Homotopy coherent Gysin functoriality

TL;DR

This paper develops a homotopy coherent framework for Gysin morphisms in algebraic geometry, enabling higher coherence and strict functoriality for complex morphisms.

Contribution

It introduces a new construction of homotopy coherent Gysin pullbacks using deformation spaces, extending classical Gysin morphisms to a higher categorical setting.

Findings

Constructed homotopy coherent Gysin pullbacks for weak Borel-Moore theories.

Proved a representability theorem for Rost-Schmid complexes.

Extended Gysin morphisms to a strict simplicial functor.

Abstract

We construct homotopy coherent Gysin pullbacks for weak Borel-Moore theories on smooth schemes, addressing the higher coherence problem for Gysin morphisms associated with closed immersions and lci-type factorizations. The construction uses the higher deformation spaces of Dubouloz-Mayeux attached to flags of closed immersions, from which we build higher Gysin simplices and their simplicial identities up to contractible choices. A rigidification procedure then turns this coherent system into a strict contravariant simplicial functor extending both smooth pullbacks and closed-immersion Gysin morphisms. As an application, we prove a representability theorem for Rost-Schmid complexes associated with homodules over general noetherian excellent bases: these complexes form weak Borel-Moore theories, and hence are represented by motivic objects obtained from the main construction.