two-dimensionalmodules M_i, i = 1, 2, ..., such that for i equal to 0, j and j + 1, and otherwise. This is probably the most straightforward examplegiving a negative answer to a question raised by Maurice Auslander.2000 Mathematics Subject Classification 16D10, 16E10, 16E30,16G10, 16G20.     

伪Smarandache函数的上下界

对于正整数n,设Z(n)=min{m|m∈N,1/2m(m+1)≡0(modn)},称为n的伪Smarandache函数.设r是正整数.根据广义Ramanujan-Nagell方程的结果,运用初等数论方法证明了下列结果:i)1/2(-1+(8n+1)≤Z(n)≤2n-1.ii)当r≠1,2,3或5时,Z(2~r+1)≥1/2(-1+(2~(r+3)·5+41)).iii)当r≠1,2,3,4或12时,Z(2~r-1)≥1/2(-1+(2~(r+3)·3-23).