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# Topic: Koszul complex

###### In the News (Sun 21 Apr 19)

 Koszul (print-only) Koszul was appointed as Maître de Conférences at the University of Strasbourg in 1949. In 1950 Koszul published a major 62 page paper Homologie et cohomologie des algèbres de Lie in which he studied the connections between the homology and cohomology (with real coefficients) of a compact connected Lie group G and purely algebraic problems on the Lie algebra associated with G. Koszul was honoured with election to the Academy of Sciences in Paris on 28 January 1980. www-groups.dcs.st-and.ac.uk /history/Printonly/Koszul.html   (816 words)

 [No title] Although it turns out that the Schur complex is exactly this gen- eralization of the Koszul complex we need, it seems, surprisingly enough, that nobody has tried to generalize the De-Rham complex in a similar man- ner. I focus on the most fundamental case of the complex associated to the identity map (this is the case important for the applications I have in mind). Similarly, d equipped 8 with a homological differential is dual to the Koszul complex. hopf.math.purdue.edu /Chalupnik/cohsdr.txt   (8238 words)

 Computing the homology of Koszul complexes, by Bernhard Koeck   (Site not responding. Last check: 2007-10-12) Let R be a commutative ring and I an ideal in R which is locally generated by a regular sequence of length d. In this paper, we compute the homology of the n-th Koszul complex associated with the homomorphism P_1 --> P_0 for all n, if d = 1. Furthermore, if d = 2, we compute the homology of the complex N Sym^2 K(P.) where K and N denote the functors occurring in the Dold-Kan correspondence. www.math.uiuc.edu /K-theory/0303   (135 words)

 Springer Online Reference Works The Koszul complex defined by these data then consists of the modules The general Koszul complex can be viewed as built up from these elementary constituents as This (and the above) makes Koszul complexes an important tool in commutative and homological algebra, for instance in dimension theory and the theory of multiplicities (and intersection theory), cf. eom.springer.de /k/k055810.htm   (166 words)

 list We find a place in the minimal free resolution of a module of finite CI-dimension after which asymptotically stable behavior develops; in particular, beyond this place the sequence of Betti numbers is either constant or strictly increasing, and gaps between consecutive numbers grow polynomially. Castelnuovo-Mumford regularity is an important measure of the complexity of the differential in a graded minimal free resolution. Vanishing of the regularity of the residue field defines the class of Koszul algebras, which have received considerable attention due to their extraordinary homological properties and to their appearance in many cases of interest in algebraic geometry, algebraic topology, combinatorics, commutative algebra, and representation theory. www.math.cornell.edu /~irena/list.html   (2476 words)

 koszul -- a differential in a Koszul map   (Site not responding. Last check: 2007-10-12) koszul -- a differential in a Koszul map -- provides the i-th differential in the Koszul complex associated to f. Here f should be a 1 by n matrix. www.msri.org /about/computing/docs/macaulay/2-0.9/1089.html   (29 words)

 Representations of Algebras 2003 Abstract: For a large class of rings, including all those encountered in algebraic geometry, we establish the conjectured Morita-like equivalence, known as Foxby equivalence, between the full subcategory of complexes of finite Gorenstein flat dimension and that of complexes of finite Gorenstein injective dimension. Abstract: The theory of Koszul duality for operads is analogous to the theory of Koszul duality for algebras. The reason for introducing this notion is to extend the theory of Koszul duality for operads to the situation involving both operations and cooperations. mystic.math.neu.edu /alexmart/MADL/RT2003.html   (1220 words)

 AMCA: Noncommutative spectral mapping theorem by Anar Dosiev Let E be a finite-dimensional Lie algebra embedded into the algebra B(X) of bounded linear operators on a complex Banach space X. X is Banach A-module), such that the Koszul complex generated by E-module (X, \alpha) is a Banach complex of A-modules. Assume that the image of L on each member of the Koszul complex is finite-dimensional. at.yorku.ca /c/a/e/o/17.htm   (484 words)

 3.3 Non-regular Extensions constitutes a double complex the total complex of which is a resolution of is the natural generalization of the Koszul complex. The resolution constructed via Proposition 3.7 is not minimal in general: Going back to the proof of Lemma 3.6 we see that a chosen representation www.mathematik.uni-kl.de /~zca/Reports_on_ca/28/paper_html/node6.html   (186 words)

 SMF - Publications - Bulletin de la SMF - Parutions - 123 - pages 87-105 Nous rappelons dans la première partie quelques résultats de base sur le complexe de Koszul gradué. The central theorem of this paper is a result on the degree where the higher non-zero homology module of the Koszul complex constructed with homogeneous polynomials over a field becomes non trivial. This result has a straightforward corollary on the complexity of the determination of the dimension of a projective variety. smf.emath.fr /Publications/Bulletin/123/html/smf_bull_123_87-105.html   (833 words)

 Commutative Algebra and Algebraic Geometry Seminar   (Site not responding. Last check: 2007-10-12) Although the question was not fully answered in the Bourbaki series, its pursuit opened up new fields of mathematics. In our new approach, we begin with an infinitesimal algebraic formulation of Stokes' theorem via the Koszul complex over a normed vector space. This induces a family of norms over pointed chains of the Koszul complex. math.berkeley.edu /alg/talks/F05/1129-2   (203 words)

 Citebase - Noether identities of a generic differential operator. The Koszul-Tate complex   (Site not responding. Last check: 2007-10-12) This construction is generalized to arbitrary differential operators on a smooth fiber bundle. Namely, if a certain necessary and sufficient condition holds, one can associate to a differential operator the exact chain complex with the boundary operator whose nilpotency condition restarts all the Noether identities characterizing the degeneracy of an original differential operator. Users are cautioned not to use it for academic evaluation yet. www.citebase.org /abstract?id=oai:arXiv.org:math/0506103   (142 words)

 Newsgroops - Re: Where did the Koszul complex come from?   (Site not responding. Last check: 2007-10-12) Newsgroops - Re: Where did the Koszul complex come from? Re: Where did the Koszul complex come from? I just want to know how one can come up with the Koszul complex www.newsgroops.org /group/sci.math/article-545688.html   (139 words)

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