A general construction of n-angulated categories using periodic injective resolutions

The axioms for n-angulated categories

Triangulated categories were introduced independently in algebraic geometry by Verdier [7, 8], based on ideas of Grothendieck, and in algebraic topology by Puppe [6]. These constructions have since played a crucial role in representation theory, algebraic geometry, commutative algebra, algebraic topology and other areas of mathematics (and even theoretical physics). Recently, Geiss, Keller and ...

Periodic Resolutions and Self-injective Algebras of Finite Type

We say that an algebra A is periodic if it has a periodic projective resolution as an (A, A)bimodule. We show that any self-injective algebra of finite representation type is periodic. To prove this, we first apply the theory of smash products to show that for a finite Galois covering B → A, B is periodic if and only if A is. In addition, when A has finite representation type, we build upon res...

Copure injective resolutions, flat resolvents and dimensions

In this paper, we show the existence of copure injective preenvelopes over noetherian rings and copure flat preenvelopes over commutative artinian rings. We use this to characterize n-Gorenstein rings. As a consequence, if the full subcategory of strongly copure injective (respectively flat) modules over a left and right noetherian ring R has cokernels (respectively kernels), then R is 2-Gorens...

Path categories and resolutions

The path category P (X) of a simplicial set X defines a functor X 7→ P (X) which is left adjoint to the nerve functor B : cat → sSet. The first section of this paper contains an elementary description of this functor, with some of its basic properties. This construction specializes to a path category functor for cubical sets which appears in the second section. The path category construction is...

Injective objects in triangulated categories