Contravariant finiteness and iterated strong tilting

Let ${\mathcal{P}}^{<\mathrm{\infty }}(\mathrm{\Lambda }$-mod$)$ be the category of finitely generated left modules of finite projective dimension over a basic Artin algebra $\mathrm{\Lambda }$. We develop an applicable criterion that reduces the test for contravariant finiteness of ${\mathcal{P}}^{<\mathrm{\infty }}(\mathrm{\Lambda }$ -mod$)$ in $\mathrm{\Lambda }$-mod to corner algebras $e\mathrm{\Lambda }e$ for suitable idempotents $e \in \Lambda$. The reduction substantially facilitates access to the numerous homological benefits entailed by contravariant finiteness of ${\mathcal{P}}^{<\mathrm{\infty }}(\mathrm{\Lambda }$-mod$)$. The consequences pursued hinge on the fact that this finiteness condition is known to be equivalent to the existence of a strong tilting object in $\mathrm{\Lambda }$-mod. We characterize the situation in which the process of strongly tilting $\mathrm{\Lambda }$-mod allows for arbitrary iteration: This occurs precisely when, in the strongly tilted module category mod-$\tilde{\mathrm{\Lambda }}$, the subcategory of modules of finite projective dimension is in turn contravariantly finite; the latter can, once again, be tested on suitable corners $e\mathrm{\Lambda }e$ of the original algebra $\mathrm{\Lambda }$. In the (frequently occurring) positive case, the sequence of consecutive strong tilts, $\tilde{\mathrm{\Lambda }}$, $\widetilde{\stackrel{~}{\mathrm{\Lambda }}}$, $\stackrel{~}{\widetilde{\stackrel{~}{\mathrm{\Lambda }}}},\dots$, is shown to be periodic with period $2$ (up to Morita equivalence); moreover, any two adjacent categories in the sequence ${\mathcal{P}}^{<\mathrm{\infty }}($mod-$\tilde{\mathrm{\Lambda }})$, ${\mathcal{P}}^{<\mathrm{\infty }}(\widetilde{\stackrel{~}{\mathrm{\Lambda }}}-mod)$, ${\mathcal{P}}^{<\mathrm{\infty }}($ mod-$\widetilde{\widetilde{\widetilde{\Lambda}}}), \dots$ are dual via contravariant Hom-functors induced by tilting bimodules which are strong on both sides.