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• W4310363577 abstract "Abstract Let $mathcal {P}^{<infty } ({Lambda }text {-mod})$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msup> <mml:mo>(</mml:mo> <mml:mi>Λ</mml:mi> <mml:mtext>-mod</mml:mtext> <mml:mo>)</mml:mo> </mml:math> be the category of finitely generated left modules of finite projective dimension over a basic Artin algebra Λ. We develop a widely applicable criterion that reduces the test for contravariant finiteness of $mathcal {P}^{<infty } ({Lambda }text {-mod})$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msup> <mml:mo>(</mml:mo> <mml:mi>Λ</mml:mi> <mml:mtext>-mod</mml:mtext> <mml:mo>)</mml:mo> </mml:math> in Λ-mod to corner algebras e Λ e for suitable idempotents e ∈Λ. The reduction substantially facilitates access to the numerous homological benefits entailed by contravariant finiteness of $mathcal {P}^{<infty } ({Lambda }text {-mod})$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msup> <mml:mo>(</mml:mo> <mml:mi>Λ</mml:mi> <mml:mtext>-mod</mml:mtext> <mml:mo>)</mml:mo> </mml:math> . The consequences pursued here 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 moreover characterize the situation in which the process of strongly tilting Λ-mod allows for unlimited iteration: This occurs precisely when, in the category $text {mod-}widetilde {Lambda }$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mtext>mod-</mml:mtext> <mml:mover> <mml:mrow> <mml:mi>Λ</mml:mi> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> </mml:math> of right modules over the strongly tilted algebra $widetilde {Lambda }$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mover> <mml:mrow> <mml:mi>Λ</mml:mi> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> </mml:math> , the subcategory of modules of finite projective dimension is in turn contravariantly finite; the latter condition can, once again, be tested on suitable corners e Λ e of the original algebra Λ. In the (frequently occurring) positive case, the sequence of consecutive strong tilts, $widetilde {Lambda }$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mover> <mml:mrow> <mml:mi>Λ</mml:mi> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> </mml:math> , $widetilde {widetilde {Lambda }}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mover> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>Λ</mml:mi> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> </mml:math> , $widetilde {widetilde {widetilde {Lambda }}}, dots $ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mover> <mml:mrow> <mml:mover> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>Λ</mml:mi> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mspace /> </mml:math> , is shown to be periodic with period 2 (up to Morita equivalence); moreover, any two adjacent categories in the sequence $mathcal {P}^{<infty } (text {mod-}widetilde {Lambda })$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msup> <mml:mo>(</mml:mo> <mml:mtext>mod-</mml:mtext> <mml:mover> <mml:mrow> <mml:mi>Λ</mml:mi> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:math> , $mathcal {P}^{<infty }(widetilde {widetilde {Lambda }}text {-mod})$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msup> <mml:mo>(</mml:mo> <mml:mover> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>Λ</mml:mi> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> <mml:mtext>-mod</mml:mtext> <mml:mo>)</mml:mo> </mml:math> , $mathcal {P}^{<infty }(text {mod-}widetilde {widetilde {widetilde {Lambda }}}), dots $ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msup> <mml:mo>(</mml:mo> <mml:mtext>mod-</mml:mtext> <mml:mover> <mml:mrow> <mml:mover> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>Λ</mml:mi> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> </mml:mrow> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mspace /> </mml:math> , alternating between right and left modules, are dual via contravariant Hom-functors induced by tilting bimodules which are strong on both sides. Our methods rely on comparisons of right $mathcal {P}^{<infty }$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mo><</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msup> </mml:math> -approximations in the categories Λ-mod, e Λ e -mod and the Giraud subcategory of Λ-mod determined by e ; these interactions hold interest in their own right. In particular, they underlie our analysis of the indecomposable direct summands of strong tilting modules." @default.
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• W4310363577 date "2022-11-29" @default.
• W4310363577 modified "2023-10-18" @default.
• W4310363577 title "Contravariant Finiteness and Iterated Strong Tilting" @default.
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