Related papers: Comparing Dualities in the $K(n)$-local Category

We study several duality isomorphisms between equivariant bivariant K-theory groups, generalising Kasparov's first and second Poincare duality isomorphisms. We use the first duality to define an equivariant generalisation of Lefschetz…

If k is an arbitrary field, we construct a category of k-1-motives in which every commutative algebraic k-group G has a dual object $G^{\vee}$. When k is a local field of arbitrary characteristic, we establish Pontryagin duality theorems…

The work is my Ph D thesis (dissertation for obtaining candidate of sciences degree in Russia) fulfilled under direction of D. A. Raikov and defended under supervision of N. Ya. Vilenkin and S. V. Ptchelintsev. In the dissertatin I gave…

Classical Spanier-Whitehead duality was introduced for the stable homotopy category of finite CW complexes. Here we provide a comprehensive treatment of a noncommutative version, termed Spanier-Whitehead $K$-duality, which is defined on the…

Let $k$ be a field. We show that locally presentable, $k$-linear categories $\mathcal{C}$ dualizable in the sense that the identity functor can be recovered as $\coprod_i x_i\otimes f_i$ for objects $x_i\in \mathcal{C}$ and left adjoints…

This is the first in a series of papers that deals with duality statements such as Mukai-duality (T-duality, from algebraic geometry) and the Baum-Connes conjecture (from operator $K$-theory). These dualities are expressed in terms of…

We determine a substantial part of the unitary representation theory of the Drinfeld double of a $q$-deformation of a compact Lie group in terms of the complexification of the compact Lie group. Using this, we show that the dual of every…

We study the structure of the formal groups associated to the Morava $K$-theories of integral Eilenberg-Mac Lane spaces. The main result is that every formal group in the collection $\{K(n)^*K({\mathbb Z}, q), q=2,3,...\}$ for a fixed $n$…

We reduced Rudyak's conjecture that a degree one map between closed manifolds cannot raise the Lusternik-Schnirelmann category to the computation of the category of the product of two lens spaces $L^n_p\times L_q^n$ with relatively prime…

Let \(G\) be a reductive group over a field \(k\), and let \(\mu\) be a cocharacter of \(G\). We prove that Viehmann's double coset spaces associated with \((G, \mu)\) are representable by certain Lusztig varieties, and establish a similar…

C*-algebras generalizing Cuntz-Krieger algebras can be associated to hyperbolic homeomorphisms of compact metric spaces. They satisfy a non-commutative form of Spanier-Whitehead duality with respect to K-theory. We prove this for the case…

In general, if M is a moduli space of stable sheaves on X, there is a unique alpha in the Brauer group of M such that a pi_M^* alpha^{-1}-twisted universal sheaf exists on X times M. In this paper we study the situation when X and M are K3…

We establish equivalences of derived categories of the following 3 categories: (1) Principal block of representations of the quantum at a root of 1; (2) G-equivariant coherent sheaves on the Springer resolution; (3) Perverse sheaves on the…

We prove that two cusps of the same dimension in the Baily-Borel compactification of some classical series of modular varieties are linearly dependent in the rational Chow group of the compactification. This gives a higher dimensional…

In this paper, we establish the global analogues of some dualities and equivalences in local algebra by developing the theory of relative Cohen-Macaulay modules. Let R be a commutative Noetherian ring (not necessarily local) with identity…

Following a suggestion of Hovey and Strickland, we study the category of $K(k) \vee K(k+1) \vee \cdots \vee K(n)$-local spectra. When $k = 0$, this is equivalent to the category of $E(n)$-local spectra, while for $k = n$, this is the…

A well-known and old result of Hazewinkel and Koszul states that the cohomology of a finite-dimensional Lie algebra is isomorphic, up to a suitable shift, to its twisted homology, a Lie-theoretical version of Poincare duality. This paper…

We are going to show that the sheafication of graded Koszul modules $% K_{\Gamma}$ over $\Gamma_{n}=K[ x_{0},x_{1}...x_{n}] $ form an important subcategory $\overset{\wedge}{K}_{\Gamma}$ of the coherents sheaves on projective space,…

Two rings A and B are said to be derived Morita equivalent if their derived categories of modules are equivalent. By results of Rickard, if A and B are derived Morita equivalent algebras over a field k, then there is a complex of bimodules…

In this paper, we construct a version of Auslander-Reiten sequences for the $K(n)$-local stable homotopy category. In particular, the role of the Auslander-Reiten translation is played by the local Brown-Comenetz duality functor. As an…