Localization in tame and wild coalgebras

Abstract

We apply the theory of localization for tame and wild coalgebras in order to prove the following theorem: "Let Q be an acyclic quiver. Then any tame admissible subcoalgebra of K Q is the path coalgebra of a quiver with relations".

Tame comodule type, roiter bocses, and a geometry context for coalgebras

“…3. In particular, it shows that the coalgebra C is fc-tame if and only if every socle-finite colocalization C E ∼ = R • E of C (in the sense of [11,25]) is fc-tame.…”

“…Since E 0 is a direct summand of E Uv , by [11] and [25] (Proposition 2.7 and Theorem 2.10) there is an isomorphism E 0 ∼ = E 0 , the socle of res Ev (E 1 ) is a finite-dimensional subcomodule of the coalgebra e v Ce v and the socle of E ∨ 1 = e v C evCev res Ev (E 1 ) is a finite direct sum of comodules S(a), with a ∈ U v . It follows that the injective envelope E 1 = E C (E ∨ 1 ) of the C-comodule E ∨ 1 lies in add(E Uv ).…”

“…where res Ev : C-Comod fc −→ e v Ce v -Comod fc is the exact restriction functor and e v C evCev (−) : e v Ce v -Comod fc −→ C-Comod fc is the left exact cotensor product functor defined in [11] and [25] ((2.9), see also [29]). …”

“…We recall from [11] and [25] (Sec. 2) that the restriction functor res Ev : C-Comod fc −→ e v Ce v -Comod fc is exact and the cotensor product functor e v C evCev (−) : e v Ce v -Comod fc −→ C-Comod fc is left exact.…”

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

“…Apply the proof of Theorem 5.7(b). The following corollary is an immediate consequence of Theorem 6.2. for n ≤ m. Here we follow the localisation technique for coalgebras studied in [11], [19], [29], [37]. It follows that µ I n (S I (a), S I (b)) = µ J n (S J (a), S J (b)) for 1 ≤ n ≤ m, and the computation of r I (a, b) in I reduces to the computation of r J (a, b) in the finite subposet J = J ab of b ⊆ I.…”