Gysin specialization is bivariant and compatible with base change

The cone embedding. Under a base change $T^{\mathrm{\prime }}\to T$ the ideal of $Z^{\mathrm{\prime }}$ is $\mathcal{I}{\mathcal{O}}_{T^{\mathrm{\prime }}}$ and its powers are generated by the images of the powers of $\mathcal{I}$, so the Rees algebra of $\mathcal{I}{\mathcal{O}}_{T^{\mathrm{\prime }}}$ is a quotient of the base change of the Rees algebra of $\mathcal{I}$; taking the symmetric algebra of the pulled-back conormal surjection and composing with the canonical map to the associated graded gives $\mathrm{Sym}(N^{\mathrm{\prime }\vee })\twoheadrightarrow {\mathrm{gr}}_{\mathcal{I}{\mathcal{O}}_{T^{\mathrm{\prime }}}}{\mathcal{O}}_{T^{\mathrm{\prime }}}$, and hence a closed immersion $a:C_{Z^{\mathrm{\prime }}}{T}^{\mathrm{\prime }}\hookrightarrow N^{\mathrm{\prime }}$ of the normal cone over $Z^{\mathrm{\prime }}$ into the vector bundle. This holds for arbitrary, including nonflat, base change, since only surjectivity of the graded maps is used.

Bivariant axioms. For each base $T^{\mathrm{\prime }}$ the operation is $c(Z,T,N{)}_{T^{\mathrm{\prime }}}=s_{N^{\mathrm{\prime }}}^{!}{a}_{\ast }{\sigma }_{Z^{\mathrm{\prime }}\mathrm{/}{T}^{\mathrm{\prime }}}$, a composite of the well-defined specialization ${\sigma }_{Z^{\mathrm{\prime }}\mathrm{/}{T}^{\mathrm{\prime }}}:A_m(T^{\mathrm{\prime }})\to A_m(C_{Z^{\mathrm{\prime }}}{T}^{\mathrm{\prime }})$ of [L1], the proper pushforward $a_{\ast }$, and the bijection $s_{N^{\mathrm{\prime }}}^{!}=(p^{\mathrm{\prime }\ast }{)}^{-1}$ of [L2]. Proper compatibility: push a lift of the pulled-back class forward along the given proper morphism and use that Cartier Gysin commutes with proper pushforward [L3]. Flat compatibility: pull the lift back and use flat/Cartier compatibility. Divisor compatibility: apply the Cartier Gysin to the lift and use that two Cartier Gysins commute [L3], noting that the ordinary restriction of the result is the lift of the desired input. Base restriction is simply the reindexing of the family. These verifications are exactly the three bivariant axioms of [L2].

Base change. Under a base change the blowup of $\mathcal{I}{\mathcal{O}}_{T^{\mathrm{\prime }}}$ admits a closed immersion into the base change of the original blowup, induced by the surjection of Rees algebras of step 1.1, and this immersion is an isomorphism over the ordinary part ${\mathbb{A}}_{T^{\mathrm{\prime }}}^1$; the induced proper morphism between the two deformation families is therefore an isomorphism over the ordinary fibres, and pushing a lift along it exhibits the two infinity operations as equal after pushforward. Consequently the cone portions of the two operations define the same class after pushforward to $N^{\mathrm{\prime }}$, and since proper pushforward, open restriction and $(p^{\mathrm{\prime }\ast }{)}^{-1}$ are bivariant, the operation is compatible with arbitrary base change.

The excess factor. Suppose the cone embedding factors through a vector subbundle $N_0^{\mathrm{\prime }}\subseteq N^{\mathrm{\prime }}$ of rank $d_0$; write the cone specialization pushed into $N_0^{\mathrm{\prime }}$ as $p_0^{\mathrm{\prime }\ast }\beta$ for a class $\beta$ on $Z^{\mathrm{\prime }}$, using the bundle-pullback bijection for $N_0^{\mathrm{\prime }}$ from [L2]. Applying the excess identity $s_{N^{\mathrm{\prime }}}^{!}{a}_{\ast }^{\mathrm{\prime }}{p}_0^{\mathrm{\prime }\ast }\beta =c_{\mathrm{rank}(N^{\mathrm{\prime }}\mathrm{/}{N}_0^{\mathrm{\prime }})}(N^{\mathrm{\prime }}\mathrm{/}{N}_0^{\mathrm{\prime }})\cap \beta$ of [L2] to the vector subbundle inclusion $a^{\mathrm{\prime }}:N_0^{\mathrm{\prime }}\hookrightarrow N^{\mathrm{\prime }}$ gives that the operation defined with $N^{\mathrm{\prime }}$ equals the operation defined with $N_0^{\mathrm{\prime }}$ multiplied by the top Chern operator of the quotient bundle $N^{\mathrm{\prime }}\mathrm{/}{N}_0^{\mathrm{\prime }}$; this is the asserted excess factor.