We generalize Kuznetsov's theory of homological projective duality to the setting of noncommutative algebraic geometry. Simultaneously, we develop the theory over general base schemes, and remove the usual smoothness, properness, and transversality hypotheses.
During the preparation of this paper, two other works on HPD appeared. @cite_1 , Jiang--Leung--Xie build on the argument from @cite_16 to prove a generalization of Theorem , where the the pair @math is replaced by any HPD pair. We independently discovered (a more general form of) this result, which appears as an application in @cite_3 .
@cite_16 @cite_1 @cite_3
2618665309 2068169489 2963702702 2062162367
Homological Projective duality (HP-duality) theory, introduced by Kuznetsov [42], is one of the most powerful frameworks in the homological study of algebraic geometry. The main result (HP-duality theorem) of the theory gives complete descriptions of bounded derived categories of coherent sheaves of (dual) linear sections of HP-dual varieties. We show the theorem also holds for more general intersections beyond linear sections. More explicitly, for a given HP-dual pair @math , then analogue of HP-duality theorem holds for their intersections with another HP-dual pair @math , provided that they intersect properly. We also prove a relative version of our main result. Taking @math to be dual linear subspaces (resp. subbundles), our method provides a more direct proof of the original (relative) HP-duality theorem. Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz's, based on reduction to cut-free form in sequent systems, but he also suggested understanding identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally the rules involved in them they yield the same premises and conclusions up to a renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic. The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it did not fare well. The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to low-dimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic. ?1. General proof theory. When a question is not in the main stream of mathematical investigations, and is also of a certain general, conceptual, kind, it runs the danger of being dismissed as "philosophical". Before the advent of recursion theory, many mathematicians would presumably have dismissed the question "What is a computable function?" as a philosophical question (or perhaps even as a psychological, empirical, question). It required something like the enthusiasm of a young discipline on the rise, which logic was in the first half of the twentieth century, for such a question to be embraced as legitimate, and seriously treated by mathematical means-with excellent results. An outsider might suppose that the question "What is a proof?" should be important for a field called proof theory, and then he would be surprised to find that this and related questions, one of which will occupy us here, Received on May 27, 2003; revised July 4, 2003; revised August 26, 2003. 2000 Mathematics Subject Classification. 03F03, 03F07, 03A05, 03-03. We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tet ali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree The aim of this note is to give short algebraic proofs of theorems of Handelman, Polya and Schmudgen concerning the algebraic structure of polynomials being positive on certain subsets of ℝ n . The main ingredient of the proofs is the representation theorem of Kadison–Dubois. The proof of the latter is elementary and algebraic but tricky.