Gts original k direct sum framing oriented gysin

The supported Thom operation. Let $(E,q,o)$ be an oriented complex orthogonal bundle of complex rank $2n$ over a finite-type complex space $Y$. Choose a positive real form $E_R$, so $E=E_R{\otimes }_{\mathbb{R}}\mathbb{C}$ and $q$ is positive definite on $E_R$. Such forms exist and their choices are contractible. One direct construction starts with a Hermitian metric: the symmetric nondegenerate $q$ determines an invertible antilinear self-adjoint operator $A$, whose square is positive Hermitian. Then $J=A(A^2{)}^{-1\mathrm{/}2}$ is an antilinear involution, and its fixed real bundle is positive for $q$. Positive square roots are unique, so this construction varies continuously with the metric; the space of Hermitian metrics is convex. Averaging over a compact torus gives equivariant choices when needed. The orientation $o$ selects the real orientation. For an isotropic vector $v=x+iy$ with $x,y\in E_R$, the equality $q(v,v)=0$ implies $q(x,x)=q(y,y)$ and $q(x,y)=0$. In particular $x=0$ implies $v=0$. Thus the real projection $p_R:E\to E_R$ has no nonzero zero on any isotropic cone $C\subset E$. Pulling back the oriented Thom class of $E_R$ along $p_R{\tau }_C$ gives a class $u_C\in H^{2n}(C,C\setminus Y;\mathbb{Q}),$ where ${\tau }_C$ is the tautological vector and the zero section is identified with its support in $C$. Capping Borel–Moore homology with this supported class defines $$\begin{array}{cccc}& G_{E,C}:H_k^{BM}(C;\mathbb{Q})⟶H_{k-2n}^{BM}(Y;\mathbb{Q})\mathrm{.}& & \text{(9)}\end{array}$$ The Thom class used here can itself be characterized without any transversality of $C$: on each real oriented trivialization it is the generator of $H^{2n}({\mathbb{R}}^{2n},{\mathbb{R}}^{2n}\setminus 0;\mathbb{Q})$. These generators glue, since oriented transition maps act by degree one. Pullback along $p_R{\tau }_C$ specifies a natural supported class on $C$; its definition does not require the other relative cohomology groups of an arbitrary cone fiber to vanish or the class to be unique in its degree. Naturality under pullback follows from naturality of the ambient Thom class. A homotopy of positive real forms gives a supported homotopy through maps whose zero locus remains the zero section, by the same positivity argument. Hence (9) is independent of the chosen form. Projection formula for cap products proves commutation with proper pushforward. For direct sums, the oriented fiber generator is the exterior product of the generators; therefore the Thom classes multiply and the corresponding operations compose. These assertions apply equivariantly by finite-dimensional Borel approximations in every fixed degree. If $C$ lies in a maximal isotropic bundle $\mathrm{\Lambda }$, the map $p_R:{\mathrm{\Lambda }}_{\mathbb{R}}\to E_R$ is an isomorphism: it is injective by the isotropy argument and the real ranks agree. Its sign is precisely the specified orientation comparison. Thus $G_{E,C}$ is the ordinary complex zero-section Gysin of $\mathrm{\Lambda }$, with that sign. In particular, on the zero section $G_{E,Y}$ is cap product with $e(E_R)$, and, when an isotropic subbundle $K$ is present, $$\begin{array}{cccc}& e(E_R)=c_{\mathrm{rk}K}(K)e((K^{\perp }\mathrm{/}K{)}_R)\mathrm{.}& & \text{(10)}\end{array}$$ Indeed an orthogonal positive metric splits the positive real bundle into the real projection of $K$ and its real orthogonal complement, and the latter is the positive real form of the quadratic reduction. Multiplicativity of the ordinary oriented real Euler class proves (10). There is a useful concrete formula on fundamental cycles. For an integral isotropic cone $C$ not contained in the zero section, blow up its zero ideal. Since $C$ is a degree-one generated cone in a vector bundle, the blowup is the total space of the tautological line on $\mathbb{P}(C)$, with exceptional zero divisor $D=\mathbb{P}(C)$. Pullback of $E$ has an isotropic line $L$, and its tautological section cuts out $D$. Splitting off the positive real projection of $L$, (9) and ordinary Thom multiplicativity give $$\begin{array}{cccc}& G_{E,C}[C]=({\rho }_D{)}_{\ast }\left(e\right((L^{\perp }\mathrm{/}L{)}_R)\cap [D])\mathrm{.}& & \text{(11)}\end{array}$$ The same assertion for a cycle is obtained by blowing up its intersection with the zero section; its section generates a line after that blowup. Cycles already in the zero section instead contribute $e(E_R)$ capped with that cycle. Proper birational pushforward of the strict transform is the original fundamental cycle, so this computation does not discard any multiplicity. Formula (11) is a direct computation of the homological operation, not a theorem identifying two independently defined operations.

