nLab
associated graded object

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Homological algebra

homological algebra

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nonabelian homological algebra

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Definition

For

X (n)X (n+1)X \cdots \hookrightarrow X_{(n)} \hookrightarrow X_{(n+1)} \hookrightarrow \cdots \hookrightarrow X

a filtered object in an abelian category 𝒞\mathcal{C}, the associated graded object Gr(X)Gr(X) is the graded object which in degree nn is the cokernel of the nnth inclusion, fitting into a short exact sequence

0X (n1)X (n)Gr n(X)0, 0 \to X_{(n-1)} \to X_{(n)} \to Gr_n(X) \to 0 \,,

hence the quotient of the nnth layer of XX by the next lower one:

Gr(X) n:=X (n)/X (n1), Gr(X)_n := X_{(n)}/X_{(n-1)} \,,

Examples

Revised on October 18, 2012 15:22:52 by Zoran Škoda (161.53.130.104)