共查询到20条相似文献,搜索用时 15 毫秒
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Amnon Yekutieli James J. Zhang 《Proceedings of the American Mathematical Society》1997,125(3):697-707
We prove the Serre duality theorem for the noncommutative projective scheme when is a graded noetherian PI ring or a graded noetherian AS-Gorenstein ring.
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Let be a projective unitary representation of a countable groupG on a separable Hilbert space H. If the set B of Bessel vectorsfor is dense in H, then for any vector x H the analysis operatorx makes sense as a densely defined operator from B to 2(G)-space.Two vectors x and y are called -orthogonal if the range spacesof x and y are orthogonal, and they are -weakly equivalent ifthe closures of the ranges of x and y are the same. These propertiesare characterized in terms of the commutant of the representation.It is proved that a natural geometric invariant (the orthogonalityindex) of the representation agrees with the cyclic multiplicityof the commutant of (G). These results are then applied to Gaborsystems. A sample result is an alternate proof of the knowntheorem that a Gabor sequence is complete in L2( d) ifand only if the corresponding adjoint Gabor sequence is 2-linearlyindependent. Some other applications are also discussed. 相似文献
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Edgar E. Enochs Overtoun M. G. Jenda Jinzhong Xu 《Transactions of the American Mathematical Society》1996,348(8):3223-3234
In 1966, Auslander introduced the notion of the -dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of -dimensions. It seemed appropriate to call the modules with -dimension 0 Gorenstein projective, so the basic problem was to define Gorenstein injective modules. These were defined in Math. Z. 220 (1995), 611--633 and were shown to have properties predicted by Auslander's results. The way we define Gorenstein injective modules can be dualized, and so we can define Gorenstein projective modules (i.e. modules of -dimension 0) whether the modules are finitely generated or not. The investigation of these modules and also Gorenstein flat modules was continued by Enochs, Jenda, Xu and Torrecillas. However, to get good results it was necessary to take the base ring Gorenstein. H.-B. Foxby introduced a duality between two full subcategories in the category of modules over a local Cohen-Macaulay ring admitting a dualizing module. He proved that the finitely generated modules in one category are precisely those of finite -dimension. We extend this result to modules which are not necessarily finitely generated and also prove the dual result, i.e. we characterize the modules in the other class defined by Foxby. The basic result of this paper is that the two classes involved in Foxby's duality coincide with the classes of those modules having finite Gorenstein projective and those having finite Gorenstein injective dimensions. We note that this duality then allows us to extend many of our results to the original Auslander setting.
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Various topological properties of projective duality between real projective varieties and their duals are obtained by making use of the microlocal theory of (subanalytically) constructible sheaves developed by Kashiwara [M. Kashiwara, Index theorem for constructible sheaves, Astérisque 130 (1985) 193-209] and Kashiwara-Schapira [M. Kashiwara, P. Schapira, Sheaves on Manifolds, Grundlehren Math. Wiss., vol. 292, Springer, Berlin-Heidelberg-New York, 1990]. In particular, we prove in the real setting some results similar to the ones proved by Ernström in the complex case [L. Ernström, Topological Radon transforms and the local Euler obstruction, Duke Math. J. 76 (1994) 1-21]. For this purpose, we describe the characteristic cycles of topological Radon transforms of constructible functions in terms of curvatures of strata in real projective spaces. 相似文献
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Audun Holme 《manuscripta mathematica》1988,61(2):145-162
In this paper we investigate some fundamental geometric and numerical properties ofduality for projective varieties inP
k
N
=P
N
. We take a point of view which in our opinion is somewhat moregeometric and lessalgebraic andnumerical than what has been customary in the literature, and find that this can some times yield simpler and more natural proofs, as well as yield additional insight into the situation. We first recall the standard definitions of thedual variety and theconormal scheme, introducing classical numerical invariants associated with duality. In section 2 we recall the well known duality properties these invariants have, and which was noted first byT. Urabe. In section 3 we investigate the connection between these invariants andChern classes in the singular case. In section 4 we give a treatment of the dual variety of a hyperplane section of X, and the dual procedure of taking the dual of a projection of X. This simplifies the proofs of some very interesting theorems due toR. Piene. Section 5 contains a new and simpler proof of a theorem ofA. Hefez and S. L. Kleiman. Section 6 contains some further results, geometric in nature. 相似文献
