# nLab Calabi-Yau algebra

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

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

## Definition

###### Definition

For $A$ a dg-algebra and $N$ a dg-bimodule over $A$, write

$N^! := RHom_{A Bimod}(N, A \otimes A)$

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

###### Definition

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

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

such that

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

This is (Ginzburg, def. 3.2.3).

## Properties

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

###### Definition

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

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

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

3. the endomorphism algebra $End(\mathcal{E}) = Hom(\mathcal{E},\mathcal{E})$ 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

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

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

###### Proposition

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

Then $End \mathcal{E}$ is a Calabi-Yau algebra of dimension $d$ precisely if $X$ is a Calabi-Yau manifold of dimension $d$.

This appears as (Ginzburg, prop. 3.3.1).

## References

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