Derived equivalence induced by infinitely generated n -tilting - - PowerPoint PPT Presentation

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

• nly 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 → Pn → ... → P1 → P0 → T → 0; (T2) Exti

R(T, T (α)) = 0 for each i > 0 and each cardinal α;

(T3) there exists a coresolution of right R-modules 0 → R → T0 → T1 → ... → Tm → 0, with Ti’ in Add T.

◮ T is a classical n-tilting module if Pi’s in (T1) are finitely

generated. T = {M ∈ Mod-R | Exti

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]

TR classical 1-tilting module. S = EndR(T) T = GenT = Ker(Ext1

R(T, −)),

F = Ker(HomR(T, −)). (T , F) a torsion pair in Mod-R Y = Ker(TorS

1 (−, T))

X = Ker(− ⊗S T) (X, Y) torsion pair in Mod-S T

HomR(T,−)

Y

−⊗ST

• F

Ext1

R(T,−)

X

TorS

1 (−,T)

• Silvana Bazzoni

Derived equivalence induced by n-Tilting modules

Classical equivalences for n > 1

TR a classical n-tilting module. S = EndR(T) KEi =

• 0≤j=i

Ker(Extj

R(T, −))

0 ≤ i ≤ n KTi =

• 0≤j=i

Ker(TorS

j (−, T))

0 ≤ i ≤ n

Theorem [Miyashita, ’86]

There are equivalences: KEi

Exti

R(T,−)

KTi

TorS

i (−,T)

• 0 ≤ i ≤ n

If TR 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]

TR a classical n-tilting module with endomorphism ring S. There is a derived equivalence: Db(R)

RHomR(T,−)

Db(S)

−

L

⊗ST

• Silvana Bazzoni

Derived equivalence induced by n-Tilting modules

Good n-tilting modules

TR 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 TR with endomorphism ring S is good if condition (T3) can be replaced by [(T3′)] 0 → R → T0 → T1 → ... → Tn → 0 where the Ti’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 TR be a good n-tilting module, S = EndR(T). Then, (T1) there exists 0 → Qn → ... → Q0 → ST → 0 Qi finitely generated projective left S-modules, (T2) Exti

S(T, T) = 0 for each i ≥ 0, and R ∼

= End(ST). Thus, ST is a partial classical n-tilting S-module.

Lemma Miyashita

Let TR be a good n-tilting module with endomorphism ring S. Then, for each injective module IR

◮ HomR(T, I) ⊗S T ∼

= I;

◮ HomR(T, I) is an (− ⊗S T)-acyclic right S-module;

For each projective right S-module PS

◮ P ⊗S T is an HomR(T, −)-acyclic right R-module.

Silvana Bazzoni Derived equivalence induced by n-Tilting modules

Generalization of the derived equivalence

◮ TR R-module, End(T) = S. ◮ D(R), D(S) derived categories of Mod-R and Mod-S. ◮ The adjoint pair

H = HomR(T, −): Mod-R − → ← − Mod-S : G = − ⊗S T induces an adjoint pair of total derived functors RH = RHomR(T, −): D(R) − → ← − D(S): LG = −

L

⊗S T

Silvana Bazzoni Derived equivalence induced by n-Tilting modules

Theorem

TR a good n-tilting module, End(T) = S. RH = RHomR(T, −), LG = −

L

⊗S T The following hold: (1) The counit of the adjunction ψ: LG ◦ RH → IdD(R) is invertible. (2) There is a triangle equivalence Θ: D(S)/Ker(LG) → D(R) D(S)

q

• LG

D(R)

D(S)/Ker(LG)

Θ ∼ =

• (3) Σ : system of morphisms u ∈ D(S) such that LG(u) is

invertible in D(R). Σ admits a calculus of left fractions and D(S)[Σ−1] ∼ = D(S)/Ker(LG) .

Silvana Bazzoni Derived equivalence induced by n-Tilting modules

The key fact is that the counit of the adjunction

◮ ψ: LG ◦ RH → IdD(R) is invertible. ◮ Obtained by using: ◮ the functors HomR(T, −) and − ⊗S T have

finite homological dimension and their total derived functors can be computed on complexes with acyclic components.

◮ RHom(T, I •) L

⊗S T = Hom(T, I •) ⊗S T, I • complex whose terms are injective right R-modules. RHom(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 : 1D(S) → RH ◦ LG the unit of the adjunction ψ : LG ◦ RH → 1D(R) the counit of the adjunction is invertibe.

Proposition

The functor L := RH ◦ LG : D(S) → D(S) is a Bousfield localization So the kernel L, i.e. E = KerLG is a localizing subcategory, and if E⊥ is the perpendicular category E⊥ := {X ∈ D(S) : HomD(S)(E, X) = 0} L factorizes as D(S)

q

→ D(S)/KerLG

ρ

− →

∼ = E⊥ j

֒ → 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 TR be a good n-tilting R-module and S = End(T). Let E be the kernel of LG we have triangle equivalence: D(R)

RH

E⊥

LG

• (RH and LG corestriction and restriction) and

we have a commutative diagram: D(R)

RH ∼ =

• ∼

= Θ

• E⊥

LG

• D(S)

KerLG ρ ∼ =

• Silvana Bazzoni

Derived equivalence induced by n-Tilting modules

Back to the classical derived equivalence

Proposition

The following are equivalent.

◮ TR 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

• f Brenner-Blutler, Colby-Fuller and Miyashita equivalences:

KEi

Exti

R(T,−)

KTi ∩ E⊥

TorS

i (−,T)

• Silvana Bazzoni

Derived equivalence induced by n-Tilting modules