In 1966 [1], Auslander introduced a class of finitely generated modules having a certain complete resolution by projective modules. Then using these modules, he defined the G-dimension (G ostensibly for Gorenstein) of finitely generated modules. It seems appropriate then to call the modules of G-dimension 0 the Gorenstein projective modules. In [4], Gorenstein projective modules (whether finite...

When R is a commutative ring, the minimal free resolution of a map Ra → Rb and symmetric and skew-symmetric maps Ra → Ra, under suitable generality conditions, are well known and have been developed by a series of authors. See [2, A2.6] for an overview. In this note we do an analog for the exterior algebra E = ⊕ ∧i V on a finite dimensional vector space V and general graded maps Ea → E(1)b, and...

In this paper we prove that the 3-fold injective tensor product ℓ1⊗ˆεℓ1⊗ˆεℓ1 is not isomorphic to any subspace of ℓ1⊗ˆεℓ1. This result provides a new solution problem Diestel on projective products c0. Moreover, implies for infinite countable compact metric space K, C(K)⊗ˆπC(K)⊗ˆπC(K) quotient C(K)⊗ˆπC(K).

We study the representation theory of increasing monoid. Our results provide a fairly comprehensive picture category: for example, we describe Grothendieck group (including effective cone), classify injective objects, establish properties and projective resolutions, construct derived auto-duality, so on. work is motivated by numerous connections this to other areas, such as stability, commutati...

Serre [8; 9] has established the rudiments of a dictionary for translating the language of projective modules into that of vector bundles. With this point of departure we have attempted to adapt some of the results and methods of homotopy theory to certain purely arithmetic and even noncommutative settings. Detailed proofs of our results will appear elsewhere. The authors are very grateful to A...

For an (n− 1)-Auslander algebra Λ with global dimension n, we give some necessary conditions for Λ admitting a maximal (n − 1)-orthogonal subcategory in terms of the properties of simple Λ-modules with projective dimension n − 1 or n. For an almost hereditary algebra Λ with global dimension 2, we prove that Λ admits a maximal 1orthogonal subcategory if and only if for any non-projective indecom...

Let X be a projective bundle. We prove that X admits an endomorphism of degree > 1 and commuting with the projection to the base, if and only if X trivializes after a finite covering. When X is the projectivization of a vector bundle E of rank 2, we prove that it has an endomorphism of degree > 1 on a general fiber only if E splits after a finite base change. It is clear that, for a complex pro...