Mathematics

We show that a finite dimensional monomial algebra satisfies the finite generation conditions of Snashall-Solberg for Hochschild cohomology if and only if it is Gorenstein. This gives, in the case of monomial algebras, the converse to a theorem of Erdmann-Holloway-Snashall-Solberg-Taillefer. We also give a necessary and sufficient combinatorial criterion for finite generation.

We present two explicit combinatorial constructions of finitely summable reduced "Gamma"-elements γ r ∈KK( C ∗ r (Γ),C) for any word-hyperbolic group (Γ,S) and obtain summability bounds for them in terms of the cardinality of the generating set S⊂Γ and the hyperbolicity constant of the associated Cayley graph.

In this paper, we show that the maximal divisible subgroup of groups K 1 and K 2 of an elliptic curve E over a function field is uniquely divisible. Further those K -groups modulo this uniquely divisible subgroup are explicitly computed. We also calculate the motivic cohomology groups of the minimal regular model of E , which is an elliptic surface over a finite field.

We give an overview of the general framework of forms of Bak, Tits and Wall, when restricting to vector spaces over fields, and describe its relationship to the classical notions of Hermitian, alternating and quadratic forms. We then prove a version of Witt's lemma in this context, showing in particular that the action of the group of isometries of a space equipped with a form is transitive on isometric subspaces.

We extend the results of G.~Garkusha and I.~Panin on framed motives of algebraic varieties [4] to the case of a finite base field, and extend the computation of the zeroth cohomology group H 0 (ZF( Δ ∙ k , G ∧n m ))= K MW n by A.~Neshitov [8] to perfect fields k of positive characteristic different from 2.

The twisted equivariant K-theory given by Freed and Moore is a K-theory which unifies twisted equivariant complex K-theory, Atiyah's `Real' K-theory, and their variants. In a general setting, we formulate this K-theory by using Fredholm operators, and establish basic properties such as the Bott periodicity and the Thom isomorphism. We also provide formulations of the K-theory based on Karoubi's gradations in both infinite and finite dimensions, clarifying their relationship with the Fredholm formulation.

A t -structure t=( C t≤0 , C t≥0 ) on a triangulated category C is right adjacent to a weight structure w=( C w≤0 , C w≥0 ) if C t≥0 = C w≥0 ; then t can be uniquely recovered from w and vice versa. We prove that if C satisfies the Brown representability property then t that is adjacent to w exists if and only if w is smashing (i.e., coproducts respect weight decompositions); then the heart Ht is the category of those functors H w op →Ab that respect products. The dual to this statement is related to results of B. Keller and P. Nicolas. We also prove that an adjacent t exists whenever w is a bounded weight structure on a saturated R -linear category C (for a noetherian ring R ); for C= D perf (X) , where the scheme X is regular and proper over R , this gives 1-to-1 correspondences between bounded weights structures on C and the classes of those bounded t -structures on it such that Ht has either enough projectives or injectives. We generalize this existence statement to construct (under certain assumptions) a t -structure t on a triangulated category C ′ such that C and C ′ are subcategories of a common triangulated category D and t is right orthogonal to w . In particular, if X is proper over R but not necessarily regular then one can take C= D perf (X) , C ′ = D b coh (X) or C ′ = D − coh (X) , and D= D qc (X) . We also study hearts of orthogonal t -structures and their restrictions, and prove some statements on "reconstructing" weight structures from orthogonal t -structures. The main tool of this paper are virtual t -truncations of (cohomological) functors; these are defined in terms of weight structures and "behave as if they come from t -truncations" whether t exists or not.

Let N be a closed spin manifold with positive scalar curvature and D N the Dirac operator on N . Let M 1 and M 2 be two Galois covers of N such that M 2 is a quotient of M 1 . Then the quotient map from M 1 to M 2 naturally induces maps between the geometric C ∗ -algebras associated to the two manifolds. We prove, by a finite-propagation argument, that the \emph{maximal} higher rho invariants of the lifts of D N to M 1 and M 2 behave functorially with respect to the above quotient map. This can be applied to the computation of higher rho invariants, along with other related invariants.

The main result of this paper is the G -homotopy invariance of the G -index of signature operator of proper co-compact G -manifolds. If proper co-compact G manifolds X and Y are G -homotopy equivalent, then we prove that the images of their signature operators by the G -index map are the same in the K -theory of the C ∗ -algebra of the group G . Neither discreteness of the locally compact group G nor freeness of the action of G on X are required, so this is a generalization of the classical case of closed manifolds. Using this result we can deduce the equivariant version of Novikov conjecture for proper co-compact G -manifolds from the Strong Novikov conjecture for G .

Extending constructions by Gabriel and Zisman, we develop a functorial framework for the cohomology and homology of simplicial sets with very general coefficient systems given by functors on simplex categories into abelian categories. Furthermore we construct Leray type spectral sequences for any map of simplicial sets. We also show that these constructions generalise and unify the various existing versions of cohomology and homology of small categories and as a bonus provide new insight into their functoriality.