Rigid families and endomorphism algebras of Kronecker modules

Abstract

LetR be a commutative ring with an identity element and let$\Gamma _2 \left( R \right) = \left( {_{0R}^{RR^2 } } \right)$ be the the KroneckerR-algebra. One of our main results is Theorem 1.2 asserting that for anyR-algebraA generated by λ elements, where λ is an infinite cardinal number, there exists a rigid direct system$\mathbb{F} = \left\{ {\mathbb{F}_\beta ,f_{\beta \gamma } } \right\}_{\beta \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \subset } \gamma \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \subset } \lambda }$ (see Definition 2.8) of fully faithfulR-linear exact functors$\mathbb{F}_\beta :Mod(A) \to Mod(\Gamma _2 (R))$ connected byR-splitting functorial monomorphisms$f_{\beta \gamma } :\mathbb{F}_\beta \to \mathbb{F}_\gamma$ satisfying some extra conditions. In particular, ifR is a field then everyR-algebra generated by at most λ elements is isomorphic to an endomorphism algebra EndX of a Kronecker moduleX=(X ^′,X ^′′,ϕ^′,ϕ^′′) in ModΓ₂(R) such that dim _R X′=dim _R X″=λ, theR-linear maps ϕ^′,ϕ^′′:X ^′→X ^′′ are injective andX ^′′ ≡ Imϕ ^′ + Imϕ ^′′.