Intrinsic localized pushforward for a proper fixed locus

Let $T=({\mathbb{C}}^{\times }{)}^r$ act on a smooth separated finite-type complex variety $X$ with a finite invariant affine open cover. Let $F=X^T$ be proper, $i:F\hookrightarrow X$, $p_F:F\to \mathrm{p}\mathrm{t}$, and $S_T$ the character multiplicative set. Every notation $S_T^{-1}{H}_j^{\mathrm{B}\mathrm{M},T}(Y;\mathbb{Q})$ below means ${\mathcal{H}}_j^T(Y)$, the degree-$j$ piece of the localization of the total graded module. By the finite-affine-atlas concentration isomorphism, $i_{\ast }:S_T^{-1}{H}_j^{\mathrm{B}\mathrm{M},T}(F;\mathbb{Q})\to S_T^{-1}{H}_j^{\mathrm{B}\mathrm{M},T}(X;\mathbb{Q})$ is invertible. Define the intrinsic localized pushforward

$$p_{\ast }^{\mathrm{l}\mathrm{o}\mathrm{c}}:=p_{F\ast }\circ (i_{\ast }{)}^{-1}:S_T^{-1}{H}_j^{\mathrm{B}\mathrm{M},T}(X;\mathbb{Q})⟶S_T^{-1}{H}_j^{\mathrm{B}\mathrm{M},T}(\mathrm{p}\mathrm{t};\mathbb{Q}),$$

where $p_{F\ast }$ is the genuine proper Borel–Moore pushforward. For a homogeneous $\alpha \in H_T^q(X;\mathbb{Q})$ and smooth complex dimension $d$, an allowed insertion is $\alpha \cap [X{]}_T\in H_{2d-q}^{\mathrm{B}\mathrm{M},T}(X;\mathbb{Q})$; the fundamental class $[X{]}_T$ is the complex-oriented class, not the cohomological unit $1\in H_T^0(X;\mathbb{Q})$. This construction does not define an ordinary pushforward along a nonproper map $X\to \mathrm{p}\mathrm{t}$.

[Equivariant pushforward by the fixed-locus inverse] Edidin–Graham, Section 4

[Borel–Moore proper pushforward and fundamental classes] Ginzburg, Section 1