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arXiv · 2509.11858

Cohen-Macaulay Type via Lattice Homology and the Motivic Poincar\'e Series

Alex Hof·András Némethi
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Abstract

We give results on reduced complex-analytic curve germs which relate their indecomposable maximal Cohen-Macaulay (MCM) modules to their lattice homology groups and related invariants, thereby providing a connection between the algebraic theory of MCM modules and techniques arising from low-dimensional topology. In particular, we characterize the germs $(C, o)$ of finite Cohen-Macaulay type in terms of the lattice homology $\mathbb{H}_*(C, o)$, and those of tame type in terms of the lattice homologies and associated spectral sequences of $(C, o)$ and its subcurves, including the distinction between germs of finite and infinite growth. As a consequence of these results, we obtain corresponding characterizations of a germ's Cohen-Macaulay type in terms of the motivic Poincar\'e series.

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Alex Hof, András Némethi. 2025-09-15. Cohen-Macaulay Type via Lattice Homology and the Motivic Poincar\'e Series. https://arxiv.org/abs/2509.11858

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