Costello's pushforward formula: errata and generalization

TL;DR

This paper corrects and generalizes Costello's Pushforward Formula, verifying its validity under properness or purity conditions and examining its application in previous works.

Contribution

It identifies the correct conditions for the formula's validity and reviews prior uses to ensure their applicability.

Findings

The formula holds under properness or purity assumptions.

Many previous papers' applications are validated under these conditions.

The correction broadens the formula's applicability.

Abstract

K. Costello's Pushforward Formula is not correct as stated. We verify it holds under an assumption of properness or of a weaker notion we call "purity." We then check many papers that used this formula to make sure it holds in their context.