Affine perfect triangle classes and bundle Euler classes

PDF p. 12: Bundle classes in spectral K zero

Assume AC and suitable universes. Let $A$ be a commutative unital ring and $X=\mathrm{Spec}A$. The triangle Grothendieck group of the homotopy category of $\mathrm{Perf}(X)$ is isomorphic to the vector-bundle Grothendieck group of $X$. Under this isomorphism, the class of the associated sheaf complex of a bounded finite-projective complex $P$ maps to ${\sum }_n(-1{)}^n[\widetilde{P^n}]$, and the class of a degree-zero bundle complex $E[0]$ maps to $[E]$.

Here $\mathrm{Perf}(X)$ is the full stable derived subcategory of locally perfect complexes. The triangle group uses one additive relation for each distinguished triangle; the vector-bundle group uses short exact sequence relations. A bounded complex has only finitely many nonzero terms. No Noetherian or regularity hypothesis is imposed.

Perfect complexes on schemes and their support

Affine bundle K zero is the projective split group

The current online Stacks tag 0AFY, Lemma 15.121.1, has a missing exponent in the first sum of its acyclic-complex display: the intended term is $(-1{)}^n[M^n]$. For the identity complex $\mathbb{Z}\to \mathbb{Z}$ in degrees zero and one, the literal unweighted negative sum is $-2[\mathbb{Z}]$ rather than zero. This localized source typo does not refute Lemma 15.121.2 or [F2], whose supplied local proof uses the alternating Euler class. This affine result does not assert global vector-bundle representation of perfect classes on arbitrary qcqs schemes.