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Calabi-Yau algebra

Contents

Idea

The notion of Calabi-Yau algebra is an algebraic incarnation of the notion of Calabi-Yau manifold .

Definition

Definition

For AA a dg-algebra and NN a dg-bimodule over AA, write

N !:=RHom ABimod(N,AA) N^! := RHom_{A Bimod}(N, A \otimes A)

for the dual AA-bimodule, where RHomRHom denotes the right derived hom-functor with respect to the model structure on dg-modules.

Definition

A homologically smooth dg-algebra AA is a Calabi-Yau algebra of dimension dd if there is a quasi-isomorphism of AA-bimodules

f:AA ![d] f : A \stackrel{\simeq}{\to} A^![d]

such that

ff ![d]. f \simeq f^![d] \,.

This is (Ginzburg, def. 3.2.3).

Properties

Let XX be a smooth quasi-projective variety. Write D b(CohX)D^b(Coh X) for the derived category of bounded chain complexes of coherent sheaves over XX.

Definition

An object D b(CohX)\mathcal{E} \in D^b(Coh X) is called a tilting generator if the Ext-functor satisfies

  1. Ext i(,)=0Ext^i(\mathcal{E}, \mathcal{E}) = 0 for all i>0i \gt 0;

  2. Ext (,)=0Ext^\bullet(\mathcal{E},\mathcal{F}) = 0 implies =0\mathcal{F} = 0;

  3. the endomorphism algebra End()=Hom(,)End(\mathcal{E}) = Hom(\mathcal{E},\mathcal{E}) has finite Hochschild dimension.

This appears as (Ginzburg, def. 7.1.1).

Remark

For \mathcal{E} a tilting generator there is an equivalence of triangulated categories

D b(CohX)D b(End()Mod) D^b(Coh X) \stackrel{\simeq}{\to} D^b(End(\mathcal{E})Mod)

to the derived category of modules over End()End(\mathcal{E}).

Proposition

For XX smooth connected variety which is projective over an affine variety, let inD b(CohX)\mathcal{E} in D^b(Coh X) be a tilting generator, def. 3.

Then EndEnd \mathcal{E} is a Calabi-Yau algebra of dimension dd precisely if XX is a Calabi-Yau manifold of dimension dd.

This appears as (Ginzburg, prop. 3.3.1).

References

Revised on May 29, 2011 21:30:57 by Urs Schreiber (89.204.153.82)