Perfect determinant with true-triangle additivity: precise external input

PDF p. 8: Proposition3.2, perfect determinant; precise KM true-triangle input

Under AC and fixed universes for the sheaf-derived model, let $X$ be a scheme and let its perfect complexes mean objects locally represented by bounded finite locally free complexes. There is a global graded determinant functor on their derived isomorphisms. Its value is a line with locally constant integer Euler rank. For every actual short-exact complex diagram $0\to P\to Q\to R\to 0$ with perfect terms, its true-triangle additivity supplies an isomorphism $\det (P)\otimes \det (R)\cong \det (Q)$. The determinant of zero is the trivial graded line. On a bounded finite locally free representative its underlying line is ${⨂}_i(\det P^i{)}^{(-1{)}^i}$. It commutes with derived pullback, with the source's composition compatibility.

Perfect complexes on schemes and their support

Exterior powers and determinant of an algebraic bundle

The Axiom of Choice

Knudsen–Mumford Definition4 and Theorem2 are the external input, under the exact hypotheses and reading limits in the source record. Their perfect-complex definition agrees with the supplied local sheaf-derived definition. Their true triangles retain actual short-exact complex diagrams; one cannot replace all their morphisms by arbitrary commuting ordinary derived-triangle diagrams. The graded-line symmetry is retained before taking Picard classes.

For applications to cofiber classes, an actual complex map has the termwise split short-exact diagram from its target to its mapping cone to its shifted source. This is within the imported true-triangle contract even when no global bounded bundle representative exists. No resolution property, separatedness, regularity or finite-type condition is imposed on $X$. This input supplies the global line and its descent isomorphisms; it supplies no enhanced K-space map or unrestricted ordinary-triangle naturality. No local proof of the imported determinant theorem is claimed.