In previous papers ([1, 2]) we defined the C*-algebra and the longitudinal pseudodifferential calculus of any singular foliation (M,). In the current paper we construct the analytic index of an elliptic operator as a KK-theory element, and prove that this element can be obtained from an “adiabatic foliation” on M×ℝ, which we introduce here.

We prove an index theorem for families of pseudodifferential operators generalizing those studied by C. Callias, N. Anghel and others. Specifically, we consider operators on a manifold with boundary equipped with an asymptotically conic (scattering) metric, which have the form D + iΦ, where D is elliptic pseudodifferential with Hermitian symbols, and Φ is a Hermitian bundle endomorphism which is invertible at the boundary and commutes with the symbol of D there. The index of such operators is completely determined by the symbolic data over the boundary. We use the scattering calculus of R. Melrose in order to prove our results using methods of topological K-theory, and we devote special attention to the case in which D is a family of Dirac operators, in which case our theorem specializes to give family versions of the previously known index formulas.

Let U be a connected noetherian scheme of finite étale cohomological dimension in which every finite set of points is contained in an affine open subscheme. Suppose that α is a class in H2(Uét,ℂm)tors. For each positive integer m, the K-theory of α-twisted sheaves is used to identify obstructions to α being representable by an Azumaya algebra of rank m2. The étale index of α, denoted eti(α), is the least positive integer such that all the obstructions vanish. Let per(α) be the order of α in H2(Uét,ℂm)tors. Methods from stable homotopy theory give an upper bound on the étale index that depends on the period of α and the étale cohomological dimension of U; this bound is expressed in terms of the exponents of the stable homotopy groups of spheres and the exponents of the stable homotopy groups of B(ℤ/(per(α))). As a corollary, if U is the spectrum of a field of finite cohomological dimension d, then , where [] is the integer part of , whenever per(α) is divided neither by the characteristic of k nor by any primes that are small relative to d.

Let be a moduli space of stable parabolic vector bundles of rank n ≥ 2 and fixed determinant of degree d over a compact connected Riemann surface X of genus g(X) ≥ g(X) = 2, then we assume that n > 2. Let m denote the greatest common divisor of d, n and the dimensions of all the successive quotients of the quasi–parabolic filtrations. We prove that the Brauer group Br is isomorphic to the cyclic group ℤ/mℤ. We also show that Br is generated by the Brauer class of the Brauer–Severi variety over obtained by restricting the universal projective bundle over X × .

This paper examines the relation between the Euler class group of a Noetherian ring and the Euler class group of its polynomial extension. When the ring is a smooth affine domain, the two groups are canonically isomorphic. This is a consequence of a theorem of Bhatwadekar-Sridharan, which they proved in order to answer a question of Nori on sections of projective modules over such rings. If the smoothness assumption is removed, the result of Bhatwadekar-Sridharan is no longer valid and also the Euler class groups above are not in general isomorphic. In this paper we investigate a variant of Nori's question for arbitrary Noetherian rings and derive several consequences to understand the relation between various groups in the theory of Euler classes.

Let S be a degree six del Pezzo surface over an arbitrary field F. Motivated by the first author's classification of all such S up to isomorphism [3] in terms of a separable F-algebra B×Q×F, and by his K-theory isomorphism Kn(S) ≅ Kn(B×Q×F) for n ≥ 0, we prove an equivalence of derived categories

where A is an explicitly given finite dimensional F-algebra whose semisimple part is B×Q×F.

Using the classical universal coefficient theorem of Rosenberg-Schochet, we prove a simple classification of all localizing subcategories of the Bootstrap category Boot ⊂ KK of separable complex C*-algebras. Namely, they are in a bijective correspondence with subsets of the Zariski spectrum Specℤ of the integers – precisely as for the localizing subcategories of the derived category D(ℤ) of complexes of abelian groups. We provide corollaries of this fact and put it in context with the similar classifications available in the literature.

We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show the existence of a new structure in the above quotient category, which we call a triangle of recollements. Moreover, we show that this quotient category is triangle equivalent to the stable module category of Cohen-Macaulay T2(R)-modules.