Generalised Igusa-Todorov functions and Lat-Igusa-Todorov algebras (Q831280)

The finitistic dimension of an algebra \(\Lambda\) is defined by \[\text{fin}(\Lambda) := \sup\{ \text{pd(M)} : M \in \text{mod} (\Lambda) \text{ and }\text{pd (M)} < \infty\},\] where \(\text{pd(M)}\) is the projective dimension of the \(\Lambda\)-module \(\text{M}\). It was conjectured by H. Bass in the 60's that the finitistic dimension of an algebra is finite and it is not solved for Artin algebras. Since then, much work has been done toward the proof of this conjecture. For instance, \textit{K. Igusa} and \textit{G. Todorov} defined in [Fields Inst. Commun. 45, 201--204 (2005; Zbl 1082.16011)] two functions \(\phi, \psi:\text{mod} \Lambda \rightarrow \mathbb{N}\), which turned out to be useful to prove that \(\text{fin}(\Lambda)\) is finite for some classes of Artin algebras. One of these classes is the family of Igusa-Todorov algebras, introduced by \textit{J. Wei} in [Adv. Math. 222, No. 6, 2215--2226 (2009; Zbl 1213.16007)]. However, not every Artin algebra is Igusa-Todorov. In fact \textit{T. Conde} showed in [On certain strongly quasihereditary algebras. Oxford: University of Oxford (PhD Thesis) (2015)] that some selfinjective algebras are not Igusa-Todorov. In this article, the authors generalize the functions \(\Phi\) and \(\Psi\) defined by Igusa and Todorov (Definitions 3.5 and 3.8) and the \(\Phi\)-dimension (\(\Phi\)dim) and \(\Psi\)-dimension (\(\Psi\)dim) defined by \textit{F. Huard} and \textit{M. Lanzilotta} [Algebr. Represent. Theory 16, No. 3, 765--770 (2013; Zbl 1270.16006)] (Definition (1.1)). These new functions, called generalised Igusa-Todorov functions (\(\Phi_{[\mathcal{D}]}\) and \(\Psi_{[\mathcal{D}]}\)), verifies similar propierties to the Igusa-Todorov functions, as can be seen in Proposition 3.9 and Proposition 3.12. The functions \(\Phi_{[\mathcal{D}]}\) and \(\Psi_{[\mathcal{D}]}\) also bound the Igusa-Todorov functions, under certain conditions, as follows. Let \(\Lambda\) be an Artin algebra and \(\mathcal{D} \subset \text{mod}(\Lambda)\) be a class of \(\Lambda\)-modules satisfying that \(\text{add} \mathcal{D} = \mathcal{D}\) and \(\Omega(\mathcal{D}) \subset \mathcal{D}\). \begin{itemize} \item (Theorem 3.5) Then, \(\Phi(X) \leq \Phi_{[\mathcal{D}]}(X)+\Phi\text{dim}(\mathcal{D})\). \item (Proposition 3.10 (a)) If in addition \(\Phi\text{dim}(\mathcal{D}) = 0\), then \(\Psi(X) \leq \Psi_{[\mathcal{D}]}(X)\). \end{itemize} Some applications are obtained for the finitistic dimension conjecture using these new functions and dimensions. For instance, for an Artin algebra \(\Lambda\) it follows \begin{itemize} \item (Theorem 4.4) \(\text{fin}(\Lambda) \leq \text{resdim}_{\mathcal{D}} (\Omega^n (\text{mod} (\Lambda))) + n \), for a left saturated subclass \(\mathcal{D}\) with \(\Phi\dim(\mathcal{D})=0\). \item (Theorem 4.5) \(\text{fin}(\Lambda) = \Phi\dim (\Lambda) = \Psi\dim (\Lambda) = \sup\{\text{Gpd}(M)\}\), where \(\text{Gpd} (M)\) is the Gorenstein projective dimension of \(M\) and the supremum is finite. \item (Theorem 4.9) \(\text{fin} (\Lambda) \leq \text{id(T)} + \Phi \dim( ^\perp \text{T})\), for \(\text{T}\) a cotilting module. \end{itemize} On the other hand the authors define the Lat-Igusa-Todorov algebras (Definition 5.1), which generalize the concept of Igusa-Todorov algebra. The authors prove that this family of algebras also verifies the finitistic dimension conjecture (Theorem 5.4) using the generalized Igusa-Todorov functions, and show that every selfinjective algebra is Lat-Igusa-Todorov (Example 5.2). This theorem is a significant extension of Wei's result (Theorem 2.3 from [loc. cit.]).