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《Journal of Pure and Applied Algebra》2023,227(2):107188
We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well as a general local duality theorem which extends, to a much broader class of rings, previous results by Herzog-Zamani and Suzuki. As an application, we establish a prescribed upper bound for the projective dimension of a module satisfying suitable cohomological conditions, and we derive some freeness criteria and questions of Auslander-Reiten type. Along the way, we prove a new characterization of Cohen-Macaulay modules which truly relies on generalized local cohomology, and in addition we introduce and study a generalization of the notion of canonical module. 相似文献
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R. Sauer 《Geometric And Functional Analysis》2006,16(2):476-515
Building upon work of Y. Shalom we give a homological-algebra flavored definition of an induction map in group homology associated
to a topological coupling. As an application we obtain that the cohomological dimension cdR over a commutative ring R satisfies the inequality
if Λ embeds uniformly into Γ and
holds. Another consequence of our results is that the Hirsch ranks of quasi-isometric solvable groups coincide. Further,
it is shown that the real cohomology rings of quasi-isometric nilpotent groups are isomorphic as graded rings. On the analytic
side, we apply the induction technique to Novikov-Shubin invariants of amenable groups, which can be seen as homological invariants,
and show their invariance under quasi-isometry.
Received: November 2004 Revision: April 2004 Accepted: April 2004 相似文献
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Supported by Ministerio de Education y Ciencia, grant #BE91-031 相似文献
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Tom Braden Anthony Licata Nicholas Proudfoot Ben Webster 《Advances in Mathematics》2010,225(4):2002-2049
Given a hyperplane arrangement in an affine space equipped with a linear functional, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of our algebras, which are in many ways analogous to integral blocks of category O. 相似文献
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Summary. To a pair consisting of an elliptic curve and a point on it, Odeskii and Feigin associate certain quadratic algebras (“Sklyanin
algebras”), having the Hilbert series of a polynomial algebra.
In this paper we show that Sklyanin algebras have good homological properties and we obtain some information about their so-called
linear modules. We also show how the construction by Odeskii and Feigin may be generalized so as to yield other “Sklyanin-type”
algebras.
Oblatum 25-XI-1993 相似文献
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Vladimir Hinich 《代数通讯》2013,41(10):3291-3323
We study the varieties that parametrize trigonal curves with assigned Weierstrass points; we prove that they are irreducible and compute their dimensions. To do so, we stratify the moduli space of all trigonal curves with given Maroni invariant. 相似文献
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On Towers Approximating Homological Localizations 总被引:2,自引:0,他引:2
Our object of study is the natural tower which, for any givenmap f:AB and each space X, starts with the localization of Xwith respect to f and converges to X itself. These towers canbe used to produce approximations to localization with respectto any generalized homology theory E*, yielding, for example,an analogue of Quillen's plus-construction for E*. We discussin detail the case of ordinary homology with coefficients inZ/p or Z[1/p]. Our main tool is a comparison theorem for nullificationfunctors (that is, localizations with respect to maps of theform f:Apt), which allows us, among other things, to generalizeNeisendorfer's observation that p-completion of simply-connectedspaces coincides with nullification with respect to a Moorespace M(Z[1/p], 1). 相似文献
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Kok-Ming Teo 《代数通讯》2013,41(9):3027-3035
In their recent paper [13], Tate and Van den Bergh studied certain quadratic algebras, called the “Sklyanin algebras”. They proved that these algebras have the Hilbert series of a polynomial algebra, are Noetherian and Koszul, and satisfy the Auslander-Gorenstein and Cohen-Macaulay conditions. This paper gives an alternative proof of these results, as suggested in [13], and thereby answering a question in their paper. 相似文献
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