On the original overlap put $V=V_1\oplus V_2$ and $P=\mathbb{P}(V)$ with tautological line $L$. Let $P_i=\mathbb{P}(V_i)$ be the closed zero-component axes and let $J_i\subset P_i$ be the actual original stable pair opens. Openness is part of the given original one-frame hypothesis, not an assertion for a new collective locus. The locus $P^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$ of trivial normalized framing-preserving endomorphisms is open: the space of such endomorphisms is the kernel of the finite relative Hom/evaluation matrix, whose fibre dimension is upper semicontinuous. Every $J_i$ lies in it. Indeed any nonzero frame-killing endomorphism would give $1+th$ as a nontrivial frame-preserving automorphism for generic $t$, contradicting the given normalized algebraic-space property.

Cosection-localized zero Gysin on cycles and rational equivalences. Let $F\to X$ have rank $r$ and let $\sigma :F\to {\mathcal{O}}_X$ be a cosection with zero scheme $S$. Put $F(\sigma )=F{\mathrm{\mid }}_S\cup \ker (\sigma {\mathrm{\mid }}_{X\setminus S})$, with its reduced cycle support. For an integral cycle $B\subset F(\sigma )$ contained over $S$, set $G_{\sigma }[B]=0_{F{\mathrm{\mid }}_S}^{!}[B]$. Otherwise let $X_B=\overline{\pi (B)}$ and blow up the image ideal of $\sigma {\mathrm{\mid }}_{X_B}$, taking the component dominating $X_B$. Denote this proper birational map by $\rho :{\tilde{X}}_B\to X_B$. There is a surjective map ${\rho }^{\ast }F⟶{\mathcal{O}}_{{\tilde{X}}_B}(-D),F^{\mathrm{\prime }}=\ker ({\rho }^{\ast }F⟶\mathcal{O}(-D))\mathrm{.}$ The strict transform $\tilde{B}$ lies in $F^{\mathrm{\prime }}$: it does so on the dense open where $\sigma$ is nonzero, and $F^{\mathrm{\prime }}$ is closed. Define $$\begin{array}{cccc}& G_{\sigma }[B]=-({\rho }_D{)}_{\ast }(D\cdot 0_{F^{\mathrm{\prime }}}^{!}[\tilde{B}])\mathrm{.}& & \text{(12)}\end{array}$$ The sign is essential. The quotient line in this convention is $\mathcal{O}(-D)$; Kiem–Li write a divisor whose line bundle is the quotient, so their divisor is $-D$. More generally a proper generically finite regularization and a cycle lift of degree $k$ give the same expression divided by $k$. To verify this, take a common integral component of the fiber product dominating both cycle lifts and regularize once more. Its two pulled-back quotient lines agree as sublines of the rational trivial line, since both are the image ideal of the same cosection. Their Cartier divisors therefore agree, including multiplicities. Their kernels agree as the saturated kernel of the same generic functional in a locally free sheaf. Ordinary Gysin commutes with proper pushforward, and the projection formula for the common divisor shows that both expressions equal the expression on the common refinement divided by its degree over $B$. This proves independence of the regularization actually used in (12). A cycle lift whose proper image has smaller dimension contributes zero. Indeed compare with a regularization generically finite over the image's base, as above. On the common refinement its image in that regularization has smaller dimension, so its proper cycle pushforward is zero. Commutation of ordinary zero Gysin with that pushforward and the projection formula make (12) zero as well. This is stronger than merely knowing that its final pushforward into $X$ vanishes. For clarity, the exact ordinary excess identity used below is $$\begin{array}{cccc}& 0_F^{!