Hom-computable coalgebras, a composition factors matrix and the Euler bilinear form of an Euler coalgebra

Abstract

A composition factors matrix C F is studied for any basic Hom-computable K-coalgebra C over an arbitrary field K, in connection with a Cartan matrix C F of C. Left Euler K-coalgebras C are defined. They are studied by means of an Euler integral bilinear form b C :the Euler characteristic χ C (M, N ) of Euler pairs of C-comodules M and N , and an Euler defect ∂ C : K 0 (C) × K 0 (C) → Z of C. It is shown that b C (lgth M, lgth N) = χ C (M, N ) + ∂ C (M, N ), for all M, N in C-comod, and ∂ C = 0, if all simple C-comodules are of finite injective dimension. A diagrammatic characterisation of representationdirected hereditary Hom-computable coalgebras is given.

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

“…b K I (v) > 0 for every non-zero vector v ∈ Z (I) with non-negative coordinates. They are just the representation-directed coalgebras in the sense of [30,Section 6]. We also show in [32] that every such coalgebra K I is tame of discrete comodule type [27] and gl.dim K I ≤ 2.…”

“…Then M = p∈I M p is of finite K-dimension and one easily shows that the linear maps q ϕ p induce a left C-comultiplication δ M : M → C ⊗ M on M such that M is a comodule and Φ(M ) = (M p , q ϕ p ) p≺q . Hence, by simple limit arguments, Φ is dense, and consequently it is an equivalence of categories (see also [30,Proposition 3.3…”

“…We show that the coalgebra C = K I is basic, c -hereditary, Homcomputable in the sense of [30], left locally artinian (hence left cocoherent), the Cartan matrix C F = C F ∈ M I (Z) [30, (4. I · y tr for all x, y ∈ Z (I) , where Z (I) is the direct sum of I-copies of the group Z. Propositions 2.9 and 2.12 give simple formulae for c − ab in terms of the poset I (see (2.11) and (2.13)). We also show that, for any intervally finite poset I without infinitely many pairwise incomparable elements, C is an Euler coalgebra, the Euler defect ∂ C : Z (I) × Z (I) → Z is zero [30, (4.14)], the Euler characteristic…”

“…Following [29] and [30], a comodule N in C-Comod is said to be finitely cogenerated (or socle-finite) if N is a subcomodule of a finite direct sum of indecomposable injective comodules, or equivalently, dim K soc N is finite. We say that N is finitely copresented if there is an exact sequence 0 → N → E → E in C-Comod, where E and E are each a finite direct sum of indecomposable injective comodules.…”

“…Similarly, the Euler quadratic form of a coalgebra or a finite-dimensional algebra is one of the basic tools in determining the representation type (see [1], [4], [15], [21], [25]- [30], [32], [33], [34]). …”