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1994•Bulletin of the Korean Mathematical SocietyRequires access

BETTI NUMBERS OVER ARTINIAN LOCAL RINGS

Sang‐Ki Choi

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Abstract

In this paper we study exponential growth of Betti numbers over artinian local rings. By the Change of Tor Formula the results in the paper extend to the asymptotic behavior of Betti numbers over Cohen-Macaulay local rings. Using the length function of an artinian ring we calculate an upper bound for the number of generators of modules, this is then used to maximize the number of generators of sygyzy modules. Finally, applying a filtration of an ideal, which we call a Loewy series of an ideal, we derive an invariant B(R) of an artinian local ring R, such that if B(R)>1, then the sequence (M) of Betti numbers is strictly increasing and has strong exponential growth for any finitely generated non-free R-module M (Theorem 2.7).).

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What this paper is about

In this paper we study exponential growth of Betti numbers over artinian local rings. By the Change of Tor Formula the results in the paper extend to the asymptotic behavior of Betti numbers over Cohen-Macaulay local rings. Using the length function of an artinian ring we calculate an upper bound for the number of generators of modules, this is then used to maximize the number of generators of sygyzy modules. Finally, applying a filtration of an ideal, which we call a Loewy series of an ideal, we derive an invariant B(R) of an artinian local ring R, such that if B(R)>1, then the sequence (M) of Betti numbers is strictly increasing and has strong exponential growth for any finitely generated non-free R-module M (Theorem 2.7).).

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Available abstract

In this paper we study exponential growth of Betti numbers over artinian local rings. By the Change of Tor Formula the results in the paper extend to the asymptotic behavior of Betti numbers over Cohen-Macaulay local rings. Using the length function of an artinian ring we calculate an upper bound for the number of generators of modules, this is then used to maximize the number of generators of sygyzy modules. Finally, applying a filtration of an ideal, which we call a Loewy series of an ideal, we derive an invariant B(R) of an artinian local ring R, such that if B(R)>1, then the sequence (M) of Betti numbers is strictly increasing and has strong exponential growth for any finitely generated non-free R-module M (Theorem 2.7).).

Key concepts: Mathematics, Betti number, Local ring, Ideal (ethics), Pure mathematics, Discrete mathematics, Finitely-generated abelian group, Exponential function

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