Derived equivalence induced by infinitely generated n -tilting modules Silvana Bazzoni (Joint work with Francesca Mantese and Alberto Tonolo) Universit` a di Padova Trieste, February 1-5, 2010 Silvana Bazzoni Derived equivalence induced by n -Tilting modules
Outline ◮ Why Infinitely generated n -tilting modules? ◮ Equivalences induced by a classical n -tilting module. ◮ Derived equivalences in the infinitely generated case. ◮ Application to module categories. Silvana Bazzoni Derived equivalence induced by n -Tilting modules
Why infinitely generated modules? ◮ “Generic modules” “Generic” modules appear in the Ziegler closure of direct limits of finitely generated modules. They parametrize families of finite dimensional modules, (Crawley-Boevey, Ringel, Krause, Herzog) ◮ Approximation theory Classical notion: covariantly or contravariantly finite classes of finitely generated modules approximations via preenvelopes or precovers allowing infinitely generated modules are somehow easier to handle. Application: tilting classes are always preenveloping. Silvana Bazzoni Derived equivalence induced by n -Tilting modules
Why infinitely generated modules? ◮ Finitistic dimension conjectures (Angeleri, Trlifaj ’02) The little finitistic dimension of a noetherian ring is finite if and only if there is a tilting module representing the category of finitely generated modules of finite projective dimension. Even in the case of finite dimensional algebra it may happen that such a tilting module cannot be chosen to be finitely generated. ◮ n -tilting classes are of finite type (B, Herbera, ˇ Sˇ tov´ ıˇ cek ’07) Every tilting class is determined by finitely presented data: it is the right Ext-orthogonal of a set of finitely presented modules. Silvana Bazzoni Derived equivalence induced by n -Tilting modules
Infinitely generated n -tilting modules R associative ring with 1. Definition A right R -module T is n -tilting module, if (T1) there exists a projective resolution of right R -modules 0 → P n → ... → P 1 → P 0 → T → 0; (T2) Ext i R ( T , T ( α ) ) = 0 for each i > 0 and each cardinal α ; (T3) there exists a coresolution of right R -modules 0 → R → T 0 → T 1 → ... → T m → 0, with T i ’ in Add T . ◮ T is a classical n -tilting module if P i ’s in (T1) are finitely generated. T = { M ∈ Mod- R | Ext i R ( T , M ) = 0 , ∀ i > 0 } is called the n -tilting class. Silvana Bazzoni Derived equivalence induced by n -Tilting modules
� � Classical equivalences for the case n = 1 Theorem [Brenner-Butler ’80, Colby-Fuller ’90] T R classical 1-tilting module. S = End R ( T ) T = Gen T = Ker ( Ext 1 R ( T , − )) , F = Ker ( Hom R ( T , − )) . ( T , F ) a torsion pair in Mod- R Y = Ker ( Tor S 1 ( − , T )) X = Ker ( − ⊗ S T ) ( X , Y ) torsion pair in Mod- S Hom R ( T , − ) � Y T −⊗ S T Ext 1 R ( T , − ) � X F Tor S 1 ( − , T ) Silvana Bazzoni Derived equivalence induced by n -Tilting modules
� Classical equivalences for n > 1 T R a classical n -tilting module. S = End R ( T ) Ker ( Ext j � KE i = R ( T , − )) 0 ≤ i ≤ n 0 ≤ j � = i � Ker ( Tor S KT i = j ( − , T )) 0 ≤ i ≤ n 0 ≤ j � = i Theorem [Miyashita, ’86] There are equivalences: Ext i R ( T , − ) � KT i KE i 0 ≤ i ≤ n Tor S i ( − , T ) If T R is a infinitely generated, the equivalences can be generalized at the cost of intersecting with particular subcategories of Mod- S . Silvana Bazzoni Derived equivalence induced by n -Tilting modules
� The classical derived equivalences Theorem [Happel ’87, Cline-Parshall-Scott ’87] T R a classical n -tilting module with endomorphism ring S . There is a derived equivalence: R Hom R ( T , − ) � D b ( S ) D b ( R ) L − ⊗ S T Silvana Bazzoni Derived equivalence induced by n -Tilting modules
Good n -tilting modules T R and T ′ R n -tilting modules are equivalent if they induce the same n -tilting class, or if Add T ′ = Add T . Definition An n -tilting module T R with endomorphism ring S is good if condition (T3) can be replaced by [( T 3 ′ )] 0 → R → T 0 → T 1 → ... → T n → 0 where the T i ’s are in add T . Each classical n -tilting module is good. Proposition Every n -tilting module admits an equivalent good n -tilting module. Silvana Bazzoni Derived equivalence induced by n -Tilting modules
