Injective object

Depends on

• Abelian category

Used by

• Injectives have costandard filtrations Corollary
• A category with enough projectives and with enough injectives Definition
• Effaceable cohomological delta functor in positive degrees Definition
• Injective resolutions in an abelian category Definition
• An acyclic resolution that is not an injective resolution Example
• Cartan-Eilenberg only resolves terms False statement
• FALSE: an acyclic resolution is the same thing as an injective resolution False statement
• Cartan–Eilenberg comparisons preserve both filtrations Lemma
• Cofinal Čech vanishing implies derived acyclicity Lemma
• Extension from subobjects of a generator detects injectivity Lemma
• Finite biproducts of injective objects are injective Lemma
• Finite-dimensional tensoring preserves projectives in category O Lemma
• Injective abelian sheaves are flasque Lemma
• Injective modules are flasque and Ext from the structure sheaf is cohomology Lemma
• Lifting a morphism from an exact complex into an injective resolution Lemma
• Long exact global sheaf Ext sequence in the first variable Lemma
• N-invariants send injective G-modules to Q-acyclics Lemma
• One-step extension of a partial injective resolution Lemma
• Pushforward along a closed immersion preserves sheaf cohomology Lemma
• Restriction of injective group modules is injective Lemma
• Contravariant derived functors are derived on the opposite category Proposition
• Positive right derived functors vanish on injective objects Proposition
• Translation functors are exact and biadjoint Proposition
• A bounded above acyclic complex of injective objects is contractible when its cycle monomorphisms split Theorem
• A bounded below complex of injectives is homotopically injective Theorem
• An object is projective exactly when Hom out of it is exact Theorem
• Cohomology of a finite disjoint union Theorem
• Flasque abelian sheaves are Γ-acyclic Theorem
• Hyper-Ext spectral sequence Theorem
• Injective object characterisations Theorem
• Lyndon-Hochschild-Serre spectral sequence Theorem
• Mayer–Vietoris sequence for sheaf cohomology Theorem

Dependency tree · two levels

3 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

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, Definition 1.6.5 (standard reference, not scraped)