Algebras and modules
Model category presentations
Geometry on formal duals of algebras
The notion of Calabi-Yau algebra is an algebraic incarnation of the notion of Calabi-Yau manifold .
A homologically smooth dg-algebra is a Calabi-Yau algebra of dimension if there is a quasi-isomorphism of -bimodules
This is (Ginzburg, def. 3.2.3).
Let be a smooth quasi-projective variety. Write for the derived category of bounded chain complexes of coherent sheaves over .
An object is called a tilting generator if the Ext-functor satisfies
for all ;
the endomorphism algebra has finite Hochschild dimension.
This appears as (Ginzburg, def. 7.1.1).
For smooth connected variety which is projective over an affine variety, let be a tilting generator, def. 3.
Then is a Calabi-Yau algebra of dimension precisely if is a Calabi-Yau manifold of dimension .
This appears as (Ginzburg, prop. 3.3.1).
Revised on May 29, 2011 21:30:57
by Urs Schreiber