Author

Tony J. Puthenpurakal

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Preprint Aug 2026

Bounds on multiplicity of MCM modules having non-extremal growth of betti-numbers

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and residue field $k$. Let $M$ be a maximal Cohen-Macaulay $A$-module. Let $e(M)$ be the multiplicity of $M$ and let $\mu(M)$ denote the number of its minimal generators. (1) Assume $A$ is not a complete intersection. If $\text{curv}(M)<\text{curv}(k)$ then we prove that under mild conditions, $e(M) \geq \mu(M)(1 + \text{curv}(k))$. (2) Assume $A$ is a complete intersection. If $\text{cx}(M)<\text{cx}(k)$ then we prove that $e(M) \geq 2\mu(M)$. In both cases we give examples which shows our results are sharp.

Tony J. Puthenpurakal · 0 citations

Preprint Jul 2026

On generalization of two results of Foxby

Let $(A,\mathfrak{m})$ be a Noetherian local ring of dimension $d$ and let $M$ be a finitely generated $A$-module. Assume $M$ has rank $r>0$. We show that if $M$ is NOT Cohen-Macaulay then $\mu_d(\mathfrak{m}, M)>r$. If further $A$ is unmixed and $\mu_n(\mathfrak{m}, M) \leq 1$ for some $n \geq d$ then we prove $\text{injdim} \ M<\infty$ and $A$ is Cohen-Macaulay.

Tony J. Puthenpurakal · 0 citations