(also nonabelian homological algebra)
For
a filtered object in an abelian category $\mathcal{C}$, the associated graded object $Gr(X)$ is the graded object which in degree $n$ is the cokernel of the $n$th inclusion, fitting into a short exact sequence
hence the quotient of the $n$th layer of $X$ by the next lower one:
For $\mathcal{A}$ an abelian category and $C_{\bullet, \bullet}$ a double complex in $\mathcal{A}$, let $X = Tot(C)$ be the corresponding total complex. This is naturally filtered by either row-degree or by column-degree. The corresponding associated graded complex is what the terms in the spectral sequence of a filtered complex compute.
Discussion of the universal property of the associated graded construction on mathoverflow
Last revised on November 25, 2019 at 09:34:08. See the history of this page for a list of all contributions to it.