Contravariant Finiteness and Iterated Strong Tilting

Abstract Let $\mathcal {P}^{<\infty } ({\Lambda }\text {-mod})$ P < ∞ ( Λ -mod ) be the category of finitely generated left modules finite projective dimension over a basic Artin algebra Λ. We develop widely applicable criterion that reduces test for contravariant finiteness in Λ-mod to corner algebras e Λ suitable idempotents ∈Λ. The reduction substantially facilitates access numerous homological benefits entailed by . consequences pursued here hinge on fact this condition is known equivalent existence strong tilting object Λ-mod. moreover characterize situation which process strongly allows unlimited iteration: This occurs precisely when, $\text {mod-}\widetilde {\Lambda }$ mod- ~ right tilted $\widetilde , subcategory turn contravariantly finite; latter can, once again, tested corners original In (frequently occurring) positive case, sequence consecutive tilts, {\widetilde }}$ }}}, \dots $ , … shown periodic with period 2 (up Morita equivalence); moreover, any two adjacent categories (\text })$ }(\widetilde }}\text }(\text }}}), alternating between and modules, are dual via Hom-functors induced bimodules both sides. Our methods rely comparisons -approximations Λ-mod, -mod Giraud determined ; these interactions hold interest their own right. particular, they underlie our analysis indecomposable direct summands modules.