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1990•Transactions of the American Mathematical SocietyRequires access

Boundedness versus periodicity over commutative local rings

Vesselin Gasharov, Irena Peeva

Open publisher page 64 citations

Abstract

Over commutative graded local artinian rings, examples are constructed of periodic modules of arbitrary minimal period and modules with bounded Betti numbers, which are not eventually periodic. They provide counterexamples to a conjecture of D. Eisenbud, that every module with bounded Betti numbers over a commutative local ring is eventually periodic of period 2 2 . It is proved however, that the conjecture holds over rings of small length.

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What this paper is about

Over commutative graded local artinian rings, examples are constructed of periodic modules of arbitrary minimal period and modules with bounded Betti numbers, which are not eventually periodic. They provide counterexamples to a conjecture of D. Eisenbud, that every module with bounded Betti numbers over a commutative local ring is eventually periodic of period 2 2 . It is proved however, that the conjecture holds over rings of small length.

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Available abstract

Over commutative graded local artinian rings, examples are constructed of periodic modules of arbitrary minimal period and modules with bounded Betti numbers, which are not eventually periodic. They provide counterexamples to a conjecture of D. Eisenbud, that every module with bounded Betti numbers over a commutative local ring is eventually periodic of period 2 2 . It is proved however, that the conjecture holds over rings of small length.

Key concepts: Betti number, Mathematics, Conjecture, Counterexample, Bounded function, Commutative property, Commutative ring, Pure mathematics

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