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Incidence coalgebras of intervally finite posets, their integral quadratic forms and comodule categories

Tom 115 / 2009

Daniel Simson Colloquium Mathematicum 115 (2009), 259-295 MSC: 16G20, 16G60, 16W30, 16W80. DOI: 10.4064/cm115-2-9

Streszczenie

The incidence coalgebras of intervally finite posets I and their comodules are studied by means of their Cartan matrices and the Euler integral bilinear form b_C:\mathbb Z^{(I)}\times\mathbb Z^{(I)}\rightarrow \mathbb Z. One of our main results asserts that, under a suitable assumption on I, C is an Euler coalgebra with the Euler defect \partial_C:\mathbb Z^{(I)}\times\mathbb Z^{(I)}\rightarrow \mathbb Z zero and b_C ({\bf lgth}\, M,{\bf lgth}\, N) =\chi_C(M,N) for any pair of indecomposable left C-comodules M and N of finite K-dimension, where \chi_C(M,N) is the Euler characteristic of the pair M, N and {\bf lgth}\, M\in \mathbb Z^{(I)} is the composition length vector. The structure of minimal injective resolutions of simple left C-comodules is described by means of the inverse {\mathfrak C}_I^{-1}\in {\mathbb M}^\preceq_I(\mathbb Z) of the incidence matrix {\mathfrak C}_I \in {\mathbb M}_I(\mathbb Z) of the poset I. Moreover, we describe the Bass numbers \mu^I_m(S_I(a), S_I(b)), with m\geq 0, for any simple K^{\square} I-comodules S_I(a), S_I(b) by means of the coefficients of the bth row of {\mathfrak C}_I^{-1}. We also show that, for any poset I of width two, the Grothendieck group {\bf K}_0(K^{\square} I\hbox{-\rm Comod}_{\rm fc}) of the category of finitely copresented K^{\square} I-comodules is generated by the classes [S_I(a)] of the simple comodules S_I(a) and the classes [E_I(a)] of the injective covers E_I(a) of S_I(a), with a\in I.

Autorzy

  • Daniel SimsonFaculty of Mathematics and Computer Science
    Nicolaus Copernicus University
    Chopina 12/18
    87-100 Toruń, Poland
    e-mail

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