Behrend–Fantechi algebraic virtual class from a perfect obstruction theory

Write the chosen resolution as $F^{\bullet }=[F^{-1}\to F^0]$ and set $F_1=(F^{-1}{)}^{\vee }$. The cartesian cone presentation $C(F^{\bullet })=C_X{\times }_{h^1\mathrm{/}{h}^0((E^{\bullet }{)}^{\vee })}{F}_1$ is a closed cone in $F_1$. Its map to $C_X$ is smooth of relative dimension $\mathrm{rk}{F}^0$, so it has pure dimension $\mathrm{rk}{F}^0$; the zero-section Gysin map lowers dimension by $\mathrm{rk}{F}^{-1}$. Hence $0^{!}[C(F^{\bullet })]$ has dimension $\mathrm{rk}{F}^0-\mathrm{rk}{F}^{-1}=n$ and is the class in the Statement.

Choose another global two-term resolution $H^{\bullet }$. Both resolution maps can be represented simultaneously as strict quasi-isomorphisms to a single two-term complex $D=[D^{-1}\to D^0]$ of sheaves representing $E^{\bullet }$. Indeed, represent $E^{\bullet }$ by an injective complex $J$ bounded below, represent both derived maps by chain maps to $J$, and truncate $J$ to $[\mathrm{coker}(J^{-2}\to J^{-1})\to \ker (J^0\to J^1)]$. The maps from the two-term complexes factor through this truncation, which is quasi-isomorphic to $J$ because $E^{\bullet }$ has cohomology only in degrees $-1,0$. Set $K^0=F^0\oplus H^0$ and $K^{-1}=D^{-1}{\times }_{D^0}(F^0\oplus H^0)$, using the sum of the two maps to $D^0$. Its differential is projection to $K^0$, and its map to $D$ is the first projection in degree $-1$ and the sum in degree zero.

Projection to $H^0$ gives an exact sequence $0\to F^{-1}\to K^{-1}\to H^0\to 0$. For surjectivity, locally lift the cohomology class of the image of a section of $H^0$ to $F^0$ using $h^0(F)\simeq h^0(D)$; the difference is a boundary in $D$ and supplies the required $D^{-1}$ coordinate. Its kernel is $D^{-1}{\times }_{D^0}{F}^0\simeq F^{-1}$: for a pair $(d,f)$, injectivity on $h^0$ makes $f$ a boundary in $F$; after lifting that boundary the remaining difference is in $\ker d_D$ and lifts uniquely from $\ker d_F$ by the isomorphism on $h^{-1}$. Thus $K^{-1}$ is a vector bundle, since the displayed extension of vector bundles locally splits. The quotient complex $K\mathrm{/}F$ is the identity complex $[H^0\stackrel{1}{\to }{H}^0]$, so $F\to K$ is a quasi-isomorphism. Exchanging $F$ and $H$ gives the analogous conclusion. This constructs the common global resolution with degreewise subbundles and acyclic vector-bundle quotients.

Dualizing $F^{-1}\hookrightarrow K^{-1}$ gives a vector-bundle surjection $\alpha :K_1\to F_1$ with kernel $(H^0{)}^{\vee }$. The cone presentations satisfy $C(K)=K_1{\times }_{F_1}C(F)$: the acyclic quotient $K\mathrm{/}F$ identifies their quotient-stack presentations with the same vector-bundle stack, and both cones pull back its intrinsic cone. Consequently ${\alpha }^{\ast }[C(F)]=[C(K)]$. Write $p_F:F_1\to X$ and $p_K:K_1\to X$. For any Chow class $\gamma$ on $F_1$, homotopy invariance gives a unique $\delta$ with $\gamma =p_F^{\ast }\delta$. Then ${\alpha }^{\ast }\gamma =p_K^{\ast }\delta$, so inverse bundle pullback gives $0_K^{!}{\alpha }^{\ast }\gamma =\delta =0_F^{!}\gamma$. Apply this to $[C(F)]$, and repeat with $H$. The resulting equality $0_F^{!}[C(F)]=0_K^{!}[C(K)]=0_H^{!}[C(H)]$ proves resolution independence. Every construction is in algebraic rational Chow groups, giving the claimed analytic limitation.