}{j}_{\ast }\alpha =c_{\mathrm{rk}Q}(Q)0_{F_1}^{!}\alpha \text{for }0\to F_1\stackrel{j}{\to }F\to Q\to 0.& & \text{(13)}\end{array}$$ Pull back a base cycle along the vector-bundle projection, which is an isomorphism on Chow groups. The identity becomes the top-degree part of $c(F)c(F_1{)}^{-1}=c(Q)$, proving (13). This also proves it for every cycle, by the same bundle homotopy isomorphism. To verify rational equivalence for (12), let $W\subset F(\sigma )$ be integral and $f$ a rational function. Regularize once over the base of $W$, lift $W$, and pull back $f$. Proper pushforward of its zero and pole divisors is the original zero and pole divisor times the generic degree. Split the lifted divisors into three groups: components whose image dimension drops, components over $D$, and the remaining components. The first group contributes zero by the preceding paragraph. For the second group, (13) with quotient $\mathcal{O}(-D)$ identifies the divisor term in (12) with the ordinary $0_{F{\mathrm{\mid }}_S}^{!}$ term. The remaining group is exactly the strict-transform contribution of (12). Their total difference is zero, since ordinary zero Gysin and intersection with $D$ send the principal divisor of the lifted $f$ to zero. Hence $G_{\sigma }(\mathrm{div}f)=0$ as a class on $S$. This proves that (12) defines a map on Chow groups, not only on cycles. The same common-refinement and degree argument proves commutation with proper pushforward. It uses actual zero-Gysin and divisor projection formulas, and never assumes injectivity of $A_{\ast }(S)\to A_{\ast }(X)$. One special evaluation fixes the later signs. Suppose $s$ is a section of a line subbundle $L\subset \mathrm{\Lambda }$, with zero divisor $D$. The associated cosection of ${\mathrm{\Lambda }}^{\vee }$ has quotient $L^{\vee }=\mathcal{O}(-D)$ and kernel $(\mathrm{\Lambda }\mathrm{/}L{)}^{\vee }$. Formula (12), applied to the zero-section cycle, gives $$\begin{array}{cccc}& G_{\sigma }[0_{{\mathrm{\Lambda }}^{\vee }}]=(-1{)}^n{c}_{n-1}(\mathrm{\Lambda }\mathrm{/}L)\cap [D],n=\mathrm{rk}\mathrm{\Lambda }\mathrm{.}& & \text{(14)}\end{array}$$ Multiplying by the $(-1{)}^n$ in the positive-maximal-isotropic square-root convention leaves $c_{n-1}(\mathrm{\Lambda }\mathrm{/}L)\cap [D]$. This is exactly the isotropic-line blowup expression (11) once an adapted maximal isotropic flag has been justified. The remaining step is that adapted-flag comparison for the source's prescribed flag construction; (14) alone does not yet assert that comparison.

Define the ACTUAL common open $U=P^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}\setminus ((P_1\setminus J_1)\cup (P_2\setminus J_2))\mathrm{.}$ The removed subsets are closed in $P$, because the axes are closed and $J_i$ is open in its axis. Thus $U$ is an algebraic-space open with trivial normalized inertia, contains both original $J_i$, and has zero-component schemes exactly $J_i$. It is not defined by an unproved higher collective-generation condition. In particular no properness of this common open is required.

