Rational Chow groups and zero-section Gysin maps

For a separated Deligne–Mumford stack $X$ of finite type over a field, let $Z_d(X)$ be the rational vector space generated by integral closed substacks of dimension $d$. The rational Chow group $A_d(X{)}_{\mathbb{Q}}$ is $Z_d(X)$ modulo rational equivalence in the stack intersection theory of Vistoli; $A_{\ast }(X{)}_{\mathbb{Q}}={⨁}_d{A}_d(X{)}_{\mathbb{Q}}$. This fixes the rational Chow convention used in the following virtual-class construction.

If $E\to X$ is a rank-$r$ vector bundle and $0_E:X\to E$ its zero section, the zero-section Gysin map is the refined intersection operation $0_E^{!}:A_d(E{)}_{\mathbb{Q}}⟶A_{d-r}(X{)}_{\mathbb{Q}}\mathrm{.}$ It is the bivariant pullback associated with the regular zero section; it lowers cycle dimension by the rank and is compatible with smooth pullback. The Behrend–Fantechi virtual class uses this map only after the intrinsic cone has been represented as a cycle in a vector-bundle presentation.

For a regular closed immersion $i:Z\hookrightarrow Y$ of codimension $c$, the refined Gysin pullback $i^{!}$ is the bivariant Chow operation on any base change of $Y$ along a morphism $Y^{\mathrm{\prime }}\to Y$; it sends a class on $Y^{\mathrm{\prime }}$ to a class on $Z^{\mathrm{\prime }}=Z{\times }_Y{Y}^{\mathrm{\prime }}$ and lowers dimension by $c$. For a smooth finite-type scheme $B$ over the field and a closed point $b\in B$, the embedding $b:\mathrm{Spec}\kappa (b)\hookrightarrow B$ is regular of codimension $\dim {\mathcal{O}}_{B,b}$. For a smooth family $Y\to B$, its fibre inclusion $Y_b\hookrightarrow Y$ is regular by flat base change. More generally, for any family $M\to B$ in the stated category, the fibre operation used in the PT deformation statement is the refinement $b_M^{!}$ of this regular base embedding; $M_b\hookrightarrow M$ need not itself be regular. No regular-closed-immersion fibre operation is asserted for nonclosed points of $B$.

Under the Axiom of Choice inherited from this supplier, the construction in Rational DM Chow groups and the deformation construction of refined Gysin specifies the rational-equivalence relation here as the rational span of the principal divisor cycles ${\mathrm{div}}_W(h)$, where $W$ is an integral closed substack of one higher dimension and $h\in k(W{)}^{\ast }$. It proves finite-scheme-cover descent of this quotient, including generic inertia degree factors, and proves vector-bundle homotopy invariance on this quotient. Thus the unrefined zero-section operation is the inverse of $p^{\ast }:A_{d-r}(X{)}_{\mathbb{Q}}\to A_d(E{)}_{\mathbb{Q}}$.

For every base map $T\to Y$ in the separated finite-type DM category, the same supplier constructs the deformation algebra ${\mathcal{O}}_T[t,I_T\mathrm{/}t]$, its special normal-cone fiber, the well-defined specialization ${\sigma }_{Z_T\mathrm{/}T}$, and the cone inclusion $a_T:C_{Z_T}T\hookrightarrow N_{Z\mathrm{/}Y}{\mathrm{\mid }}_{Z_T}$. It then defines the refined operation by $(p_T^{\ast }{)}^{-1}(a_T{)}_{\ast }{\sigma }_{Z_T\mathrm{/}T}$ and proves its proper, flat and Cartier compatibilities. Restriction along arbitrary further base maps means reindexing this family; it requires neither a general Chow pullback along arbitrary maps nor regularity of the base-changed immersion. For the closed-point assertion in the Definition, the regular local ring ${\mathcal{O}}_{B,b}$ has a regular system of parameters generating its maximal ideal. These parameters define the point immersion near $b$, and flat pullback preserves their regularity for a smooth family. For a possibly nonflat moduli family, the operation is instead the supplier's refinement of that base immersion.