# Spectral sequence of a filtration: a possible mistake

$$\require{AMScd}$$The following is taken from these notes by Daniel Murfet.

Let $$\cdots \subseteq F^{p + 1}(C) \subseteq F^p(C) \subseteq F^{p - 1}(C) \subseteq \cdots$$ be a filtration of a complex $$C$$ in an abelian category. There is either a mistake or I don't understand something. I think that $$\ddot{A^{pq}_r} \subseteq \ddot{A^{pq}_{r + 1}}$$. Indeed, $$A^{pq}_r$$ defined by the following pullback

$$\begin{CD} A^{pq}_r @>>> F^p(C^{p + q}) \\ @VVV @VVV \\ F^{p + r}(C^{p + q + 1}) @>>> F^p(C^{p + q + 1}) \end{CD}$$

where the bottom morphism is a subobject inclusion and the left morphism is a differential of $$F^p(C)$$. Concretely, $$A^{pq}_r$$ is a pullback of $$d^{p,p+q}$$ along the subobject inclusion $$F^{p + r}(C^{p + q + 1}) \subseteq F^p(C^{p + q + 1})$$. Then $$\ddot{A^{pq}_r}$$ is the image of the composition in the following diagram

$$\begin{CD} A^{p - r + 1, q + r - 2}_{r - 1} @>>> F^{p - r + 1}(C^{p + q - 1}) \\ @VVV @VVV \\ F^p(C^{p + q}) @>>> F^{p - r + 1}(C^{p + q}) \end{CD}$$

and $$\ddot{A^{pq}_{r + 1}}$$ is the image of the composition in the following diagram

$$\begin{CD} A^{p - r, q + r - 1}_r @>>> F^{p - r}(C^{p + q - 1}) \\ @VVV @VVV \\ F^p(C^{p + q}) @>>> F^{p - r}(C^{p + q}) \end{CD}$$

To have a map from from one image to another, we need a map between their domains and codomains. But the pullback universal property only gives a morphism from $$A^{p - r + 1, q + r - 2}$$ to $$A^{p - r, q + r - 1}$$. Similarly, for the following screenshot I only see how to construct a map from $$A^{p + r, q - r + 1} \to A^{pq}_r$$, for similar reasons.

So, my question is: is there is a mistake? If yes, can the proof be salvaged? If not, what am I missing?

• What follows "I think that" is unclear because of a probable typo (more generally, your "dot dot" on top of $A$ is not always well written). Are you saying you think there is a mistake in the order of inclusion of the "A dot dot"'s ? May 10 '20 at 7:57
• @MaximeRamzi Exactly. In a similar vein, I think a domain and a codomain of a morphism $A^{pq}_r \to A^{p + r, q - r + 1}_r$ Murfet defines are reversed. May 10 '20 at 8:07
• I haven't read all of it yet, but have you tried forgetting you're in a general abelian category, pretending you're in $R-\mathbf{Mod}$ and reading all images, intersections, pre-images etc. as the usual set-theoretic constructions and seeing what happens ? If you do this, you should either be able to confirm your doubts, or see where you're mistaken, and what's actually happening. Then you can translate back to general abelian categories May 10 '20 at 8:20
• @MaximeRamzi Yeah, I've tried and I think this confirms my suspicion, though I may be making a similar mistake in the category of modules. Besides, does this mean the proof is completely wrong, or it can be salvaged? I don't know. May 10 '20 at 18:14

We have $$\ddot{A^{p,q}_r}=\partial \newcommand\of{\left({#1}\right)} \of{ F^{p-r+1}C^{p+q-1}\cap \partial^{-1} \of{ F^{p+1}C^{p+q} } }.$$ Note that $$\partial^{-1}(F^{p+1}C^{p+q})$$ is independent of $$r$$, and as $$r$$ increases, the filtration index decreases, and therefore the groups get larger, so you appear to be correct that the order of inclusions should be $$\ddot{A}^{p,q}_r\subseteq \ddot{A}^{p,q}_{r+1}$$.
This doesn't really affect the proof in any way (at least the visible part), because the excerpted discussion doesn't use the ordering at all. However, I will say that this ordering should be the correct one, since we want $$Z^{p,q}_{r+1}\subseteq Z^{p,q}_r$$ and $$B^{p,q}_{r+1}\supseteq B^{p,q}_r$$ so that $$E_{r+1}^{p,q}$$ is a subquotient of $$E_r^{p,q}$$.
$$r$$-cocycles should get smaller and $$r$$-coboundaries should get larger so that we have a sensible notion of convergence.
• Thanks for you answer. But what do you think about morphism $A^{pq}_r \to A^{p + r, q - r + 1}$? Does the order of the domain and the codomain reversed too? But one needs this morphism to construct a morphism $d^{pq}_r\colon E^{pq}_r \to E^{p + r, q - r + 1}_r$ (a differential of a spectral sequence) May 10 '20 at 19:47