Comparison on the actual supported cycle classes. First the global Euler class in the source flag construction has cycle-class image $e(E_R)$. To verify every normalization, form the bundle of isotropic flags of lengths $1,\dots ,n-1$. Successively its fibers are smooth quadrics of complex dimensions $2n-2,2n-4,\dots ,2$. At stage $j$ let $h_j$ be the hyperplane class of the tautological isotropic line in the quadratic reduction. Put $h={\prod }_{j=1}^{n-1}{h}_j^{2(n-j)}\mathrm{.}$ Each quadric's hyperplane degree is two, as follows by intersecting its single degree-two equation with complementary generic hyperplanes in projective space. Iterated projective pushforward therefore gives $p_{\ast }h=2^{n-1}$. The final oriented rank-two quadratic quotient chooses one of its two isotropic lines, producing the positive maximal isotropic bundle $\mathrm{\Lambda }$ on the flag space. Projection to the positive real form identifies ${\mathrm{\Lambda }}_{\mathbb{R}}$ with $p^{\ast }{E}_R$ with its prescribed positive orientation. Hence $c_n(\mathrm{\Lambda })=p^{\ast }e(E_R)$ in cohomology. It follows directly that $$\begin{array}{cccc}& \mathrm{cl}(2^{1-n}{p}_{\ast }(h{c}_n(\mathrm{\Lambda })\cap \alpha ))=e(E_R)\cap \mathrm{cl}(\alpha )\mathrm{.}& & \text{(15)}\end{array}$$ This proves the required cycle-class statement; it does not need an unproved equality of all characteristic classes in integral Chow. For an isotropic section $s$ and a positive maximal isotropic $\mathrm{\Lambda }$, put $s^{\ast }={\pi }_{{\mathrm{\Lambda }}^{\vee }}s$ and $Z^{\ast }=Z(s^{\ast })$. On $Z^{\ast }$, $s$ lies in $\mathrm{\Lambda }$ and defines a cosection of ${\mathrm{\Lambda }}^{\vee }{\mathrm{\mid }}_{Z^{\ast }}$. This cosection annihilates the normal cone $C_{Z^{\ast }\mathrm{/}Y}$. Indeed locally split $E=\mathrm{\Lambda }\oplus {\mathrm{\Lambda }}^{\vee }$, write $s=s_{\mathrm{\Lambda }}+s^{\ast }$, and use isotropy $⟨s^{\ast },s_{\mathrm{\Lambda }}⟩=0$. The linear function defined by $s_{\mathrm{\Lambda }}$ vanishes on every graph of $t{s}^{\ast }$. Passing to its scheme-theoretic Rees limit gives its vanishing on the normal cone, including nilpotents. This is a polynomial identity on the Rees algebra, not just a set-theoretic limit. The source localized square-root operator is, on this flag cover, $(-1{)}^n$ times the cosection-localized ordinary zero Gysin applied to the normal-cone specialization. Pushing from $Z(s)$ into $Y$ and using (12)–(13) gives ordinary zero Gysin of the normal cone: the quotient-line top Chern factor is exactly the divisor in (12). Ordinary normal-cone specialization and zero Gysin compute $c_n({\mathrm{\Lambda }}^{\vee })\cap \alpha$. Multiplying by $(-1{)}^n$ changes this to $c_n(\mathrm{\Lambda })\cap \alpha$. Applying the flag pushforward and (15) proves $$\begin{array}{cccc}& i_{\ast }\mathrm{cl}({\sqrt{e}}_{\mathrm{a}\mathrm{l}\mathrm{g}}(E,s)(\alpha ))=e(E_R)\cap \mathrm{cl}(\alpha )\mathrm{.}& & \text{(16)}\end{array}$$ Both the construction and this identity commute with restriction to open subsets, by the blowup formula (12) and the Rees specialization construction. That open compatibility is the only localization property needed in the next step. Let now $C\subset E{\mathrm{\mid }}_Y$ be an isotropic cone. Its canonical projective completion is $\bar{C}=\mathbb{P}(C\oplus \mathcal{O})$, with projection $\bar{\pi }:\bar{C}\to Y$. In ${\bar{\pi }}^{\ast }(E\oplus {\mathcal{O}}^2)$ give the last two summands the hyperbolic quadratic form. The tautological line $L={\mathcal{O}}_{\bar{C}}(-1)$ is isotropic, since