The Stacks project

Remark 12.20.3 (Variant). It is often the case that the terms of a spectral sequence have additional structure, for example a grading or a bigrading. To accommodate this (and to get around certain technical issues) we introduce the following notion. Let $\mathcal{A}$ be an abelian category. Let $(T_ r)_{r \geq 1}$ be a sequence of translation or shift functors, i.e., $T_ r : \mathcal{A} \to \mathcal{A}$ is an isomorphism of categories. In this setting a spectral sequence is given by a system $(E_ r, d_ r)_{r \geq 1}$ where each $E_ r$ is an object of $\mathcal{A}$, each $d_ r : E_ r \to T_ rE_ r$ is a morphism such that $T_ rd_ r \circ d_ r = 0$ so that

\[ \xymatrix{ \ldots \ar[r] & T_ r^{-1}E_ r \ar[r]^-{T_ r^{-1}d_ r} & E_ r \ar[r]^-{d_ r} & T_ rE_ r \ar[r]^{T_ r d_ r} & T_ r^2E_ r \ar[r] & \ldots } \]

is a complex and $E_{r + 1} = \mathop{\mathrm{Ker}}(d_ r)/\mathop{\mathrm{Im}}(T_ r^{-1}d_ r)$ for $r \geq 1$. It is clear what a morphism of spectral sequences means in this setting. In this setting we can still define

\[ 0 = B_1 \subset B_2 \subset \ldots \subset B_ r \subset \ldots \subset Z_ r \subset \ldots \subset Z_2 \subset Z_1 = E_1 \]

and $Z_\infty $ and $B_\infty $ (if they exist) as above.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AMI. Beware of the difference between the letter 'O' and the digit '0'.