Homotopy coherent Gysin functoriality

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.