its vectors have the form $(v,t,0)$ and $q(v,v)=0$ on $C$. The oriented quadratic reduction $\bar{E}=L^{\perp }\mathrm{/}L$ has rank $2n$. The constant isotropic vector $(0,1,0)$ lies in $L^{\perp }$ and induces a section $\bar{s}$ of $\bar{E}$. Its zero scheme is precisely the zero section $Y\subset C$: being zero means that $(0,1,0)$ lies in $L$, forcing $v=0$ and $t\mathrm{\ne }0$. On the open chart $t\mathrm{\ne }0$, the pair $(\bar{E},\bar{s})$ identifies with $({\pi }^{\ast }E,-{\tau }_C)$. This follows by subtracting multiples of $(v,1,0)$ to set the second coordinate to zero in the orthogonal quotient. Replacing a complex section by its negative preserves the oriented localized class. For an algebraic cycle $\alpha$ on $C$, take its closure $\bar{\alpha }$ in $\bar{C}$. Apply (16) on $\bar{C}$ and then push by $\bar{\pi }$. Since $\bar{\pi }$ composed with the zero section is the identity, and the zero locus lies wholly in the open chart $C$, open compatibility gives $$\begin{array}{cccc}& \mathrm{cl}\left(\sqrt{0_E^{!}}\alpha \right)={\bar{\pi }}_{\ast }(e({\bar{E}}_R)\cap \mathrm{cl}(\bar{\alpha }))\mathrm{.}& & \text{(17)}\end{array}$$ No injectivity of the zero-section pushforward has been invoked: its proper left inverse ${\bar{\pi }}_{\ast }$ is what recovers the supported class here. Apply the same reasoning to the Thom operation (9). Its supported class extends to the isotropic section $\bar{s}$ on the proper completion, and its pushforward into $\bar{C}$ is the ordinary Euler class of ${\bar{E}}_R$. Open compatibility of the Thom class and the same proper left inverse show $$\begin{array}{cccc}& G_{E,C}\mathrm{cl}(\alpha )={\bar{\pi }}_{\ast }(e({\bar{E}}_R)\cap \mathrm{cl}(\bar{\alpha }))\mathrm{.}& & \text{(18)}\end{array}$$ Equations (17) and (18) prove the exact supported comparison on the cycle classes used in the virtual construction. The calculation is equivariant: make it over a finite-dimensional Borel approximation before taking each fixed degree. It is independent of which cycle closure is used. Indeed a difference is supported on the boundary $\mathbb{P}(C)$. There the section $\bar{s}$ is nowhere zero and isotropic; its positive real projection is nowhere zero, so $e({\bar{E}}_R)$ restricts to zero. The projection formula annihilates every boundary contribution. The same formula proves commutation with a refined regular pullback: projective completion and its tautological orthogonal reduction commute with the Cartesian base change, refined pullback commutes with proper pushforward, and cap product with the pulled-back Euler class commutes with that pullback. A refined pullback of a chosen cycle closure restricts to the refined pullback of the cycle on the open chart, so (17) applies without any flatness assertion about the cone.

The projections of the tautological line give sections $s_i$ of $B_i=L^{\vee }\otimes V_i$ on $U$. The zero scheme of $s_j$ is precisely $J_i$ for $i\mathrm{\ne }j$. On the chart where component $i$ is nonzero the pair is a line in $V_i$ plus a vector in $B_j$. Over $J_i$ every such vector has trivial normalized stabilizer, since component $i$ already does, and it cannot belong to the removed bad axis. Thus this neighbourhood is the full vector bundle $\mathrm{Tot}(B_j{\mathrm{\mid }}_{J_i})$. The section is its fibre coordinate. Therefore $i_i:J_i\to U$ is a regular closed embedding of codimension $f_j$, with normal bundle $B_j{\mathrm{\mid }}_{J_i}$. Its Koszul regularity holds over singular and nonreduced old bases, being ordinary independent polynomial coordinates.

