Decompositions of degenerations
Let \(k\) be an algebraically closed field and let \(\Lambda\) be a finite dimensional associative \(k\)-algebra with unit. By definition, a \(d\)-dimensional \(\Lambda\)-module is the vector space \(k^d\) endowed with a multiplication by \(\Lambda\). Making use of a purely algebraic description of the partial order \(\leq_{\text{deg}}\) [\textit{G. Zwara}, Compos. Math. 121, No. 2, 205-218 (2000; Zbl 0957.16007)], the author develops a method of decomposition of a given degeneration of \(\Lambda\)-modules into finer ones. He then applies this method to the case when \(\Lambda\) has a directed Auslander-Reiten quiver and gives a new proof of the equivalence of the orders \(\leq_{\text{deg}}\) and \(\leq_{\text{ext}}\) for such algebras. In fact, this result has been earlier obtained in a different context by \textit{K. Bongartz} [in Algebras and modules I. Workshop on representations of algebras and related topics, Trondheim, Norway, 1996. CMS Conf. Proc. 23, 1-27 (1998; Zbl 0915.16008)].
scientific article; zbMATH DE number 1491674