Extension classes for almost-split sequences #
For a map f : P₁ → P₀ between finite projectives over a finite-dimensional algebra,
assume that T = D Tr(f) has local endomorphism ring and is not injective. There is a
nonzero class in Ext¹(coker f,T) killed by pushforward along every map out of T
which is not a split monomorphism.
Such a class is the extension-theoretic input for constructing a left almost-split sequence: vanishing of its pushout is equivalent to factorization through the first map of a representing short exact sequence. The result constructs the class itself; it does not choose a representing sequence or assert the right lifting property.
The construction combines LinearMap.auslanderReitenExtEquiv with a functional on
injective stable endomorphisms that vanishes on the image of the radical and takes
the value one on the identity. Finite-dimensional biduality turns this functional
into an extension class. The test modules need not be finite-dimensional.
References #
- M. Auslander, I. Reiten, S. O. Smalø, Representation Theory of Artin Algebras, Cambridge University Press (1995), Sections IV.2 and V.1.
A noninjective translate with local endomorphism ring admits a nonzero extension class annihilated by every pushforward along a map that is not a split monomorphism. No minimality of the presentation or algebraic closedness of the field is required.