Serial coalgebras

Abstract

In this paper we extend the theory of serial and uniserial ÿnite dimensional algebras to coalgebras of arbitrary dimension. Nakayama-Skorniakov Theorems are proved in this new setting and the structure of such coalgebras is determined up to Morita-Takeuchi equivalences. Our main structure theorem asserts that over an algebraically closed ÿeld k the basic coalgebra of a serial indecomposable coalgebra is a subcoalgebra of a path coalgebra k where the quiver is either a cycle or a chain (ÿnite or inÿnite). In the uniserial case, is either a single point or a loop. For cocommutative coalgebras, an explicit description is given, serial coalgebras are uniserial and these are isomorphic to a direct sum of subcoalgebras of the divided power coalgebra.

When Does the Rational Torsion Split Off for Finitely Generated Modules

“…Teply; we also include alternate direct proofs. Chain coalgebras were also studied recently in [13] and also briefly in [4] and [5]. However, our interest in chain coalgebras is of a different nature; it is a representation theoretic one and is directed towards our main result of this paper, that generalizes a result previously obtained [4] in the commutative case: we characterize the coalgebras having the f.g. Rat-splitting property and that are colocal, and show that they are exactly the chain coalgebras (Section 4, Theorem 4.4), a result that will involve quite technical arguments.…”

“…Part of the following proposition is a somewhat different form of Lemma 2.1 from [5]. However, we will need to use some of the other equivalent statements bellow.…”

“…The coalgebra n∈N K n having a basis c n , n ∈ N and comultiplication and counit given by these equations is called the divided power coalgebra (see [6]). Part of the following Lemma is discussed in [5] Theorem 3.2; also part of it in the cocommutative case is observed in [4], 3.5 and 3.6. The same result appears in [13], but with a different proof.…”

Triangular matrix coalgebras and applications

“…We recall a few definitions used in [C,CGT04,GN,IO,LS]; the terminology is derived from serial modules. Given a coalgebra C, a C-comodule M is called uniserial if M has a unique composition series, equivalently, the lattice of submodules of M is totally ordered.…”

“…A coalgebra is called left (right) serial if it is serial as a left (right) comodule. It is called serial if it is serial as left and right comodule (see [CGT04]). …”