A Quadratically Enriched Pushforward for Deligne--Mumford Stacks

We define a quadratically enriched pushforward for proper morphisms between smooth, proper Deligne–Mumford stacks over an arbitrary field using Grothendieck–Serre duality. We apply this pushforward to define quadratically enriched Euler classes and Euler numbers of relatively oriented vector bundles on smooth, proper Deligne–Mumford stacks. Finally, we define quadratically enriched fundamental classes for smooth, proper schemes over an arbitrary field, and we use these to propose a definition of quadratically enriched quantum $K$-invariants when the moduli space of stable maps is smooth, tame, and relatively oriented.