This book contains a detailed exposition of the nonhomogeneous Koszul duality theory in relative situation over noncentral, noncommutative, nonsemisimple base ring, as announced Section 0.4 arXiv:0708.3398. We prove Poincare-Birkhoff-Witt theorem this context and construct triangulated equivalences derived duality. The between ring differential operators de Rham DG-algebra, with functions is th...

Let K be a field of characteristic 0 and S = K[x1, . . . , xn] the polynomial ring over K with graded maximal ideal m = (x1, . . . , xn). Denote by βi(M) = Tor S i (K,M) the ith Betti number of a finitely generated graded module M and by Gin(I) the generic initial ideal of a graded ideal I with respect to the reverse lexicographical order. In this paper we answer (positively) a question raised ...

This paper provides a rigorous account on the geometry of forms supermanifolds, with focus its algebraic-geometric aspects. First, we introduce de Rham complex differential and compute cohomology. We then discuss three intrinsic definitions Berezinian sheaf supermanifold - as quotient sheaf, via cohomology super Koszul or total complex. Further, study properties showing in particular that it de...

Let I be an ideal generated by quadrics in a standard graded polynomial ring S over field. A question of Avramov, Conca, and Iyengar asks whether the Betti numbers R = / can bounded above binomial coefficients on minimal number generators if is Koszul. This has been answered affirmatively for Koszul algebras defined three almost complete intersections with any generators. We give strong affirma...

In this paper we construct, for F1 and F2 subbundles of a vector bundle E, a “Koszul duality” equivalence between derived categories of Gm-equivariant coherent (dg-)sheaves on the derived intersection F1 R ∩EF2, and the corresponding derived intersection F ⊥ 1 R ∩E∗F ⊥ 2 . We also propose applications to Hecke algebras.

Abstract Conca–Rossi–Valla [6] ask if every quadratic Gorenstein ring $R$ of regularity three is Koszul. In [15], we use idealization to answer their question, proving that in nine or more variables there exist rings three, which are not this paper, study the analog question when four more. Let be a having ${\operatorname {codim}} \ R = c$ and {reg}} r \ge 4$. We prove $c r+1$ then always Koszu...