Quasi-isomorphism

Depends on

• A chain map induces a well-defined map on homology

Used by

• An exact functor preserves quasi-isomorphisms and reflects them when it is conservative Corollary
• Quasi isomorphism criterion from a filtered map Corollary
• Two-out-of-three for quasi-isomorphisms in a short exact sequence diagram Corollary
• A quasi-isomorphism that is not an isomorphism of complexes Counterexample
• A quasi-isomorphism with no homotopy inverse Counterexample
• An acyclic noncontractible cone Counterexample
• Cohomological mixed Hodge complexes Definition
• Derived schemes and the cotangent complex of a morphism Definition
• K-flat complexes of abelian sheaves in the bounded-above setting Definition
• The cotangent complex of a morphism of schemes Definition
• The cotangent complex of a ring map Definition
• FALSE: an acyclic mapping cone is contractible False statement
• FALSE: every quasi-isomorphism is a chain homotopy equivalence False statement
• FALSE: every quasi-isomorphism is an isomorphism of complexes False statement
• FALSE: the homotopy category is obtained by identifying quasi-isomorphisms with identities False statement
• Acyclic directions of the Čech–Godement double complex Lemma
• Cech hypercohomology of an affine cover computes Ext of the cotangent complex Lemma
• Derived tensor product of abelian sheaves Lemma
• Finite projective complex for proper flat coherent cohomology Lemma
• Flat resolutions of abelian sheaves and K-flatness of bounded-above flat complexes Lemma
• H0 of the cotangent complex and the polynomial case Lemma
• Independence of the cotangent complex from the chosen simplicial resolution Lemma
• K-flat sheaf complexes preserve quasi-isomorphisms Lemma
• Koszul coherence of derived sheaf tensor Lemma
• Normalized simplicial chains, prism homotopies and the abelian trivial-fibration criterion Lemma
• Sheaf cohomology classes as derived morphisms Lemma
• The Khovanov-Seidel grid resolutions of the vertex modules Lemma
• The Lichtenbaum-Schlessinger complex computes Ext of the cotangent complex in degrees at most two Lemma
• Truncation, differentials and the cotangent complex of a smooth morphism Lemma
• A chain map is a quasi-isomorphism exactly when its cochain reindexing is Proposition
• A quasi-isomorphism has an acyclic cone in the triangulated language Proposition
• Isomorphisms of complexes are quasi-isomorphisms Proposition
• Quasi isomorphisms contain identities and are closed under composition Proposition
• Shift preserves homotopy equivalences, contractibility, and quasi-isomorphisms Proposition
• A chain homotopy equivalence is a quasi-isomorphism Theorem
• A chain map is a quasi-isomorphism exactly when its cone is acyclic Theorem
• Acyclic assembly lemma for a first quadrant double complex Theorem
• Canonical map from fixed-cover Čech to sheaf cohomology Theorem
• Leray acyclic-cover comparison Theorem
• Long exact sequence of sheaf cohomology Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
• The Stacks Project, Definition 12.13.4 (standard reference, not scraped)