Murthy splitting input in rank three

A finite-type algebra over a field is noetherian, and normality makes $A$ reduced. By [F1], $P=\mathrm{\Gamma }(X,E)$ is finite projective of constant rank $3$, and $E$ is its associated bundle. The affine K-zero comparison in [F1] is an isomorphism.

Let $G\subset K_0^{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}(A)$ be the subgroup in [F4] for $n=3$. A height-three smooth closed point lies on a three-dimensional component, and conversely a smooth closed point on a three-dimensional component has height three. Over the perfect field $k$, these are the point classes in [F3]. Their residue modules admit finite projective resolutions: at such a point the local ring is regular, while the residue module is supported only there; the associated perfect residue sheaf has the same class as the alternating resolution by [F1]. Hence the comparison in [F1] identifies $G$ with $F_{\mathrm{s}\mathrm{m}}^3{K}_0(X)$ in the pure case and with $F_{\mathrm{s}\mathrm{m},\mathrm{t}\mathrm{o}\mathrm{p}}^3{K}_0(X)$ in the general normal case. By [F3], multiplication by $2$ is injective on that image, and therefore on $G$. Thus $G$ has no $(3-1)!=2$ torsion.

By [F2], $\beta (E)=0$ implies that the image of $C_3^K(P)$ in perfect $K_0(X)$ is zero. The comparison is injective by [F1], so $C_3^K(P)=0$ in projective $K_0(A)$. The reducedness, dimension, field, rank and factorial-torsion hypotheses of [F4] now hold. It follows that $P\cong A\oplus Q$ for a finite projective module $Q$ of rank two. Applying the affine equivalence gives $E\cong {\mathcal{O}}_X\oplus \tilde{Q}$, where $\tilde{Q}$ is a rank-two vector bundle.

The pure smooth-point subgroup and the zero-extended top-dimensional subgroup are subgroups of $K_0(X)$ by their definitions. Thus an element trivial in either subgroup is zero in $K_0(X)$, and step 3.1 applies. The stated nonpure convention follows from [F3]. This proves the splitting implication and its subgroup formulations.