The Igusa-Todorov function for comodules

We define the Igusa-Todorov function in the context of finite dimensional comodules and prove that a coalgebra is left qcF if and only if it is left semiperfect and its Igusa-Todorov function on each right finite dimensional comodule is zero.

💡 Research Summary

The paper introduces the Igusa‑Todorov (IT) function into the setting of finite‑dimensional comodules over a coalgebra C and uses it to characterize left quasi‑co‑Frobenius (left qcF) coalgebras. After recalling that the IT‑function was originally defined for Artin algebras, where its vanishing on all finitely generated modules characterizes self‑injective rings, the authors adapt the construction to the comodule categories MC (right C‑comodules) and CM (left C‑comodules). They define a free abelian group K(C) generated by isomorphism classes of indecomposable non‑injective comodules, introduce the syzygy operator Ω⁻¹ on K(C), and for each finite‑dimensional comodule M define a subgroup h_M generated by the indecomposable non‑injective summands of M. The integer ϕ(M) is the smallest n such that the rank of Ω⁻ⁿ(h_M) stabilizes; this generalizes the notion of injective dimension. Basic properties of ϕ are collected in Lemma 2.1, including the fact that ϕ(M)=0 precisely when the ranks of the iterated syzygies of h_M remain constant.

Two invariants are then introduced: dim ϕ(MC^f) and dim ϕ(CM^f), the supremum of ϕ over all finite‑dimensional right and left comodules respectively. An explicit example based on a quiver shows that these two numbers can differ, illustrating that the left and right IT‑functions need not coincide.

The main result, Theorem 3.4, states that a coalgebra C is left qcF if and only if it is left semiperfect and dim ϕ(MC^f)=0. The proof proceeds in two directions. If C is left qcF, every injective right comodule is projective, making Ω⁻¹ an automorphism of K(C); consequently the ranks of the subgroups h_M never drop, forcing ϕ(M)=0 for all finite‑dimensional right comodules. Conversely, assuming left semiperfectness and vanishing of the right IT‑function, the authors argue by contradiction: suppose an injective right comodule E is not projective. By constructing suitable short exact sequences involving the injective envelope of a simple comodule S and using Lemma 3.2 (a technical surjectivity lemma) together with the Snake Lemma, they obtain a strict decrease in the rank of Ω⁻¹ applied to a certain subgroup, contradicting the hypothesis that dim ϕ(MC^f)=0. Hence every injective right comodule must be projective, proving left qcF.

Section 4 exploits this characterization to re‑derive several known properties of qcF coalgebras via the IT‑function. Proposition 4.1 shows that indecomposable injective right comodules and indecomposable projective left comodules have simple top and simple socle respectively. Proposition 4.2 defines a map ν_r on the set of simple right comodules by ν_r(S)=Top(E(S)) and proves it is injective; this is the Nakayama permutation in the comodule context. Proposition 4.3 demonstrates that a non‑simple indecomposable left qcF coalgebra admits no simple injective right comodules. Corollary 4.4 concludes that when a coalgebra is both left and right qcF, the Nakayama permutations ν_r and ν_l are bijective. Finally, Corollary 4.5 restates a known generation criterion: a left semiperfect coalgebra generates its left comodule category if and only if dim ϕ(MC^f)=0.

Overall, the paper successfully transports the Igusa‑Todorov homological invariant from ring theory to coalgebra theory, providing a clean homological criterion for left qcF coalgebras and offering new proofs of classical results. The work highlights the asymmetry between left and right structures in coalgebras and opens the door for further applications of IT‑functions in the study of comodule categories.