Only the following special compatibility is needed. If $p:\mathrm{Tot}(B)\to J$ is a vector bundle with the completed smooth relative obstruction diagram of [F1], its virtual class is $p^{!}[J{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}$ with the induced hyperbolic orientation. To verify this rather than assume a virtual-pullback theorem, trivialize $B$ and an explicit quadratic presentation of $J$. The new smooth ambient is the old one times this vector space; its relation section and middle orthogonal bundle are pulled back, and its intrinsic normal cone is the corresponding flat pullback. The additional outer tangent and its shifted dual are precisely the smooth relative summands of the completed diagram. The construction in step 3.1 commutes with this flat pullback: flag bundles, blowup regularizations, projective completions and their cycle formulas all base change flatly. Hence it gives the stated equality on actual virtual cycle classes. Ordinary zero-section refined Gysin then satisfies $i^{!}{p}^{!}=1$ by vector-bundle homotopy. Trivialization changes preserve this computation, the ordered relative orientation and the ordinary normal map, so it glues. This proves this smooth bundle/zero-section compatibility, not a virtual pullback theorem for every PVP diagram.

Apply [F1] to the actual $Y$ decoration with both component maps. On $U$ it gives a self-dual complex and virtual class whether or not its fixed locus is proper. The ordinary zero-component inclusion gives exactly the original $J_i$ diagram. On its actual universal family, pull back the SAME completed diagonal and trace fibre of [F1, steps11.2, 12.2, 13.2, 14.1, 15.1, 16.1 and 17.1], together with the ORIGINAL module and pair cocycles ${\kappa }_B^0,{\kappa }_I^0$ and the explicit homotopy ${\eta }_G=i_G(p_B{\alpha }_B-p_G{\alpha }_G)-H_G{\kappa }_I^0$ of its step16.1, including its scalar determinant component. First restrict their square with target $L_U{\mathrm{\mid }}_{J_i}$, and only then compose with the canonical map to $L_{J_i}$; the resulting comparison is the original $J_i$ obstruction diagram. This is restriction of a complex and its actual cotangent square on the specified base, not a claimed old-generator restriction of arbitrary completed modules. The relative summands are the vector-bundle tangent in step 3.2 and its shifted dual; their determinant factors pair by ordered evaluation. Hence the induced orientation is the original one. Reordering whole complex hyperbolic blocks gives no additional real-orientation sign. This is the same completed smooth relative diagram verified in step 4.1, so $i_i^{!}[U{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}=[J_i{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}\mathrm{.}$ The equality is supported on the actual zero scheme, including the cycle multiplicities and nilpotents; no injectivity of pushforward from that scheme is assumed.

We verify this restriction on the actual completed smooth chart, rather than deducing it from an abstract equivalence of quadratic complexes. On the neighbourhood in step 3.2 choose coordinates $w_1,\dots ,w_{f_j}$ for $B_j$. In the common underlying $Y$ diagram these are the independent added component map, with no relation involving $w$; the equation $w=0$ is precisely the original $J_i$ diagram, not a second stability condition. Write a semi-free presentation of the original diagram as $(R,d)$, and retain all its relation and gauge generators. The independent extension is $(R[w],d)$ with $dw=0$. Its actual evaluated cyclic-diagonal comparison, constructed by the full bar contractions and source trace homotopy of [F1], has the added dual generators $w^{\ast }$ of degree $-2$ and the universal evaluation pairing $\sum d{w}_{\alpha }d{w}_{\alpha }^{\ast }$. Its Hamiltonian has no new term: the old Hamiltonian depends only on old generators and the new coordinates have no differential or equation. Consequently its completed cotangent complex is the old complex plus ${\mathrm{\Omega }}_{B_j\mathrm{/}{J}_i}$ in degree zero and its shifted dual in degree $-2$. In the ORIGINAL universal module bar cocycle the new forward action contributes its coefficient derivative $d_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}{w}_{\alpha }$, and its dual action has value zero. Under the marked projection of [F1, step16.1] these give the obstruction arrow $d{w}_{\alpha }\mapsto d{w}_{\alpha }\in L_U$ on the forward variation and zero on its added dual. This computes the module cocycle itself, rather than differentiating the differential of an endomorphism complex. The original differential, relation section and middle orthogonal bundle are pulled back. Thus the actual obstruction arrow is the one used in step 4.1, not merely an isomorphic unstructured complex. The source trace homotopy fibre is the SAME one on both models. Its local projector may have the nonzero trace correction computed in [F1]; the common-bar contraction carries that correction and the original module comparison primitives ${\alpha }_B,{\alpha }_G$, hence ${\eta }_G$, as well. Auxiliary relative bundle Atiyah markings remain in their gauge rows. At $w=0$ the forward normal marking $dw$ is still present with target $L_U{\mathrm{\mid }}_{J_i}$ and becomes zero only on composition to $L_{J_i}$; removing the normal hyperbolic pair then leaves the original $J_i$ cocycle and its trace-fibre primitive.

