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Some results on the generic vanishing of Koszul cohomology via deformation theory  [PDF]
Jie Wang
Mathematics , 2011,
Abstract: We study the deformation-obstruction theory of Koszul cohomology groups of $g^r_d$'s on singular nodal curves. We compute the obstruction classes for Koszul cohomology classes on singular curves to deform to a smooth one. In the case the obstructions are nontrivial, we obtain some partial results for generic vanishing of Koszul cohomology groups.
On the vanishing of weight one Koszul cohomology of abelian varieties  [PDF]
Marian Aprodu,Luigi Lombardi
Mathematics , 2015,
Abstract: In this Note we prove the vanishing of (twisted) Koszul cohomology groups $K_{p,1}$ of an abelian variety with values in powers of an ample line bundle. It complements the work of G. Pareschi on the property $(N_p)$.
Generic vanishing and minimal cohomology classes on abelian varieties  [PDF]
Giuseppe Pareschi,Mihnea Popa
Mathematics , 2006,
Abstract: We establish a, and conjecture further, relationship between the existence of subvarieties representing minimal cohomology classes on principally polarized abelian varieties, and the generic vanishing of certain sheaf cohomology. The main ingredient is the Generic Vanishing criterion of math.AG/0608127, based on the Fourier-Mukai transform.
Koszul cohomology and singular curves  [PDF]
Edoardo Ballico,Claudio Fontanari,Luca Tasin
Mathematics , 2009,
Abstract: We investigate Koszul cohomology on irreducible nodal curves. In particular, we prove both Green and Green-Lazarsfeld conjectures for the general k-gonal nodal curve.
Koszul duality and Galois cohomology  [PDF]
Leonid Positselski,Alexander Vishik
Mathematics , 1995, DOI: 10.4310/MRL.1995.v2.n6.a8
Abstract: It it shown that the Bloch-Kato conjecture on the norm residue homomorphism $K^M(F)/l \to H^*(G_F,Z/l)$ follows from its (partially known) low-degree part under the assumption that the Milnor K-theory algebra $K^M(F)/l$ modulo $l$ is Koszul. This conclusion is a case of a general result on the cohomology of nilpotent (co-)algebras and Koszulity.
The cohomology of Koszul-Vinberg algebras  [PDF]
Michel Nguiffo Boyom
Mathematics , 2002,
Abstract: This work is devoted to an intrinsic cohomology theory of Koszul-Vinberg algebras and their modules. Our results may be regarded as improvements of the attempt by Albert Nijenhuis in [NA]. The relationships between the cohomology theory developed here and some classical problems are pointed out, e.g. extensions of algebras and modules, and deformation theory. The real Koszul-Vinberg cohomology of locally flat manifolds is initiated. Thus regarding the idea raised by M. Gerstenhaber we can state : The category of KV-algebras has its proper cohomology theory.
Koszul cohomology and applications to moduli  [PDF]
Marian Aprodu,Gavril Farkas
Mathematics , 2008,
Abstract: We discuss recent progress on syzygies of curves, including proofs of Green's and Gonality Conjectures as well as applications of Koszul cycles to the study of the birational geometry of various moduli spaces of curves. We prove a number of new results, including a complete solution to Green's Conjecture for arbitrary hexagonal curves. Finally, we propose several new conjectures on syzygies, including a Prym-Green conjecture for l-roots of trivial bundles as well as a strong Maximal Rank Conjecture for generic curves. To appear in the Proceedings of Clay Mathematical Institute.
On the vanishing of cohomology in triangulated categories  [PDF]
Petter Andreas Bergh
Mathematics , 2008,
Abstract: We study the vanishing of cohomology in triangulated categories admitting a central ring action. In particular, we study vanishing gaps and symmetry.
Koszul duality and equivariant cohomology  [PDF]
Matthias Franz
Mathematics , 2003,
Abstract: Let G be a topological group such that its homology H(G) with coefficients in a principal ideal domain R is an exterior algebra, generated in odd degrees. We show that the singular cochain functor carries the duality between G-spaces and spaces over BG to the Koszul duality between modules up to homotopy over H(G) and H^*(BG). This gives in particular a Cartan-type model for the equivariant cohomology of a G-space. As another corollary, we obtain a multiplicative quasi-isomorphism C^*(BG) -> H^*(BG). A key step in the proof is to show that a differential Hopf algebra is formal in the category of A-infinity algebras provided that it is free over R and its homology an exterior algebra.
Koszul duality and equivariant cohomology for tori  [PDF]
Matthias Franz
Mathematics , 2003, DOI: 10.1155/S1073792803206103
Abstract: Let T be a torus. We show that Koszul duality can be used to compute the equivariant cohomology of topological T-spaces as well as the cohomology of pull backs of the universal T-bundle. The new features are that no further assumptions about the spaces are made and that the coefficient ring may be arbitrary. This gives in particular a Cartan-type model for the equivariant cohomology of a T-space with arbitrary coefficients. Our method works for intersection homology as well.
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