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For an abelian category , the localization of its homotopy category of complexes at quasi-isomorphisms.
A derived category is constructed from a category of complexes, not from one individual chain complex.
wp-derived-categorynlab-derived-categorystacks-derived-modulesstacks-derived-categorystacks-derived-textEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Form complexes in , pass to the homotopy category , and localize at quasi-isomorphisms. The result is a new category, not an equivalent presentation of .
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
stacks-derived-categorystacks-derived-moduleswp-derived-categorynlab-derived-categorystacks-derived-textThis is an authored directed relation from the source endpoint to the target endpoint.
The derived category is obtained from a suitable category of complexes by localizing at quasi-isomorphisms; this is categorical localization, not an elementwise quotient.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
wp-derived-categorynlab-derived-categorynlab-quasi-isomorphismstacks-derived-categoryEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
In the derived category of an abelian category, is represented by morphisms .
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
wp-ext-functornlab-ext-functorstacks-ext-groupsThis is an authored directed relation from the source endpoint to the target endpoint.
In derived categories of modules, the homology of computes , with grading conventions stated explicitly.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
wp-tor-functornlab-tor-functorstacks-tor-groupsThis is an authored directed relation from the source endpoint to the target endpoint.
A derived category carries a canonical triangulated structure.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
wp-derived-categorynlab-derived-categorystacks-derived-modulesstacks-triangulatedwp-triangulatednlab-triangulatedstacks-derived-text