BM concentration from a finite invariant affine atlas

Fix a member $U=\mathrm{Spec}A$ of the invariant affine cover. Choose finitely many algebra generators of $A$. The algebraic $T$-action makes $A$ a rational $T$-comodule, so the coaction of each generator is a finite Laurent sum and the generator is a finite sum of character eigenfunctions. Replacing the generators by these finitely many eigencomponents still generates $A$, since their sums recover the original generators. The fixed ideal in $A$ is generated by its nontrivial-weight subspace: the coaction sends a weight-$\chi$ element $f$ to $f\otimes \chi$, so the equalizer with the trivial coaction kills $f$ when $\chi \mathrm{\ne }1$, and the quotient by all such elements has only weight-zero functions and hence trivial action. The ideal generated by the nontrivial-weight algebra generators contains every nonzero-weight polynomial in the generators, since each monomial of nonzero total weight has a nonzero-weight factor. Thus $U\setminus F$ is covered by the finitely many invariant principal opens $D(f)$ for those generators.

Taking these principal opens over the finitely many affine charts gives a finite invariant open cover $\{D_a\}$ of $X\setminus F$. Every nonempty finite intersection is a $T$-invariant open subset of at least one $D_a$; [F2] therefore makes its localized Borel–Moore groups zero in every degree.

For a finite invariant open cover of $X\setminus F$, the augmented Čech complex with terms $j_{I!}{\mathbb{Q}}_{D_I}$, where $D_I$ is an intersection and $j_I:D_I\hookrightarrow X\setminus F$ is open, resolves the constant equivariant sheaf. At a point the indices containing it form a nonempty simplex, whose augmented chain complex is contracted by adjoining its least index. Applying derived Hom into the normalized dualizing object gives a finite Čech spectral sequence for Borel–Moore homology: open base change identifies each term with the Borel–Moore group of an intersection $D_I$. Those groups vanish after localization by [F2]. Take the direct sum over all homological degrees in each term, so that the spectral sequence consists of $R_T$-modules with homogeneous differentials. Since the Čech filtration is finite and localization of these total modules is exact by [F4], taking the homogeneous degree pieces gives that the abutment $S_T^{-1}{H}_j^{\mathrm{B}\mathrm{M},T}(X\setminus F;\mathbb{Q})$ is zero for every $j$.

Apply exact localization to the total graded closed/open sequence and then take homogeneous pieces. The resulting closed/open localization sequence for $i:F\hookrightarrow X$ and $X\setminus F$ contains $S_T^{-1}{H}_{j+1}^{\mathrm{B}\mathrm{M},T}(X\setminus F)\to S_T^{-1}{H}_j^{\mathrm{B}\mathrm{M},T}(F)\stackrel{i_{\ast }}{\to }{S}_T^{-1}{H}_j^{\mathrm{B}\mathrm{M},T}(X)\to S_T^{-1}{H}_j^{\mathrm{B}\mathrm{M},T}(X\setminus F)$. Both outside groups vanish by step 3.1, so $i_{\ast }$ is an isomorphism. If $F=\mathrm{\varnothing }$, the same argument gives zero localized Borel–Moore homology for $X$; if $F=X$, $i_{\ast }$ is the identity. The proof used only the finite invariant affine cover and not Sumihiro's theorem or compactness.