Mathematics

We provide a framework connecting several well known theories related to the linearity of graded modules over graded algebras. In the first part, we pay a particular attention to the tensor products of graded bimodules over graded algebras. Finally, we provide a tool to evaluate the possible degrees of a module appearing in a graded projective resolution once the generating degrees for the first term of some particular projective resolution are known.

Letting G=F/R be a finitely-presented group, Hopf's formula expresses the second integral homology of G in terms of F and R . Expanding on previous work, we explain how to find generators of H 2 (G; F p ) . The context of the problem, which is related to a conjecture of Quillen, is presented, as well as example calculations.

We describe a map from the equivariant twisted K-homology of a compact, connected, simply connected Lie group G to the Verlinde ring. Our map is described at the level of `D-cycles' for the geometric twisted K-homology of G , and is inverse to the Freed-Hopkins-Teleman isomorphism. As an application, we show that two possible definitions of the `quantization' of a Hamiltonian loop group space are compatible with each other.

Given an abelian category, we introduce a categorical concept of (strongly) Gorenstein projective (resp., injective) objects, by defining a new special class of objects. Then we study the transfer of these properties when passing to an abelian category and its n-trivial extension category and also give a characterization of Gorenstein object over it. We give, at the end, applications of this study on the category of modules over an associative ring and triangular matrix rings.

Let R⊂A be a Frobenius extension of rings. We prove that: (1) for any left A -module M , A M is Gorenstein projective (injective) if and only if the underlying left R -module R M is Gorenstein projective (injective). (2) if G- proj.dim A M<∞ , then G- proj.dim A M=G- proj.dim R M , the dual for Gorenstein injective dimension also holds. (3) if the extension is split, then G-gldim(A)=G-gldim(R) .

We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective, then its underlying module over the the base ring is Gorenstein projective; the converse holds if the Frobenius extension is either left-Gorenstein or separable (e.g. the integral group ring extension Z⊂ZG ). Moreover, for the Frobenius extension R⊂A=R[x]/( x 2 ) , we show that: a graded A -module is Gorenstein projective in GrMod(A) , if and only if its ungraded A -module is Gorenstein projective, if and only if its underlying R -module is Gorenstein projective. It immediately follows that an R -complex is Gorenstein projective if and only if all its items are Gorenstein projective R -modules.

We study the category of left unital graded modules over the Steinberg algebra of a graded ample Hausdorff groupoid. In the first part of the paper, we show that this category is isomorphic to the category of unital left modules over the Steinberg algebra of the skew-product groupoid arising from the grading. To do this, we show that the Steinberg algebra of the skew product is graded isomorphic to a natural generalisation of the the Cohen-Montgomery smash product of the Steinberg algebra of the underlying groupoid with the grading group. In the second part of the paper, we study the minimal (that is, irreducible) representations in the category of graded modules of a Steinberg algebra, and establish a connection between the annihilator ideals of these minimal representations, and effectiveness of the groupoid. Specialising our results, we produce a representation of the monoid of graded finitely generated projective modules over a Leavitt path algebra. We deduce that the lattice of order-ideals in the K 0 -group of the Leavitt path algebra is isomorphic to the lattice of graded ideals of the algebra. We also investigate the graded monoid for Kumjian--Pask algebras of row-finite k -graphs with no sources. We prove that these algebras are graded von Neumann regular rings, and record some structural consequences of this.

We study the algebraic K -theory and Grothendieck-Witt theory of proto-exact categories, with a particular focus on classes of examples of F 1 -linear nature. Our main results are analogues of theorems of Quillen and Schlichting, relating the K -theory or Grothendieck-Witt theories of proto-exact categories defined using the (hermitian) Q -construction and group completion.

Building on work by Kasparov, we study the notion of Spanier-Whitehead K-duality for a discrete group. It is defined as duality in the KK-category between two C*-algebras which are naturally attached to the group, namely the reduced group C*-algebra and the crossed product for the group action on the universal example for proper actions. We compare this notion to the Baum-Connes conjecture by constructing duality classes based on two methods: the standard "gamma element" technique, and the more recent approach via cycles with property gamma. As a result of our analysis, we prove Spanier-Whitehead duality for a large class of groups, including Bieberbach's space groups, groups acting on trees, and lattices in Lorentz groups.

Let B?�A be a left or right bounded extension of finite dimensional algebras. We use the Jacobi-Zariski long nearly exact sequence to show that B satisfies Han's conjecture if and only if A does, regardless if the extension splits or not. We provide conditions ensuring that an extension by arrows and relations is left or right bounded. Finally we give a structure result for extensions of an algebra given by a quiver and admissible relations, and examples of non split left or right bounded extensions.