In the representation theory of Artin algebras, one of the most famous conjectures is the finitistic dimension conjecture which has a close relation with the generalized Nakayama conjecture, the Wakamatsu tilting conjecture and the Gorenstein symmetry conjecture. This conjecture was verified for many class of algebras, such as algebras of finite representation type, monomial-relation algebras, radical square and cube zero algebras. To study the finitistic dimension conjecture, Igusa and Todorov introduced two powerful functions which are called Igusa-Todorov functions, and used these two functions to prove that the finitistic dimension of \(A\) is finite provided that the representation dimension of \(A\) is at most \(3\). In [J. Pure Appl. Algebra 193, No. 1–3, 287–305 (2005; Zbl 1067.16016); J. Algebra 320, No. 1, 116–127 (2008; Zbl 1160.16003); ibid. 319, No. 9, 3666–3688 (2008; Zbl 1193.16006)], C. Xi developed some new ideas to prove the finiteness of finitistic dimension of some Artin algebras using the Igusa-Todorov function. In [Adv. Math. 222, No. 6, 2215–2226 (2009; Zbl 1213.16007)], the second author used the Igusa-Todorov function to define Igusa-Todorov algebras which satisfy the finitistic dimension conjecture. And D. Bravo et al. [J. Pure Appl. Algebra 223, No. 10, 4494–4508 (2019; Zbl 1412.16005)] proved that under some conditions, the triangular matrix Artin algebras \(T=\begin{pmatrix} A & 0 \\ M & B \end{pmatrix},\) is an Igusa-Todorov algebra if and only if both \(A\) and \(B\) are Igusa-Todorov algebras. Moreover, the second author [Math. Proc. Camb. Philos. Soc. 164, No. 2, 325–343 (2018; Zbl 1446.16011)] proved that syzygy-finite algebras and Igusa-Todorov algebras are derived invariant by syzygy complexes.
In algebraic geometry and representation theory, recollement of triangulated categories, introduced by A. A. Beilinson et al. [Astérisque 100, 172 p. (1982; Zbl 0536.14011))], is an important tool. As is known, a recollement of derived categories of algebras provide a useful framework for comparing the three algebras with respect to certain homological properties, such as global dimension, finitistic dimension and Gorensteinness. Both triangular matrix algebras and derived equivalent algebras can be viewed as special cases of recollements of derived categories of algebras.
In the paper under review, inspired by the works of D. Bravo et al. [J. Pure Appl. Algebra 223, No. 10, 4494–4508 (2019; Zbl 1412.16005)] and J. Wei [Math. Proc. Camb. Philos. Soc. 164, No. 2, 325–343 (2018; Zbl 1446.16011)], the authors compare syzygy-finite properties and Igusa-Todorov properties of the algebras \(A, B\) and \(C\), under a recollement of unbounded derived categories of the form \(\mathscr{D}(\mathrm{Mod} B), \mathscr{D}(\mathrm{Mod} A)\) and \(\mathscr{D}(\mathrm{Mod} C)\). Finally, they characterize when the recollement when the functor \(i^{\ast}\) (resp., \(i^{!}\)) in a recollement \((\mathscr{D}^b(\mathrm{Mod} B), \mathscr{D}^b(\mathrm{Mod} A)\) and \(\mathscr{D}^b(\mathrm{Mod} C), i^{\ast},i_{\ast}, i^{!}, j_!, j^{\ast}, j_{\ast})\) is an eventually homological isomorphism.