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Abstract

The incidence coalgebras $ K^{\Box} I$ of interval finite posets $I$ and their comodules are studied by means of the reduced Euler integral quadratic form $q^\bullet :\mathbb Z^{(I)}\to \mathbb Z$, where $K$ is an algebraically closed field. It is shown that for any such coalgebra the tameness of the category $K^{\Box} I\mbox{-}{\rm comod}$ of finite-dimensional left $ K^{\Box} I$-modules is equivalent to the tameness of the category $K^{\Box} I{\mbox{-}{\rm Comod}_{{\rm fc}}}$ of finitely copresented left $ K^{\Box} I$-modules. Hence, the tame-wild dichotomy for the coalgebras $K^{\Box} I$ is deduced. Moreover, we prove that for an interval finite $\widetilde {\mathbb A}^*_m$-free poset $I$ the incidence coalgebra $K^{\Box} I$ is of tame comodule type if and only if the quadratic form $q^\bullet $ is weakly non-negative. Finally, we give a complete list of all infinite connected interval finite $\widetilde {\mathbb A}^*_m$-free posets $I$ such that $K^{\Box} I$ is of tame comodule type. In this case we prove that, for any pair of finite-dimensional left $K^{\Box} I$-comodules $M$ and $N$, $ \overline b_{K^{\Box} I} (\operatorname{\bf dim} M,\operatorname{\bf dim} N) = \sum _{j=0}^{\infty}(-1)^j\dim_K \operatorname{Ext}_{K^{\Box} I}^j(M,N) $, where $ \overline b_{K^{\Box} I}:\mathbb Z^{(I)}\times \mathbb Z^{(I)}\to \mathbb Z $ is the Euler $\mathbb Z$-bilinear form of $I$ and $\operatorname{\bf dim} M$, $\operatorname{\bf dim} N$ are the dimension vectors of $M$ and $N$.