Mathematics

Let (S,L) be a Lie-Rinehart algebra such that L is S -projective and let U be its universal enveloping algebra. In this paper we present a spectral sequence which converges to the Hochschild cohomology of U with values on a U -bimodule M and whose second page involves the Lie-Rinehart cohomology of the algebra and the Hochschild cohomology of S with values on M . After giving a convenient description of the involved algebraic structures we use the spectral sequence to compute explicitly the Hochschild cohomology of the algebra of differential operators tangent to a central arrangement of three lines.

In the context of the Kasparov product in unbounded KK-theory, a well-known theorem by Kucerovsky provides sufficient conditions for an unbounded Kasparov module to represent the (internal) Kasparov product of two other unbounded Kasparov modules. In this article, we discuss several improved and generalised variants of Kucerovsky's theorem. First, we provide a generalisation which relaxes the positivity condition, by replacing the lower bound by a relative lower bound. Second, we also discuss Kucerovsky's theorem in the context of half-closed modules, which generalise unbounded Kasparov modules to symmetric (rather than self-adjoint) operators. In order to deal with the positivity condition for such non-self-adjoint operators, we introduce a fairly general localisation procedure, which (using a suitable approximate unit) provides a 'localised representative' for the KK-class of a half-closed module. Using this localisation procedure, we then prove several variants of Kucerovsky's theorem for half-closed modules. A distinct advantage of the localised approach, also in the special case of self-adjoint operators (i.e., for unbounded Kasparov modules), is that the (global) positivity condition in Kucerovsky's original theorem is replaced by a (less stringent) 'local' positivity condition, which is closer in spirit to the well-known Connes-Skandalis theorem in the bounded picture of KK-theory.

We introduce the notion of a Bredon-style equivariant coarse homology theory. We show that such a Bredon-style equivariant coarse homology theory satisfies localization theorems and that a general equivariant coarse homology theory can be approximated by a Bredon-style version. We discuss the special case of algebraic and topological equivariant coarse K -homology and obtain the coarse analog of Segal's localization theorem.

Let Γ be a f.g. discrete group and let M ~ be a Galois Γ -covering of a smooth closed manifold M . Let S Γ ∗ ( M ~ ) be the analytic structure group, appearing in the Higson-Roe analytic surgery sequence → S Γ ∗ ( M ~ )→ K ∗ (M)→ K ∗ ( C ∗ r Γ)→ . We prove that for an arbitrary discrete group Γ it is possible to map the whole Higson-Roe sequence to the long exact sequence of even/odd-graded noncommutative de Rham homology → H [∗−1] (AΓ)→ H del [∗−1] (AΓ)→ H e [∗] (AΓ)→ , with AΓ a dense homomorphically closed subalgebra of C ∗ r Γ . Here, H del ∗ (AΓ) is the delocalized homology and H e ∗ (AΓ) is the homology localized at the identity element. Then, under additional assumptions on Γ , we prove the existence of a pairing between H C ∗ del (CΓ) , the delocalized part of the cyclic cohomology of CΓ , and H del ∗−1 (AΓ) . This, in particular, gives a pairing between S Γ ∗ ( M ~ ) and H C ∗−1 del (CΓ) . We also prove the existence of a pairing between S Γ ∗ ( M ~ ) and the relative cohomology H [∗−1] (M→BΓ) . Both these parings are compatible with known pairings associated with the other terms in the Higson-Roe sequence. In particular, we define higher rho numbers associated to the rho class ρ( D ~ )∈ S Γ ∗ ( M ~ ) of an invertible Γ -equivariant Dirac type operator on M ~ . Finally, we provide a precise study for the behavior of all previous K-theoretic and homological objects and of the higher rho numbers under the action of the diffeomorphism group of M . Then, we establish new results on the moduli space of metrics of positive scalar curvature when M is spin.

We introduce twisted matrix factorizations for quantum complete intersections of codimension two. For such an algebra, we show that in a given dimension, almost all the indecomposable modules with bounded minimal projective resolutions correspond to such matrix factorizations.

Let L be a (pro-nilpotent) curved L-infinity algebra, and let h be a homotopy between L and a subcomplex M. Using homotopical perturbation theory, Fukaya constructed from this data a curved L-infinity structure on M. We prove that projection from L to M induces a bijection between the set of Maurer-Cartan elements x of L such that hx=0 and the set of Maurer-Cartan elements of M.

We apply a Mayer-Vietoris sequence argument to identify the Atiyah-Segal equivariant complex K-theory rings of certain toric varieties with rings of integral piecewise Laurent polynomials on the associated fans. We provide necessary and sufficient conditions for this identification to hold for toric varieties of complex dimension 2, including smooth and singular cases. We prove that it always holds for smooth toric varieties, regardless of whether or not the fan is polytopal or complete. Finally, we introduce the notion of fans with "distant singular cones," and prove that the identification holds for them. The identification has already been made by Hararda, Holm, Ray and Williams in the case of divisive weighted projective spaces, in addition to enlarging the class of toric varieties for which the identification holds, this work provides an example in which the identification fails. We make every effort to ensure that our work is rich in examples.

We study the mod p r Milnor K -groups of p -adically complete and p -henselian rings, establishing in particular a Nesterenko-Suslin style description in terms of the Milnor range of syntomic cohomology. In the case of smooth schemes over complete discrete valuation rings we prove the mod p r Gersten conjecture for Milnor K -theory locally in the Nisnevich topology. In characteristic p we show that the Bloch-Kato-Gabber theorem remains true for valuation rings, and for regular formal schemes in a pro sense.

Using combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, we give, for any monomial algebra A , an explicit description of its minimal model. This also provides us with formulas for a canonical A ∞ -structure on the Ext-algebra of the trivial A -module. We do this by exploiting the combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, and the algebraic discrete Morse theory of Jöllenbeck, Welker and Sköldberg. We then show how this result can be used to obtain models for algebras with a chosen Gröbner basis, and briefly outline how to compute some classical homological invariants with it.

We examine the cyclic homology of the monoidal category of modules over a finite dimensional Hopf algebra, motivated by the need to demonstrate that there is a difference between the recently introduced mixed anti-Yetter-Drinfeld contramodules and the usual stable anti-Yetter-Drinfeld contramodules. Namely, we show that Sweedler's Hopf algebra provides an example where mixed complexes in the category of stable anti-Yetter-Drinfeld contramodules (previously studied) are not equivalent, as differential graded categories to the category of mixed anti-Yetter-Drinfeld contramodules (recently introduced).