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Proceedings of the London Mathematical Society Advance Access published online on June 20, 2008
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdn027
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© 2008 London Mathematical Society

Finitude homotopique des dg-algèbres propres et lisses

Bertrand Toën

Laboratoire Emile Picard UMR CNRS 5580
Université Paul Sabatier Toulouse 3, Bat 1R2
31 062 Toulouse cedex 9
France

Received 30 April 2007. Revision received 1 April 2008.

Resume: On montre que toute dg-algèbre propre et lisse (sur un anneau de base k) est déterminée à quasi-isomorphisme près par sa An-algèbre sous-jacente pour un certain n. De même, tout morphisme entre dg-algèbres propres et lisses est déterminé à homotopie près par le morphisme induit sur les An-algèbres sous-jacentes. On démontre de plus que si k est local alors l’entier n peut être choisi uniformement pour toutes les dg-algèbres propres et lisses dont deux invariants numériques (le type et la dimension cohomologique) sont bornés.

We show that every smooth and proper dg-algebra (over some base commutative ring k) is determined, up to a quasi-isomorphism, by its underlying An-algebra for a certain integer n. In the same way we show that every morphism between smooth and proper dg-algebras is determined, up to homotopy, by the induced morphism on the underlying An-algebras. We show, moreover, that if the ring k is local then the integer n can be chosen uniformly for all smooth and proper dg-algebras for which two numerical invariants (the type and the cohomological dimension) are bounded.

2000 Mathematics Subject Classification 16E45.


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