The Igusa-Todorov function for comodules

The Igusa-Todorov function (IT-function) appeared first in [1] and has been considered again in [5] and [6]. It is a new homological tool that generalises the notion of injective dimension (see Lemma 2.1). In [5], the authors proved that, for artinian rings, the selfinjectivity can be characterised by the nullity of the ITfunction on each finitely generated module. In other words, they proved that a ring is quasi-Frobenius if and only if it is right artinian and its IT-function is zero on each finitely generated right module. A coalgebra C is said to be left (right) quasi-co-Frobenius (qcF) if every indecomposable injective right (left) C-comodule is projective. Since indecomposable projective comodules are finite dimensional, a left (right) qcF coalgebra is left (right) semiperfect, that is, all indecomposable injective right (left) comodules are finite dimensional. We define here an IT-function in the context of finite dimensional comodules and we prove that a coalgebra C is left qcF if and only if it is left semiperfect and its IT-function is zero. Even if this equivalence can be seen as a possible dual version of the problem treated in [5], it is not strictly the case (we deal with coalgebras while the authors in [5] deal with artinian rings) and the proof given here uses quite different ideas and tools. It is worth to remark that while the notion of quasi-Frobenius ring is left-right symmetric, the notion of qcF coalgebra is not. This applied to a coalgebra that is left and also right semiperfect will give an example of a coalgebra for which its IT-function of right comodules is zero, while its IT-function for left comodules is not (see Example 2.4). In what follows we present a brief description of the contents of this paper. In Section 2 we recall a few basic notations and give the definitions, main properties and examples needed to understand and treat the question. In Section 3 we prove the main result mentioned above and some needed previous lemmas.

In Section 4 we use some computations with the IT-function, to deduce some well known facts about qcF coalgebras.

2. The Igusa-Todorov function 2.1. Some notations. In this work, C will be a coalgebra over a field k and we will denote by M C and C M the categories of right and left comodules over C respectively and by M C f and C M f the respective complete subcategories of finite dimensional comodules. Since M C and C M are Grothendieck categories, every object in them has an injective envelope (see for example [3]).

2.2. The Igusa-Todorov function on comodules. Let C be a coalgebra and K(C) be the free abelian group generated by all symbols

is the free abelian group generated by all isomorphism classes of indecomposable non injective objects in M C . As the syzygy Ω -1 respects direct sums and sends injective comodules to 0, it gives rise to a group morphism (that we also call Ω -1 ) Ω -1 : K(C) → K(C).

f , let M denote the subgroup of K(C) generated by all the symbols [N ], where N is an indecomposable non injective direct summand of M . Since the rank of Ω -1 ( M ) is less or equal to the rank of M , which is finite, it exists a non-negative integer n such that the rank of Ω

The main properties of ϕ are summarised in the following lemma, whose version for Artin algebras has been proved, almost all in [1] and the last in [6] and can be easily adapted to obtain the version for coalgebras.

Lemma 2.1. ( [1], [6]) Let C be a coalgebra and

In a similar way it is possible to define ϕ on the category C M f . We will use the same notation for both functions, when no confusion arises. Remark 2.2. Note that ϕ(M ) = 0 mains that rkΩ -n ( M ) remains constant for all n ∈ N.

The following example shows that dim ϕ ( C M f ) and dim ϕ (M C f ) can be different. Example 2.4. Consider the quiver

• and let C be the coalgebra whose elements are all paths in kQ of length less or equal to one. Each comodule M in C M f can be seen as a kQ-representation (M i , T i ) i∈N where T i : M i+1 → M i is such that T i .T i+1 = 0, for all i ∈ N. It is easy to check that every such representation can be decomposed into a sum of indecomposable representations of the form (see [9]):

Note that representations of the first type are injective, while representations of the second type are simple. As ϕ(M ⊕ I) = ϕ(M ), whenever I is injective, in order to prove that ϕ(M ) = 0, for every comodule M in C M f , it is enough to show it for cosemisimple comodules. If we consider:

after applying Ω -1 to M we obtain the representation

Hence the ranks of Ω -1 (M ) and M are equal, and then, by induction (because Ω -1 (M ) is cosemisimple), the ranks of Ω -n (M ) and M are equal. Then ϕ(M ) = 0 for every cosemisimple object in C M f , so dim ϕ ( C M f ) = 0.

On the other hand, right C-comodules can be seen as Q ′ -representations (M i , T i ) i∈N where T i : M i → M i+1 is such that T i .T i+1 = 0, for all i ∈ N and

Now, the right C-comodule:

(where the non zero vector space k is p

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