We construct a singular homology theory on the category of schemes of finite type over a Dedekind domain and verify several basic properties. For arithmetic schemes we construct a reciprocity isomorphism between the integral singular homology in degree zero and the abelianized modified tame fundamental group.

We establish an algebra-isomorphism between the complexified Grothendieck ring F of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This provides a purely categorical proof of a conjecture by Ostrik concerning the structure of F . As a by-product we obtain a concrete expression for the structure...

We construct a canonical pseudofunctor ( ) on the category of finite-type maps of (say) connected noetherian universally catenary finite-dimensional separated schemes, taking values in the category of Cousin complexes. This pseudofunctor is a concrete approximation to the restriction of the Grothendieck Duality pseudofunctor( ) to the full subcategory of the derived category having Cohen-Macaul...

Dade [D1, Theorem 7.4] obtained an important result on the equivalence of categories, extending the classical stable Clifford theory. He used the theory of strongly graded rings. Recently, this work has been generalized to arbitrary graded rings, see E. Dade [D2], [D3], J.L. Gómez Pardo and C. Nǎstǎsescu [GN ], C. Nǎstǎsescu and F. Van Oystaeyen [NVO2]. In the classical case the stable Clifford...

We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group W = G(m, 1, n) and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category O for the rational Cherednik algebra of W , including a shift isomorphism which is proved in Appendix 1.

We study the circumstances under which one can reconstruct a stack from its associated functor of isomorphism classes. This is possible surprisingly often: we show that many of the standard examples of moduli stacks are determined by their functors. Our methods seem to exhibit new anabelian-type phenomena, in the form of structures in the category of schemes that encode automorphism data in gro...

Let Covering be the category of fuzzy coverings, and Partition, partitions. We geometrically construct an isomorphism categories between Partition a full subcategory Covering, which can used to derive bijections partitions coverings with finitely many sets. Also, we establish Covering[n], n sets, whose objects are sets satisfy certain conditions.

We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic Kand L-theory of integral group rings and to the Baum-Connes Conjecture on the topological K-theory of reduced group C∗-algebras. The approach is through spectra over the orbit category of a discrete group G. We give several points of view on the assembly map for a family of subgroups and describe suc...

Here HOM is the functor category, π∗ is induced by the natural projection π : hocolimI C −→ I, 0 is the category with only one map and id maps the only object of 0 to the identity functor. A reason for using the above definitions is that taking nerves one recovers the usual homotopy (co)limits for simplicial sets, up to homotopy in the case of hocolim ([T]) and up to isomorphism in the case of ...