Contravariant finiteness and iterated strong tilting (2111.09181v2)

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