Generalised Igusa-Todorov functions and Lat-Igusa-Todorov algebras

Abstract

In this paper we study a generalisation of the Igusa-Todorov functions which gives rise to a vast class of algebras satisfying the finitistic dimension conjecture. This class of algebras is called Lat-Igusa-Todorov and includes, among others, the Igusa-Todorov algebras (defined by J. Wei) and the self-injective algebras which in general are not Igusa-Todorov algebras. Finally, some applications of the developed theory are given in order to relate the different homological dimensions which have been discussed through the paper.

Triangular Lat-Igusa-Todorov algebras

“…First we point out that if T is a (n, V T ) Igusa-Todorov algebra, then T is a (n, {0}, V T ) LIT algebra. As in the proof of Theorem 3.1, there is no loss of generality in taking the same integer n. A final remark before continuing is that item (ii) follows immediately from (i) and Theorem 5.4 in [6].…”

“…First we point out that there is no loss of generality assuming that T and U are both n-LIT because even if they are k-LIT and m-LIT, they can be regarded as n := max{k, m}-LIT. Furthermore, item (ii) follows immediately from (i) and Theorem 5.4 in [6]. In order to prove item (i), we have to show that the class D Λ satisfies the conditions (a) and (b) of Definition 2.2.…”

“…It turns out that among LIT algebras it is possible to find algebras that are not IT, neither selfinjective, as can be seen in Example 3.4. Finally, in [6] it was proven that LIT algebras also satisfy the fin.dim. conjecture.…”

“…conjecture, namely exterior algebras of vector spaces of dimension greater or equal than 3. In [6] the authors defined generalized IT functions, which gave way to a generalization of the concept of Igusa-Todorov algebra. This new class of algebras, named Lat-Igusa-Todorov (LIT for short), is strictly bigger than IT algebras, since it includes all self-injective algebras, which we know are not IT in general because the example provided by Conde is a family of self-injective algebras.…”

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In [6] the authors gave a generalization of the concept of Igusa-Todorov algebra and proved that those algebras, named Lat-Igusa-Todorov (or LIT for short), satisfy the finitistic dimension conjecture. In this paper we explore the scope of that generalization and give conditions for a triangular matrix algebra to be LIT in terms of the algebras and the module used in its definition.

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