# Relation between extensions and filtrations

We work in an Abelian category. Consider Yoneda extensions, i.e., the Abelian groups Ext$$^n(C,A)$$ consisting (for $$n \ge 1$$) of equivalence classes of exact sequences starting at $$A$$ and ending at $$C$$ with $$n+1$$ maps in between, with the Baer sum as operation. See this Wikipedia page for reference.

The datum of a length $$1$$ extension, i.e., of an element of Ext$$^1(C,A)$$ (for some $$A$$ and $$C$$), corresponds to the datum of a filtration of length $$2$$, i.e., to the datum of an object $$B$$ and of a subobject $$A \subseteq B$$. Indeed, I am just saying that a short exact sequence $$0 \to A \to B \to C \to 0$$ gives the same information as the filtration $$0 \subseteq A \subseteq B$$, since $$C \cong B/A$$.

My question is if there is any relation between length $$n$$ extensions and length $$n+1$$ filtrations in general.

More specifically, if I have a nontrivial filtration of length $$n+1$$, can I obtain some nontrivial extension of length $$n$$? Or the converse?

This is already not so clear for $$n=2$$ (and we can stick to this case, if it makes things easier). An exact sequence

$$0 \to A \overset{\alpha}{\to} B \overset{\beta}{\to} C \overset{\gamma}{\to} D \to 0$$

is the same thing as the two short exact sequences

$$0 \to A \to B \to \ker(\gamma) \to 0 \quad \text{and} \quad 0 \to \ker(\gamma) \to C \to D \to 0,$$

but this is not the kind of thing that I am looking for, because this gives me two filtrations of length $$2$$ and not a single one of length $$3$$, which is what I would like to obtain somehow.

• In the other direction is it easier: starting from example from a filtration $C\subset B\subset A$ one gets an exact sequence $0\to C\to B\to A/C\to A/B\to 0$. But unfortunately I am not able to say something clever about the corresponding $\mathrm{Ext}^2$-class... Sep 4 '19 at 6:02
• @AurélienDjament Thanks for your comment, that’s the kind of thing I’m looking for :) Sep 4 '19 at 6:08
• @AurélienDjament The $\text{Ext}^2$-class is zero, as your extension has a map of extensions from an extension $0\to C\to C\oplus B\to A\to A/B\to0$ which is the Yoneda product of the split extension $0\to C\to C\oplus B\to B\to0$ with the extension $0\to B\to A\to A/B\to0$. Sep 4 '19 at 21:12
• Rather than looking for an interpretation of elements of $\text{Ext}^n$ as classifying length $n+1$ filttrations, you might have better luck interpreting them as obstructions to constructing length $n+1$ filtrations. Sep 4 '19 at 21:16
• @JeremyRickard This sounds very interesting! Could you please explain a bit more in detail what you mean? Sep 5 '19 at 10:20