Correction to: The Schwartz space of a smooth semi-algebraic stack

Abstract

The purpose of this note is to fix two gaps in the construction of Schwartz spaces of semi-algebraic stacks in [4], and to strengthen some statements, replacing quasiisomorphisms by homotopy equivalences. I am grateful to Avraham Aizenbud, Shachar Carmeli, and Dmitry Gourevitch for pointing out the gaps, and suggesting the stronger statements. The first gap is in the proofs of Propositions 3.1.2 and 3.1.4, where I misquote [3, Theorem A.1.1] and write a Schwartz function as a product of two Schwartz functions. There is also an obvious typo in the statement of Proposition 3.1.4: the sequence appearing should end with ∂ 0 − → S(Y) → 0. Moreover, with this gap corrected, a stronger statement is actually proven in these two propositions than claimed. Namely, the sequence of Proposition 3.1.4 (with the aforementioned typo corrected) is not just strictly exact, but homotopic to zero. I formulate this here as a proposition, which supersedes both of Propositions 3.1.2 and 3.1.4 in the paper, and indicate the corrections needed for a complete proof. Proposition 1 Let π : X → Y be a smooth surjective morphism of Nash manifolds. Let [X ] n Y = the fiber product of n copies of X over Y (whose projection map to Y is still denoted by π), and consider the complex