Proposition Let T R be a good n -tilting module, S = End R ( T ). Then, (T1) there exists 0 → Q n → ... → Q 0 → S T → 0 Q i finitely generated projective left S -modules, S ( T , T ) = 0 for each i ≥ 0, and R ∼ (T2) Ext i = End ( S T ). Thus, S T is a partial classical n -tilting S -module. Lemma Miyashita Let T R be a good n -tilting module with endomorphism ring S . Then, for each injective module I R ◮ Hom R ( T , I ) ⊗ S T ∼ = I ; ◮ Hom R ( T , I ) is an ( − ⊗ S T )-acyclic right S -module; For each projective right S -module P S ◮ P ⊗ S T is an Hom R ( T , − )-acyclic right R -module. Silvana Bazzoni Derived equivalence induced by n -Tilting modules
Generalization of the derived equivalence ◮ T R R -module, End ( T ) = S . ◮ D ( R ), D ( S ) derived categories of Mod- R and Mod- S . ◮ The adjoint pair H = Hom R ( T , − ): Mod- R − → − Mod- S : G = − ⊗ S T ← induces an adjoint pair of total derived functors L R H = R Hom R ( T , − ): D ( R ) − → − D ( S ): L G = − ⊗ S T ← Silvana Bazzoni Derived equivalence induced by n -Tilting modules
� � Theorem T R a good n -tilting module, End ( T ) = S . L R H = R Hom R ( T , − ) , L G = − ⊗ S T The following hold: (1) The counit of the adjunction ψ : L G ◦ R H → Id D ( R ) is invertible. (2) There is a triangle equivalence Θ: D ( S ) / Ker ( L G ) → D ( R ) L G � D ( R ) D ( S ) � � � � q � � � Θ ∼ � = � � � D ( S ) / Ker ( L G ) (3) Σ : system of morphisms u ∈ D ( S ) such that L G ( u ) is invertible in D ( R ). Σ admits a calculus of left fractions and D ( S )[Σ − 1 ] ∼ = D ( S ) / Ker ( L G ) Silvana Bazzoni Derived equivalence induced by n -Tilting modules .
The key fact is that the counit of the adjunction ◮ ψ : L G ◦ R H → Id D ( R ) is invertible. ◮ Obtained by using: ◮ the functors Hom R ( T , − ) and − ⊗ S T have finite homological dimension and their total derived functors can be computed on complexes with acyclic components. L ◮ R Hom ( T , I • ) ⊗ S T = Hom ( T , I • ) ⊗ S T , I • complex whose terms are injective right R -modules. R Hom ( T , P • L ⊗ S T ) = Hom ( T , P • ⊗ S T ) . P • complex whose terms are projective right S -modules. ◮ The rest follows by Proposition 1.3 in Gabriel-Zisman’s book. Silvana Bazzoni Derived equivalence induced by n -Tilting modules
The perpendicular subcategory φ : 1 D ( S ) → R H ◦ L G the unit of the adjunction ψ : L G ◦ R H → 1 D ( R ) the counit of the adjunction is invertibe. Proposition The functor L := R H ◦ L G : D ( S ) → D ( S ) is a Bousfield localization So the kernel L , i.e. E = Ker L G is a localizing subcategory, and if E ⊥ is the perpendicular category E ⊥ := { X ∈ D ( S ) : Hom D ( S ) ( E , X ) = 0 } L factorizes as j q ρ D ( S ) → D ( S ) / Ker L G − = E ⊥ → ֒ → D ( S ) ∼ where q is the canonical quotient functor and ρ is an equivalence. Silvana Bazzoni Derived equivalence induced by n -Tilting modules
� � � � � Theorem Let T R be a good n -tilting R -module and S = End ( T ). Let E be the kernel of L G we have triangle equivalence: R H � E ⊥ D ( R ) L G ( R H and L G corestriction and restriction) and we have a commutative diagram: R H ∼ = D ( R ) E ⊥ � � L G � � � � � � � � � ∼ ρ ∼ � � = Θ = � � � D ( S ) Ker L G Silvana Bazzoni Derived equivalence induced by n -Tilting modules
Back to the classical derived equivalence Proposition The following are equivalent. ◮ T R is a classical n -tilting module; ◮ E = 0 or equivalently E ⊥ = D ( S ); ◮ the class E is smashing, i.e. E ⊥ is closed under direct sums. Silvana Bazzoni Derived equivalence induced by n -Tilting modules
� Back to module categories Using the canonical embeddings Mod- R → D ( R ) Mod- S → D ( S ) we have a generalization to infinitely generated n -tilting modules of Brenner-Blutler, Colby-Fuller and Miyashita equivalences: Ext i R ( T , − ) � KT i ∩ E ⊥ KE i Tor S i ( − , T ) Silvana Bazzoni Derived equivalence induced by n -Tilting modules
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