Topological Hochschild cohomology and generalized Morita equivalence

We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an extension of the algebraic theory to include the case when $M$ is not necessarily a progenerator. Our approach is complementary to recent work of Dwyer and Greenlees and of Schwede and Shipley. A central notion of noncommutative ring theory related to Morita equivalence is that of central separable or Azumaya algebras. For such an Azumaya algebra A, its Hochschild cohomology HH^*(A,A) is concentrated in degree 0 and is equal to the center of A. We introduce a notion of topological Azumaya algebra and show that in the case when the ground S-algebra R is an Eilenberg-Mac Lane spectrum of a commutative ring this notion specializes to classical Azumaya algebras. A canonical example of a topological Azumaya R-algebra is the endomorphism R-algebra F_R(M,M) of a finite cell R-module. We show that the spectrum of mod 2 topological K-theory KU/2 is a nontrivial topological Azumaya algebra over the 2-adic completion of the K-theory spectrum widehat{KU}_2. This leads to the determination of THH(KU/2,KU/2), the topological Hochschild cohomology of KU/2. As far as we know this is the first calculation of THH(A,A) for a noncommutative S-algebra A.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-29.abs.htm

Non-commutative crepant resolutions: scenes from categorical geometry

Non-commutative crepant resolutions are algebraic objects defined by Van den Bergh to realize an equivalence of derived categories in birational geometry. They are motivated by tilting theory, the McKay correspondence, and the minimal model program, and have applications to string theory and representation theory. In this expository article I situate Van den Bergh's definition within these contexts and describe some of the current research in the area.Comment: 57 pages; final version, to appear in "Progress in Commutative Algebra: Ring Theory, Homology, and Decompositions" (Sean Sather-Wagstaff, Christopher Francisco, Lee Klingler, and Janet Vassilev, eds.), De Gruyter. Incorporates many small bugfixes and adjustments addressing comments from the referee and other

D-equivalence and K-equivalence

Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$. They are called {\it $D$-equivalent} if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories, while {\it $K$-equivalent} if they are birationally equivalent and the pull-backs of their canonical divisors to a common resolution coincide. We expect that the two equivalences coincide at least for birationally equivalent varieties. We shall provide a partial answer to the above problem in this paper.Comment: 25 pages, minor chang

The Almost Equivalence by Asymptotic Probabilities for Regular Languages and Its Computational Complexities

We introduce p-equivalence by asymptotic probabilities, which is a weak almost-equivalence based on zero-one laws in finite model theory. In this paper, we consider the computational complexities of p-equivalence problems for regular languages and provide the following details. First, we give an robustness of p-equivalence and a logical characterization for p-equivalence. The characterization is useful to generate some algorithms for p-equivalence problems by coupling with standard results from descriptive complexity. Second, we give the computational complexities for the p-equivalence problems by the logical characterization. The computational complexities are the same as for the (fully) equivalence problems. Finally, we apply the proofs for p-equivalence to some generalized equivalences.Comment: In Proceedings GandALF 2016, arXiv:1609.0364