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We show that the Grothendieck abelian category\n                    <jats:inline-formula>\n                      <jats:tex-math>$$X{{\\mathsf {-Qcoh}}}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    of quasi-coherent sheaves on\n                    <jats:italic>X<\/jats:italic>\n                    satisfies the Roos axiom\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\textrm{AB}4^*$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -\n                    <jats:italic>n<\/jats:italic>\n                    : the derived functors of infinite direct product have finite homological dimension in\n                    <jats:inline-formula>\n                      <jats:tex-math>$$X{{\\mathsf {-Qcoh}}}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . In each of the two settings, two proofs of the main result are given: a more elementary one, based on the \u010cech coresolution, and a more conceptual one, demonstrating existence of a generator of finite projective dimension in\n                    <jats:inline-formula>\n                      <jats:tex-math>$$X{{\\mathsf {-Qcoh}}}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    in the semi-separated case and using the co-contra correspondence (with contraherent cosheaves) in the Noetherian case. The hereditary complete cotorsion pair (very flat quasi-coherent sheaves, contraadjusted quasi-coherent sheaves) in the abelian category\n                    <jats:inline-formula>\n                      <jats:tex-math>$$X{{\\mathsf {-Qcoh}}}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    for a quasi-compact semi-separated scheme\n                    <jats:italic>X<\/jats:italic>\n                    is discussed.\n                  <\/jats:p>","DOI":"10.1007\/s10485-025-09845-9","type":"journal-article","created":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T15:52:39Z","timestamp":1771429959000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Roos Axiom Holds for Quasi-Coherent Sheaves"],"prefix":"10.1007","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8836-3911","authenticated-orcid":false,"given":"Leonid","family":"Positselski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,2,18]]},"reference":[{"issue":"2","key":"9845_CR1","first-page":"209","volume":"86","author":"M Boekstedt","year":"1993","unstructured":"Boekstedt, M., Neeman, A.: Homotopy limits in triangulated categories. 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