This paper is a natural extension of the previous note [Ar2]. Semiinfinite cohomology of Tate Lie algebra was defined in that note in terms of some duality resembling Koszul duality. The language of differential graded Lie algebroids was the main technical tool of the note. The present note is devoted to globalization of the main construction from [Ar2] in the following sense. The setup in [Ar2...

Given a monomial m in polynomial ring and subset L of the variables ring, principal -Borel ideal generated by is all monomials which can be obtained from successively replacing those are have smaller index. collection I = { 1 , … r } where i for (where subsets may different each ideal), we prove essence that if bipartite incidence graph among chordal bipartite, then defining equations multi-Ree...

In this paper we prove that the linear Koszul duality isomorphism for convolution algebras in K-homology of [MR3] and the Fourier transform isomorphism for convolution algebras in Borel– Moore homology of [EM] are related by the Chern character. So, Koszul duality appears as a categorical upgrade of Fourier transform of constructible sheaves. This result explains the connection between the cate...

We define analogues of homogeneous coordinate algebras for noncommutative two-tori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate algebras. We give a criterion for such an algebra to be Koszul and prove that the Koszul dual algebra also c...

We describe applications of Koszul cohomology to the BrillNoether theory of rank 2 vector bundles. Among other things, we show that in every genus g > 10, there exist curves invalidating Mercat’s Conjecture for rank 2 bundles. On the other hand, we prove that Mercat’s Conjecture holds for general curves of bounded genus, and its failure locus is a Koszul divisor in the moduli space of curves. W...

The reductions of an ideal I give a natural pathway to the properties of I , with the advantage of having fewer generators. In this paper we primarily focus on a conjecture about the reduction exponent of links of a broad class of primary ideals. The existence of an algebra structure on the Koszul and Eagon–Northcott resolutions is the main tool for detailing the known cases of the conjecture. ...

1. Flat, projective, and free modules 1 2. Review of regular local rings 2 3. Cohen-Macauley rings 2 4. The Koszul complex 3 5. The Koszul resolution detects depth 5 6. The detection of depth by use of Ext 6 7. Global dimension 6 8. Minimal resolutions and global dimension 7 9. Serre’s characterization of regular local rings 8 10. The Auslander-Buchsbaum theorem 10 11. Localizations of regular ...