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It is a well-known result of Auslander and Reiten that contravariant finiteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ (of finitely generated modules of finite projective dimension) over an Artin algebra is a sufficient condition for validity of finitistic dimension conjectures. Motivated by the fact that finitistic dimensions of an algebra can alternatively be computed by Gorenstein projective dimension, in this work we examine the Gorenstein counterpart of Auslander--Reiten condition, namely contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ (of finitely generated modules of finite Gorenstein projective dimension), and its relation to validity of finitistic dimension conjectures. It is proved that contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ implies validity of the second finitistic dimension conjecture over left artinian rings. In the more special setting of Artin algebras, however, it is proved that the Auslander--Reiten sufficient condition and its Gorenstein counterpart are virtually equivalent in the sense that contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ implies contravariant finiteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ over any Artin algebra, and the converse holds for Artin algebras over which the class $\mathcal{GP}^{\mathrm{fin}}_0$ (of finitely generated Gorenstein projective modules) is contravariantly finite., Comment: 19 pages; comments welcome

Given a non-unit, non-zero-divisor, central element $x$ of a ring $\Lambda$, it is well known that many properties or invariants of $\Lambda$ determine, and are determined by, those of $\Lambda / x \Lambda$ and $\Lambda_x$. In the present paper, we investigate how the property of "being tilting" behaves in this situation. It turns out that any tilting module over $\Lambda$ gives rise to tilting modules over $\Lambda_x$ and $\Lambda / x \Lambda$ after localization and passing to quotient respectively. On the other hand, it is proved that under some mild conditions, a module over $\Lambda$ is tilting if its corresponding localization and quotient are tilting over $\Lambda_x$ and $\Lambda / x \Lambda$ respectively., Comment: A gap in the statement of Proposition 2.5 and in the proof of Theorem 2.10 has been fixed. Minor editorial changes have been made. To appear in "Journal of Pure and Applied Algebra"