Orders of Bounded and Strongly Unbounded Lattice Type
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Orders of Bounded and Strongly Unbounded Lattice Type Fahimeh Sadat Fotouhi1 · Alex Martsinkovsky2 · Shokrollah Salarian1,3 Received: 20 August 2018 / Accepted: 30 July 2019 / © Springer Nature B.V. 2019
Abstract Brauer and Thrall conjectured that a finite-dimensional algebra over a field of bounded representation type is actually of finite representation type and a finite-dimensional algebra (over an infinite field) of infinite representation type has strongly unbounded representation type. The first conjecture was proved in full generality, and the second conjecture was proved under the additional assumption that the field be algebraically closed. These results are our motivation for studying (generalized) orders of bounded and strongly unbounded lattice type. To each lattice over an order we assign a numerical invariant, h-length, measuring Hom modulo projectives. We show that an order of bounded lattice type is actually of finite lattice type, and if there are infinitely many non-isomorphic indecomposable lattices of the same h-length, then the order has strongly unbounded lattice type. For a hypersurface R = k[[x0 , ..., xd ]]/(f ), we show that R is of bounded (respectively, strongly unbounded) lattice type if and only if the double branched cover R of R is of bounded (respectively, strongly unbounded) lattice type. This is an analog of a result of Kn¨orrer and BuchweitzGreuel-Schreyer for rings of finite mCM type. Consequently, it is proved that R has strongly unbounded lattice type whenever k is infinite. Keywords Brauer-Thrall conjectures · Order · Lattice · Bounded lattice type · Strongly unbounded lattice type Presented by: Steffen Koenig Shokrollah Salarian
[email protected] Fahimeh Sadat Fotouhi [email protected] Alex Martsinkovsky [email protected] 1
Department of Mathematics, University of Isfahan, P.O.Box: 81746-73441, Isfahan, Iran
2
Mathematics Department, Northeastern University, 360 Huntington Avenue, Boston, MA 02115, USA
3
School of Mathematics, Institute for Research in Fundamental Science (IPM), P.O.Box: 19395-5746, Tehran, Iran
F.S. Fotouhi et al.
Mathematics Subject Classification (2010) 16G30 · 16H20 · 16G60
1 Introduction In [15], Jans states that R. Brauer and R. M. Thrall conjectured that a finite-dimensional algebra over a field k of bounded representation type (meaning that there is a bound on the lengths of the indecomposable finitely generated modules) is actually of finite representation type. They also conjectured that a finite-dimensional algebra over an infinite field of infinite representation type has strongly unbounded representation type (meaning that there is an infinite sequence n1 < n2 < · · · of positive integers such that there are, for any i, infinitely many non-isomorphic indecomposable modules of k-dimension ni ). Both conjectures are now theorems. The first Brauer-Thrall conjecture was proved by Roiter [24]. Later, Ringel [22, 23] proved it for artin algebras. Bautista [8] and Bongartz [9] proved the second conjecture under the extra
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