Representations of finite partially ordered sets over commutative artinian uniserial rings

Abstract

Categories of representations of finite partially ordered sets over commutative artinian uniserial rings arise naturally from categories of lattices over orders and abelian groups. By a series of functorial reductions and a combinatorial analysis, the representation type of a category of representations of a finite partially ordered set S over a commutative artinian uniserial ring R is characterized in terms of S and the index of nilpotency of the Jacobson radical of R. These reductions induce isomorphisms of Auslander-Reiten quivers and preserve and reflect Auslander-Reiten sequences. Included, as an application, is the completion of a partial characterization of representation type of a category of representations arising from pairs of finite rank completely decomposable abelian groups.

Indecomposable (1,3)-Groups and a matrix problem

“…But then A has no unit to avoid a horizontal double cross in [β 1 | β 2 ] that displays a direct summand of rank 3, cf. Corollary 26 (4). Hence A = 0 and G is decomposable by Proposition 27.…”

“…(2) and (3). We annihilate with both, pI in the Smith Normal Form of pB 4 , and with I in the Smith Normal Form of F 1 . This can be done independently and the fill-ins can be annihilated as in (2) 1,1 AY 1,1 mod p is decomposed, contradicting the hypothesis on A.…”

“…Note the following consequences if additional blocks are present, and some properties of the entries of pB 4 , F 1 , pD 1 , pD 2 .…”

“…Moreover, the entries of pB 4 are either 0 or in pZ \ p 2 Z since the entries in p 2 Z can be annihilated by pI to the left in β 1 . The fill-ins in the L-row below pB 4 are 0 modulo p. There is no 0-row in pB 4 to avoid a vertical double cross located in β 1 .…”

Endo-wild representation type and generic representations of finite posets

“…As demonstrated in [Arnold 2000;Arnold and Simson 2002;Simson 1992], properties of the categories fspr(S, R) for a commutative uniserial ring R and fpr(S, k) for an arbitrary field k have immediate application to categories of abelian groups and lattices over orders. In particular, Theorems 2.4, 3.4 and 3.8 answer some open questions stated in [Arnold 2000] for some quasi-homomorphism categories of torsion-free abelian groups of finite rank.…”