A Riemann-Roch theorem for dg algebras

Petit, François

Bulletin de la Société Mathématique de France, Tome 141 (2013) no. 2, pp. 197-223.

Étant donnée une dg algèbre $A$, propre et lisse, un dg $A$-module parfait $M$ et un endomorphisme $f$ de $M$, nous définissons la classe de Hochschild de la paire $(M,f)$. Cette classe est à valeurs dans l’homologie de Hochschild de l’algèbre $A$. Notre principal résultat est une formule de type Riemann-Roch faisant intervenir la convolution de deux de ces classes de Hochschild.

Given a smooth proper dg algebra $A$, a perfect dg $A$-module $M$ and an endomorphism $f$ of $M$, we define the Hochschild class of the pair $(M,f)$ with values in the Hochschild homology of the algebra $A$. Our main result is a Riemann-Roch type formula involving the convolution of two such Hochschild classes.

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