7
$\begingroup$

This question is in reference to Gaitsgory's preprint Contractibility of the space of rational maps. On p. 5 of the preprint, Gaitsgory defines a prestack $\mathscr{Y}$ (say over affine $\mathbb{C}$-schemes) to be homologically contractible if the functor $\text{Vect}\longrightarrow\mathfrak{D}(\mathscr{Y})$ taking $V\mapsto V\otimes\omega_{\mathscr{Y}}$ is fully faithful. Here $\mathfrak{D}(\mathscr{Y})$ denotes the DG category of D-modules on $\mathscr{Y}$. Further down the same page, Gaitsgory mentions that homological contractibility of $\mathscr{Y}$ is equivalent to the condition that $H_{\bullet}(\mathscr{Y}(\mathbb{C})^{\text{top}},\mathbb{Q})\cong\mathbb{Q}$. What is the argument for the equivalence of these two formulations of homological contractibility? I would be happy to see the argument even in a simple case, like the case where $\mathscr{Y}$ is a smooth complex variety.

$\endgroup$
2
$\begingroup$

This is proven in some detail in section 3 Gaitsgory's writeup of his the Atiyah-Bott formula. He starts with the fully faithfulness definition, then proves the equivalance with homological statement at the very end of the section.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.