False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Every triangulated subcategory is thick

Statement

Every triangulated subcategory is closed under direct summands.

Refutation

Given: The displayed data.

1.1

In $K^{b} ({F r e e}_{Z}^{f g})$ , let $E$ be the full subcategory of bounded complexes of finitely generated free abelian groups with even Euler characteristic. Shifts negate Euler characteristic and a cone triangle makes Euler characteristics additive, so $E$ is triangulated.

given

2.1

The complex $Z [0] \oplus Z [0]$ lies in $E$ but its direct summand $Z [0]$ does not; hence $E$ is not thick.

step 1.1given∎

Depends on

Used by

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Dependency tree · two levels

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Sources

The Stacks Project, Derived Categories, Section 13.4 (standard reference, not scraped)