Around the Gysin triangle. I. (Q2906501)

This paper studies exceptional functoriality in Voevodsky's derived category of motives \(D_{gm}(k)\) over a perfect field \(k\). If \(i:Z\subseteq X\) is closed embedding of smooth \(k\)-schemes of pure codimension \(d\), Voevodsky constructed a Gysin morphism \(i^*:M(X)\to M(Z)(d)[2d]\) which fits into the Gysin triangle NEWLINE\[NEWLINE M(X-Z) \to M(X) \to M(Z)(d)[2d] \to M(X-Z)[1]. NEWLINE\]NEWLINE The main result of this paper is that the Gysin triangle is natural with respect to Gysin morphisms.NEWLINENEWLINEThis exceptional functoriality for closed immersions is generalized to a construction of Gysin morphisms \(f^*:M(X) \to M(Y)(d)[2d]\) for a projective morphism \(f:X\to Y\), of codimension \(d\) between smooth \(k\)-schemes. (When \(d=0\) this Gysin morphism agrees with the transfer morphism for finite maps which are built into Voevodsky's theory). Lastly, the Gysin morphisms are used to deduce the strong dualizability of the motive of a smooth projective \(k\)-scheme.NEWLINENEWLINEFor the entire collection see [Zbl 1242.14001].