We develop some of the ingredients needed for string theory on noncommutative spacetimes, proposing an axiomatic formulation of T-duality as well as establishing a very general formula for D-brane charges. This formula is closely related to a noncommutative Grothendieck-Riemann-Roch theorem that is proved here. Our approach relies on a very general form of Poincaré duality, which is studied her...

We study the representation theory of increasing monoid. Our results provide a fairly comprehensive picture category: for example, we describe Grothendieck group (including effective cone), classify injective objects, establish properties and projective resolutions, construct derived auto-duality, so on. work is motivated by numerous connections this to other areas, such as stability, commutati...

T-duality acts on circle bundles by exchanging the first Chern class with the fiberwise integral of the H-flux, as we motivate using E8 and also using S-duality. We present known and new examples including NS5-branes, nilmanifolds, Lens spaces, both circle bundles over RPn, and the AdS×S to AdS×CP×S with backgroundH-flux of Duff et al. When T-duality leads to M-theory on a non-spin manifold the...

The duality theory of coherent sheaves on algebraic varieties goes back to Roch’s half of the Riemann-Roch theorem for Riemann surfaces (1870s). In the 1950s, it grew into Serre duality on normal projective varieties; and shortly thereafter, into Grothendieck duality for arbitrary varieties and more generally, maps of noetherian schemes. This theory has found many applications in geometry and c...

We show that any separated essentially finite-type map f of noetherian schemes globally factors as f = hi where i is an injective localization map and h a separated finite-type map. In particular, via Nagata’s compactification theorem, h can be chosen to be proper. We apply these results to Grothendieck duality. We also obtain other factorization results and provide essentialized versions of ma...

For a noetherian scheme, we introduce its unbounded stable derived category. This leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. Some applications are included, for instance an analogue of maximal CohenMacaulay approximations, a construction of Tate cohomology, and an extension ...

In 1970 Alexander Grothendieck [6] posed the following problem: let Γ1 and Γ2 be finitely presented, residually finite groups, and let u : Γ1 → Γ2 be a homomorphism such that the induced map of profinite completions û : Γ̂1 → Γ̂2 is an isomorphism; does it follow that u is an isomorphism? In this paper we settle this problem by exhibiting pairs of groups u : P ↪→ Γ such that Γ is a direct product...