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2009 | 115 | 2 | 259-295
Tytuł artykułu

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

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The incidence coalgebras $C = K^{□} I$ of intervally finite posets I and their comodules are studied by means of their Cartan matrices and the Euler integral bilinear form $b_{C}: ℤ^{(I)} × ℤ^{(I)} → ℤ$. One of our main results asserts that, under a suitable assumption on I, C is an Euler coalgebra with the Euler defect $∂_{C}: ℤ^{(I)} × ℤ^{(I)} → ℤ$ zero and $b_{C}(lgth M,lgth} N) = χ_{C}(M,N)$ for any pair of indecomposable left C-comodules M and N of finite K-dimension, where $χ_{C}(M,N)$ is the Euler characteristic of the pair M, N and $lgth M∈ ℤ^{(I)}$ is the composition length vector. The structure of minimal injective resolutions of simple left C-comodules is described by means of the inverse $ℭ_{I}^{-1} ∈ 𝕄 ^{⪯}_{I}(ℤ)$ of the incidence matrix $ℭ_{I} ∈ 𝕄_{I}(ℤ)$ of the poset I. Moreover, we describe the Bass numbers $μₘ^{I}(S_{I}(a),S_{I}(b))$, with m ≥ 0, for any simple $K^{□} I$-comodules $S_{I}(a)$, $S_{I}(b)$ by means of the coefficients of the bth row of $ℭ_{I}^{-1}$. We also show that, for any poset I of width two, the Grothendieck group $K₀(K^{□} I-Comod_{fc})$ of the category of finitely copresented $K^{□} I$-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 ∈ I.
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  • Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
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bwmeta1.element.bwnjournal-article-doi-10_4064-cm115-2-9
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