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.
In his unpublished habilitation thesis @cite_23 , Kuznetsov developed a version of HPD which works for input categories @math that are realizable (compatibly with their @math -linear structure) as an admissible subcategory of the derived category of a variety. Due to my inadequate Russian I was not able to read @cite_23 , but its existence served as an inspiration for this paper.
@cite_23
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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. The following paper is reproduced from a Russian journal of the character of our own Proceedings of the National Academy of Sciences, Comptes Rendus (Doklady) de I'Academie des Sciences de I'URSS, 1942, Volume XXXVII, No. 7-8. The author is one of the most distinguished of Russian mathematicians. He has made very important contributions in pure mathematics in the theory of functional analysis, and has made equally important contributions to applied mathematics in numerical analysis and the theory and practice of computation. Although his exposition in this paper is quite terse and couched in mathematical language which may be difficult for some readers of Management Science to follow, it is thought that this presentation will: (1) make available to American readers generally an important work in the field of linear programming, (2) provide an indication of the type of analytic work which has been done and is being done in connection with rational planning in Russia, (3) through the specific examples mentioned indicate the types of interpretation which the Russians have made of the abstract mathematics (for example, the potential and field interpretations adduced in this country recently by W. Prager were anticipated in this paper). It is to be noted, however, that the problem of determining an effective method of actually acquiring the solution to a specific problem is not solved in this paper. In the category of development of such methods we seem to be, currently, ahead of the Russians.--A. Charnes, Northwestern Technological Institute and The Transportation Center. Let f be a polynomial of degree n in ZZ[x_1,..,x_n], typically reducible but squarefree. From the hypersurface f=0 one may construct a number of other subschemes Y by extracting prime components, taking intersections, taking unions, and iterating this procedure. We prove that if the number of solutions to f=0 in ^n is not a multiple of p, then all these intersections in ^n_ just described are reduced. (If this holds for infinitely many p, then it holds over as well.) More specifically, there is a_Frobenius splitting_ on ^n_ compatibly splitting all these subschemes Y . We determine when a Gr "obner degeneration f_0=0 of such a hypersurface f=0 is again such a hypersurface. Under this condition, we prove that compatibly split subschemes degenerate to compatibly split subschemes, and stay reduced. Our results are strongest in the case that f's lexicographically first term is i=1 ^n x_i. Then for all large p, there is a Frobenius splitting that compatibly splits f's hypersurface and all the associated Y . The Gr "obner degeneration Y' of each such Y is a reduced union of coordinate spaces (a Stanley-Reisner scheme), and we give a result to help compute its Gr "obner basis. We exhibit an f whose associated Y include Fulton's matrix Schubert varieties, and recover much more easily the Gr "obner basis theorem of [Knutson-Miller '05]. We show that in Bott-Samelson coordinates on an opposite Bruhat cell X^v_ in G B, the f defining the complement of the big cell also has initial term i=1 ^n x_i, and hence the Kazhdan-Lusztig subvarieties X^v_ w degenerate to Stanley-Reisner schemes. This recovers, in a weak form, the main result of [Knutson '08]. We describe the wavelet–vaguelette decomposition (WVD) of a linear inverse problem. It is a substitute for the singular value decomposition (SVD) of an inverse problem, and it exists for a class of special inverse problems of homogeneous type—such as numerical differentiation, inversion of Abel-type transforms, certain convolution transforms, and the Radon transform. We propose to solve ill-posed linear inverse problems by nonlinearly "shrinking" the WVD coefficients of the noisy, indirect data. Our approach offers significant advantages over traditional SVD inversion in recovering spatially inhomogeneous objects. We suppose that observations are contaminated by white noise and that the object is an unknown element of a Besov space. We prove that nonlinear WVD shrinkage can be tuned to attain the minimax rate of convergence, for L2 loss, over the entire scale of Besov spaces. The important case of Besov spaces Bσp,q, p < 2, which model spatial inhomogeneity, is included. In comparison, linear procedures— SVD included—cannot attain optimal rates of convergence over such classes in the case p < 2. For example, our methods achieve faster rates of convergence for objects known to lie in the bump algebra or in bounded variation than any linear procedure.