The first Brauer–Thrall conjecture over one-dimensional local rings

Abstract

Let [Formula: see text] be a one-dimensional commutative Noetherian complete local ring. Assume that the cohomology annihilator [Formula: see text] is [Formula: see text]-primary. We use the notion of Gabriel–Roiter (co)measure in the category of maximal Cohen–Macaulay [Formula: see text]-modules and prove that, if there is an infinite set [Formula: see text] of indecomposable maximal Cohen–Macaulay [Formula: see text]-modules of bounded multiplicity, then there are indecomposable maximal Cohen–Macaulay modules of arbitrary large multiplicity which are cogenerated by modules in [Formula: see text]. This, in particular, guarantees the validity of the first Brauer–Thrall-type theorem for the category of maximal Cohen–Macaulay [Formula: see text]-modules.