For a regular section the ordinary supported zero-section identity gives $i_{\ast }{i}^{!}a=c_{\mathrm{r}\mathrm{k}\mathrm{B}}(B)\cap a$. This follows from the excess identity (13), or from ordinary Thom multiplication on a vector-bundle chart; it glues for these actual sections. Apply it to step 5.1 to obtain $(i_1{)}_{\ast }[J_1{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}=c_{f_2}(B_2)\cap [U{]}^{\mathrm{v}\mathrm{i}\mathrm{r}},(i_2{)}_{\ast }[J_2{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}=c_{f_1}(B_1)\cap [U{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}\mathrm{.}$ Here the right sides have their supported meaning on the zero components. All pushforwards used so far are closed-embedding or projective-completion pushforwards. On passing to the source localized homology these supported classes come from the original proper fixed loci; there is no pushforward integral over a nonproper $U$.

This local calculation also identifies the actual cones and orientations on the zero component. If $I$ is the old equation ideal in a smooth chart with ring $A$, the new ideal in $A[w]$ is $IA[w]$, and $(IA[w]{)}^n\mathrm{/}(IA[w]{)}^{n+1}=(I^n\mathrm{/}{I}^{n+1}){\otimes }_{A}A[w]$ in every degree. Therefore the scheme-theoretic normal cone, with its lengths and nilpotents, is the flat pullback of the old cone. The same assertion holds after the identical auxiliary-gauge quotient; on $U$ the normalized gauge action is free. The ordered pairing of each new forward variation with its shifted dual gives the canonical hyperbolic determinant trivialization. At $w=0$ its old factor is exactly the stipulated original orientation, since the common orientation was defined by tensoring that old factor with this ordered hyperbolic factor; the stipulated compatibility of the old orientations on the overlap is used here. Changes of a trivialization $w^{\mathrm{\prime }}=Gw$ transform the dual by $(G^{-1}{)}^{\mathsf{t}}$; the two determinant factors cancel. Explicitly $d_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}(Gw)=G{d}_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}w+(d_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}G)w$; on the zero component the second term vanishes, while the normal marking transforms by $G$. The original bundle Atiyah markings and the scalar determinant primitive restrict as well. Thus the cotangent transition terms of the actual bundle and its dual preserve the same Liouville pairing and ORIGINAL universal obstruction arrow. The overlap identifications are the common-bar prism maps of [F1, step15.1], and their connection/Atiyah defect is the homotopy of its step16.1. Thus their full Čech components glue the actual cotangent diagram; an identity or horizontal splitting of old blocks has not been assumed. The resulting diagram is the completed smooth pullback diagram of step 4.1. Applying its explicit cone/zero-section computation and then the ordinary zero-section refined Gysin proves $i_i^{!}[U{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}=[J_i{]}^{\mathrm{v}\mathrm{i}\mathrm{r}}$ on the actual $J_i$, with the original orientation. This argument uses the actual completion comparison of [F1]; an ordinary arrow space with the same truncation would not suffice.

On $J_i$, component $i$ is a nowhere-zero section of $B_i$. It defines a trivial line subbundle, so $c_{f_i}(B_i){\mathrm{\mid }}_{J_i}=0$. Apply the other cap and the projection formula to either identity of step 6.2. Both double-cap classes are zero. Thus the proved comparison is an actual oriented supported Gysin identity, and its double cap has no injectivity with which to infer index independence. The original claim is exactly this construction and its stated boundary.