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

• Autorzy: Daniel Simson
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 16G20: Representations of quivers and partially ordered sets
• 16W80: Topological and ordered rings and modules
• 16G60: Representation type (finite, tame, wild, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2009
• Tom: 115
• Numer: 2
• Strony: 259-295

Daty

wydano

2009

Twórcy

autor

Daniel Simson

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

Identyfikatory

DOI

10.4064/cm115-2-9