Suite exacte

En mathématiques, plus particulièrement en algèbre homologique, une suite exacte est une suite (finie ou infinie) d'objets et de morphismes entre ces objets telle que l' de l'un est égale au noyau du suivant. Définition Dans le contexte de la théorie des groupes, on dit que la suite (finie ou infinie) de groupes et de morphismes de groupes G_0 \xrightarrow{f_0} G_1 \xrightarrow{f_1} \cdots \xrightarrow{f_{n-1}} G_n \xrightarrow{f_n} G_{n+1} \xrightarrow{f_{n+1}} \cdots
est exacte si pour tout entier naturel n on a \mathrm{Im}(f_n) = \mathrm{Ker}(f_{n+1}) . Dans ce qui précède, G_0, G_1, \dots sont des groupes et f_0, f_1, \dots des morphismes de groupes avec f_n:G_n\rightarrow G_{n+1}. Cas simples Dans la suite, 0 dénote le groupe trivial, qui est l'objet nul dans la catégorie des groupes.

La suite 0\rightarrow G_1 \xrightarrow fG_2 est exacte si et seulement si f est injectiv

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