Contravariant finiteness and iterated strong tilting

TL;DR

This paper develops criteria to determine contravariant finiteness of modules with finite projective dimension over Artin algebras, enabling iterative strong tilting with periodicity and duality properties.

Contribution

It introduces a reduction method to test contravariant finiteness via corner algebras, characterizes conditions for iterative strong tilting, and proves periodicity of tilting sequences.

Findings

Contravariant finiteness can be tested on corner algebras.

Iterated strong tilting sequences are periodic with period 2.

Adjacent categories in the tilting sequence are dual via tilting bimodules.

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.