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

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.