Incidence coalgebras of interval finite posets of tame comodule type

Zbigniew Leszczyński ¹   ; Daniel Simson ¹

¹ Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland

Colloquium Mathematicum, Tome 141 (2015) no. 2, pp. 261-295

Résumé

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 $N$.

Détail

DOI : 10.4064/cm141-2-10

Keywords: incidence coalgebras box interval finite posets their comodules studied means reduced euler integral quadratic form bullet mathbb mathbb where algebraically closed field shown coalgebra tameness category box mbox comod finite dimensional box i modules equivalent tameness category box mbox comod finitely copresented box i modules hence tame wild dichotomy coalgebras box deduced moreover prove interval finite widetilde mathbb * m free poset incidence coalgebra box tame comodule type only quadratic form bullet weakly non negative finally complete list infinite connected interval finite widetilde mathbb * m free posets box tame comodule type prove pair finite dimensional box i comodules overline box operatorname dim operatorname dim sum infty dim operatorname ext box where overline box mathbb times mathbb mathbb euler mathbb z bilinear form operatorname dim operatorname dim dimension vectors