Contravariant finiteness and iterated strong tilting

Abstract

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