Refined Gysin operations commute and compose

Commutation with bivariant operations. Test equality of two bivariant operations on the fundamental class of an integral $T^{\mathrm{\prime }}\to T$, allowing a proper birational modification by [L3]. If the inverse image of $Z$ in $T^{\mathrm{\prime }}$ is all of $T^{\mathrm{\prime }}$, the cone is the zero section and $c(Z,T,N)$ is the top Chern operator of the virtual normal bundle by [L1] with $N_0^{\mathrm{\prime }}=0$, hence central by [L2]. Otherwise blow up the nonzero ideal defining the inverse image of $Z$: on the blowup the inverse image is an effective Cartier divisor $D$, the pulled-back conormal surjection maps onto ${\mathcal{I}}_D\mathrm{/}{\mathcal{I}}_D^2$, an invertible sheaf. Locally this surjection onto a free rank-one module splits; dualizing gives a subbundle injection $N_{D\mathrm{/}{\tilde{T}}^{\mathrm{\prime }}}\hookrightarrow N{\mathrm{\mid }}_D$ with locally free quotient $Q$, and the excess formula of [L1] gives $c=c_{\mathrm{top}}(Q)\cap D^{!}$. The factor $c_{\mathrm{top}}(Q)$ commutes with every bivariant operation by [L2] and the Cartier factor $D^{!}$ commutes by the Cartier axiom [L3]; pushing the resulting identity forward along the blowup proves commutation on the original integral base.

Sections of smooth morphisms. Let $s:Z\hookrightarrow P$ be a regular immersion followed by a smooth projection $p:P\to T$ with $ps$ flat of the expected relative dimension. Locally write $T=\mathrm{Spec}A$, $P=\mathrm{Spec}B$, $Z=\mathrm{Spec}C$; the Koszul resolution of $C$ over $B$ has $A$-flat terms because $B$ is smooth hence flat over $A$, and its quotient $C$ is $A$-flat by hypothesis, so the successive kernels are $A$-flat by the Tor vanishing criterion; tensoring with any $A^{\mathrm{\prime }}$ therefore stays exact, so the regular equations remain regular after every base change, and the normal cone of the base-changed embedding is the full base-changed normal bundle with fundamental cycle the bundle pullback. Inverting bundle pullback gives $s^{!}{p}^{\ast }=(ps{)}^{\ast }$; when $ps=\mathrm{id}$ this is $s^{!}{p}^{\ast }=1$.

Composition. Apply the equality criterion of [L3] and blow up the product of the two ideals defining the inverse images of $Z$ and $Y$ to make both Cartier, splitting off the excess bundles with [L1] so that the outer normal bundle is a line bundle $N_{Y\mathrm{/}T}$ over the base; if an inverse image is the whole base the identity reduces to the excess formula of [L1]. Write $o:Y\to C_{Y}T$ for the zero section; the essential cone identity is $o^{!}[C_{Z}T]=[C_{Z}Y]$ as a class in $A_{n-1}(C_{Z}T{\times }_{C_{Y}T}Y)$ for integral $T$ of dimension $n$. On an affine open write $Z=(a)$ and $Y=(b)$ with $b=ac$ and $a,b\mathrm{\ne }0$; the $t$-charts of the two deformations are $R_a=A[t,u]\mathrm{/}(tu-a)$ and $R_b=A[t,v]\mathrm{/}(tv-b)$, both embedded in $A[t,t^{-1}]$ because their principal ideals are generated by nonzerodivisors, and the homomorphism $R_b\to R_a$, $v\mapsto cu$, defines a morphism from the $t$-chart of the $Z$-deformation to that of the $Y$-deformation which restricts on $t=0$ to the induced map of normal cones. The strict transform of $Y\times {\mathbb{P}}^1$ in the $Z$-blowup has coordinate ring $R_a\mathrm{/}((b)R_a:t^{\mathrm{\infty }})=(A\mathrm{/}(b))[t,\bar{a}\mathrm{/}t]$, the contraction of $(b)$ after inverting $t$ (saturation is essential: for $b=a^2$ the quotient has relation $u^2=0$ at infinity, not merely $\bar{c}u=0$), and its special fibre is ${\mathrm{gr}}_{(\bar{a})}(A\mathrm{/}(b))=C_{Z}Y$ with all nonreduced multiplicities. The strict transform ${\mathbb{P}}_Y^1$ in the $Y$-blowup is the Cartier divisor $v=0$ with infinity restriction the zero section $o$; pulling its Cartier operator back to the $Z$-chart cuts by $cu$, a nonzero divisor because $R_a$ is a domain, and the ordinary parts agree, so the two Cartier classes differ by classes at infinity, which the infinity Cartier Gysin kills because the normal line is trivial ($j^{!}{j}_{\ast }\eta =c_1(1)\eta =0$). Hence $o^{!}[C_{Z}T]=[C_{Z}Y]$; write $p:N_{Z\mathrm{/}T}\to Z$, $p_1:N_{Z\mathrm{/}Y}\to Z$, and $e:N_{Z\mathrm{/}Y}\hookrightarrow N_{Z\mathrm{/}T}$. After the excess reduction, $e$ is the inverse image of the zero section of the outer normal line. Proper/Cartier compatibility pushes the cone identity to $e^{!}{a}_{\ast }[C_{Z}T]=a_{1\ast }[C_{Z}Y]$, while the relative-line coordinate calculation gives $e^{!}{p}^{\ast }=p_1^{\ast }$. The classes defining the two operations are characterized by $p^{\ast }\gamma =a_{\ast }[C_{Z}T]$ and $p_1^{\ast }{\gamma }_1=a_{1\ast }[C_{Z}Y]$; hence $p_1^{\ast }\gamma =p_1^{\ast }{\gamma }_1$, and the injectivity of $p_1^{\ast }$ proves the required equality, and pushing down the modifications gives the composition formula for regular embeddings.

Graph composition. For morphisms $f:X\to Y$ and $g:Y\to W$ between smooth schemes, work in $X\times Y\times W$: the two graph pullbacks intersect in the graph of $(f,gf)$, their equations form regular sequences with independent coordinate directions, and the regular-immersion composition of step 2.1 identifies the iterated operation with the small graph Gysin followed by pullback from $W$. Comparing with the graph of $gf$ in $X\times W$ uses the regular section $X\times W\to X\times Y\times W$, $(x,w)\mapsto (x,f(x),w)$, of a smooth projection, whose Gysin cancels the projection pullback by step 1.2; composition then reduces the small graph operation to the graph of $gf$. All normal-bundle equalities used are exact sequences, not global splittings.