On the density of fiber sum functors
Fiber sum functors furnish a technique to study the category \(A\)-mod of finitely generated left \(A\)-modules over a finite dimensional \(k\)-algebra where \(k\) is a field. Namely consider \(P \in A\text{-mod}\) satisfying \(\text{Ext}^ 1_ A(P,\text{fac }P) = 0 = \text{Ext}^ 1_ A(P,\text{ sub }P)\) and the full subcategory \({\mathcal K} = {\mathcal K}(A,P)\) of all \(V \in A\text{-mod}\) such that \(\text{Ext}^ 1_ A(V, \text{fac }P) = 0\). The fiber sum functor \(F\) with respect to \(P\) has as domain the category \(\text{mat}(P,{\mathcal K})\) of all \(A\)-homomorphisms \(\delta : U \to V\) where \(U \in \text{add }P\), \(V \in {\mathcal K}\) and as target the category \(A\)-mod. It maps a given object \(\delta \in \text{mat}(P,{\mathcal K})\) to its cokernel. A very classical tool to study the category \(A\)-mod is to describe the modules by their projective presentations. A more modern device for this study is the use of one-point (co)extensions. We explain that fiber sum functors give a common access to both techniques by choosing suitable modules \(P\). But our main interest lies in the situations not covered by these special cases. It was shown by the author [J. Algebra 113, 430-437 (1988; Zbl 0659.16020)] that \(F\) is always full and has a well-behaved, small kernel whereas the density of \(F\) was proved only for \(A\) of finite representation type. The first main result of our paper is to establish the density of \(F\) in general by observing that \(\text{fac }P\) is a covariantly finite subcategory and using results of \textit{M. Auslander} and \textit{I. Reiten} [Adv. Math. 86, 111-152 (1991; Zbl 0774.16006)]. Obviously \(F\) is only useful if \(\text{mat }(P,{\mathcal K})\) is better understood than \(A\)-mod. This is the case if \(k\) is algebraically closed and \(P\) is an indecomposable projective module with \(\dim_ k \text{End}_ A(P) = 1\). Under these assumptions \(\text{mat}(P,K)\) can be identified with the subspace category of the vectorspace category \(({\mathcal K},\text{Hom}_ A(P,-))\). Subspace categories of vectorspace categories are a generalization of representations of partially ordered sets. Many questions about the representation type are easier to answer for these categories than for \(A\)-mod itself. For example there exist efficient criteria for vectorspace categories to be of tame subspace type. Our second main result is concerned with this situation. We prove that \(A\) is of tame representation type if and only if the following three conditions are satisfied: a) The vector space category \(({\mathcal K}_ P, \text{Hom}_ A(P,-))\) is of tame subspace type. b) The set of isomorphism classes of indecomposable objects \(V\) in \({\mathcal K}_ P\) satisfying \(\dim_ k V\leq n\) is finite for all \(n \in \mathbb{N}\). c) The factor algebra of \(A\) by the trace ideal of \(P\) is of tame representation type. Condition c) is harmless as this factor algebra is of smaller \(k\)- dimension so that one can work inductively. The category \({\mathcal K}_ P\) occurring in a), b) is the full subcategory of \(\mathcal K\) given by objects \(V\) admitting only direct summands \(V'\) with \(\text{Hom}_ A(P,V') \neq 0\). This reflects the fact that, if the factor algebra of \(A\) by the trace ideal of \(P\) is representation tame, then the tameness of \(A\) is decided by the modules not annihilated by this ideal.
representations of partially ordered sets
scientific article; zbMATH DE number 703850
scientific article