Projective object

Depends on

• Abelian category

Used by

• A category with enough injectives but not enough projectives Counterexample
• Finite length and finite Hom do not imply a finite category Counterexample
• A category with enough projectives and with enough injectives Definition
• Effaceable homological delta functor in positive degrees Definition
• Finite graded projective modules Definition
• Finite k-linear abelian categories Definition
• Projective resolutions in an abelian category Definition
• Small projective generators and progenerators Definition
• Superfluous subobjects and projective covers in an abelian category Definition
• An acyclic resolution that is not an injective resolution Example
• FALSE: a generator is automatically projective False statement
• A maximal-label Verma is projective in its truncation Lemma
• Diagonal bar coinvariants compute group homology Lemma
• Exact projections onto linkage blocks preserve projectives Lemma
• Extending a partial comparison homotopy by one degree Lemma
• Extending a partial comparison map by one degree Lemma
• Finite-dimensional tensoring preserves projectives in category O Lemma
• Finite-support families of finite-dimensional vector spaces are locally finite but not finite Lemma
• Fitting decomposition in a finite-length abelian category Lemma
• Generator differences form a basis of the free-group augmentation ideal Lemma
• Lifting a map through degree zero of a projective resolution Lemma
• Module diagrams have projective representables and computable derived colimits Lemma
• One-step extension of a partial projective resolution Lemma
• Projective epimorphisms onto the simples generate every finite-length object Lemma
• The degree-zero horseshoe lift Lemma
• The Khovanov-Seidel grid resolutions of the vertex modules Lemma
• A projective object has a length-zero projective resolution Proposition
• Contravariant derived functors are derived on the opposite category Proposition
• Projective covers in O are indecomposable and unique Proposition
• Translation functors are exact and biadjoint Proposition
• A bounded above complex of projectives is homotopically projective Theorem
• A bounded below acyclic complex of projective objects is contractible when its cycle epimorphisms split Theorem
• An object is projective exactly when Hom out of it is exact Theorem
• Category O has enough projectives Theorem
• Finite abelian categories admit finite-dimensional module models Theorem
• Finite graded projectives are finite shifted-free summands Theorem
• Intrinsic finite category hypotheses give a finite projective generator Theorem
• Projective object characterisations Theorem
• Schanuel's lemma in an abelian category Theorem
• Tensoring with a dualizable object preserves projectives 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)