Refined Gysin pullback for regular embeddings

Assume the Axiom of Choice (The Axiom of Choice) inherited from the smooth normal-sequence suppliers. Fix a field $k$; all schemes and base changes below are locally of finite type over $k$, and all morphisms are $k$-morphisms. Let $i:X\hookrightarrow Y$ be a regular closed embedding of codimension $d$ with normal bundle $N$. In particular a closed embedding of smooth schemes of constant codimension is regular. For any $f:Y^{\mathrm{\prime }}\to Y$, put $X^{\mathrm{\prime }}=X{\times }_Y{Y}^{\mathrm{\prime }}$ and $N^{\mathrm{\prime }}=N{\mathrm{\mid }}_{X^{\mathrm{\prime }}}$. The pulled-back ideal gives a closed immersion $a:C_{X^{\mathrm{\prime }}}{Y}^{\mathrm{\prime }}\hookrightarrow N^{\mathrm{\prime }}$. Define $i_{Y^{\mathrm{\prime }}}^{!}=(p^{\mathrm{\prime }\ast }{)}^{-1}{a}_{\ast }{\sigma }_{X^{\mathrm{\prime }}\mathrm{/}{Y}^{\mathrm{\prime }}}:A_m(Y^{\mathrm{\prime }})⟶A_{m-d}(X^{\mathrm{\prime }}),$ using Deformation to the normal cone and specialization and Homotopy invariance for vector bundles. No fibre product of deformation spaces over ${\mathbb{P}}^1$ with a fictitious map to $Y$ is used. The same construction for a regular locally closed embedding uses restriction to an open in which it is closed; the operations agree under further restriction and extension of cycles, so this is intrinsic.

For $h:Y^{\mathrm{\prime }\mathrm{\prime }}\to Y^{\mathrm{\prime }}$ and $h_X:X^{\mathrm{\prime }\mathrm{\prime }}=X{\times }_Y{Y}^{\mathrm{\prime }\mathrm{\prime }}\to X^{\mathrm{\prime }}$, properness of $h$ gives $i_{Y^{\mathrm{\prime }}}^{!}{h}_{\ast }=(h_X{)}_{\ast }{i}_{Y^{\mathrm{\prime }\mathrm{\prime }}}^{!}$, and flatness of $h$ of fixed pure relative dimension gives $i_{Y^{\mathrm{\prime }\mathrm{\prime }}}^{!}{h}^{\ast }=h_X^{\ast }{i}_{Y^{\mathrm{\prime }}}^{!}$. These do not require the base-changed embedding to be regular. If $X^{\mathrm{\prime }}\hookrightarrow Y^{\mathrm{\prime }}$ is regular with normal bundle $N_0\subset N^{\mathrm{\prime }}$ and excess bundle $Q=N^{\mathrm{\prime }}\mathrm{/}{N}_0$, then $i_{Y^{\mathrm{\prime }}}^{!}=c_{d-\mathrm{rank}{N}_0}(Q)\cap (i^{\mathrm{\prime }}{)}^{!}$. In particular $i^{!}{i}_{\ast }\alpha =c_d(N)\cap \alpha$. Here $c_j$ means the already defined operational Chern operator of Operational Chern classes and the Whitney formula; it later becomes the Chow-ring Chern class on a smooth scheme. Codimension-one Gysin equals Cartier divisor Gysin; refined Gysins commute, and $(ji{)}^{!}=i^{!}{j}^{!}$ for composed regular embeddings. They commute with Chern cap operations. For any operational class $c$ on $Y$ and $\beta \in A_{\ast }(X)$, its projection formula is $i_{\ast }(c{\mathrm{\mid }}_X\cap \beta )=c\cap i_{\ast }\beta$. Under the later smooth operational-ring identification this is the ring formula $i_{\ast }(i^{\ast }\alpha \cdot \beta )=\alpha \cdot i_{\ast }\beta$.

Well-definedness. The construction and the arbitrary-base-change bivariant axioms are proved in Gysin specialization is bivariant and compatible with base change: the specialization ${\sigma }_{X^{\mathrm{\prime }}\mathrm{/}{Y}^{\mathrm{\prime }}}$ is well defined on Chow groups by the triviality of the normal line at infinity, the cone embedding exists by the Rees-algebra surjection, and $a_{\ast }$ and $(p^{\mathrm{\prime }\ast }{)}^{-1}$ are the proper pushforward and inverse flat pullback of Homotopy invariance for vector bundles. The excess formula is the corresponding statement of Gysin specialization is bivariant and compatible with base change; for the self-intersection formula apply it to the base change $Y^{\mathrm{\prime }}=X$, where the ideal is zero, the cone is the zero section and the quotient bundle is $N$, and use proper compatibility to identify the restricted operation with $i^{!}{i}_{\ast }$. For the agreement with Cartier divisor Gysin in codimension one, test on an integral base cycle: if the cycle is not contained in the divisor its cone is the normal line and the operation is its Cartier fundamental cycle, while if it is contained the zero-cone computation gives $c_1$ of the restricted normal line, which are exactly the two cases of the Cartier Gysin of Intersection with an invertible sheaf and the first Chern class. Commutation and composition, including the necessary cone computation with the saturated strict transforms in the deformation charts, are Refined Gysin operations commute and compose. Locality for locally closed embeddings follows because on each cycle the identical ideal and normal bundle give identical operators, and in overlaps the two restrictions agree. The operational projection formula is the proper axiom of a bivariant class (Refined Gysin operations commute and compose); its ring interpretation is provided by the later smooth-ring theorem and is not a prerequisite for this construction.