$\DeclareMathOperator{\op}{\mathrm{op}}\DeclareMathOperator{\Ab}{\mathsf{Ab}}\DeclareMathOperator{\Vect}{\mathsf{Vect}}$**Question 1:** What is an example of a sequence $(X\_\alpha)\_{\alpha<\kappa}$ of abelian groups such that $\varprojlim^2\_{\alpha < \kappa} X\_\alpha \neq 0$?


Here $\varprojlim^2\_{\alpha<\kappa}$ is the second derived functor of the limit functor. Necessarily $\kappa$ will be uncountable, and of uncountable cofinality. I suspect it should be possible to give an example where the $X\_\alpha$ are vector spaces and $\kappa = \omega\_1$ is the first uncountable ordinal.




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This question likely sounds funny -- usually one only discusses $\varprojlim^n\_{\alpha<\kappa}$ for $n\geq 2$ in more exotic abelian categories than $\Ab$ or $\Vect$. This is because one usually only deals with the case where $\kappa = \omega$ or at least has cofinality $\omega$, in which case the functor $\varprojlim\_{\alpha < \kappa}^n : \Ab^{\kappa^{\op}} \to \Ab$, i.e. the $n$th derived functor of the limit functor, vanishes for $n \geq 2$. The usual [proof](https://ncatlab.org/nlab/show/lim%5E1+and+Milnor+sequences) uses a very natural 2-step resolution, which only works when $\kappa = \omega$, or by extension when $\kappa$ is of countable cofinality.


But when it comes to *longer* sequences, this resolution is not available. In fact, I've only seen $\varprojlim\_{\alpha<\kappa}^n$ discussed for $\kappa$ of uncountable cofinality in Neeman's [Triangulated Categories](https://hopf.math.purdue.edu/Neeman/triangulatedcats.pdf), appendix A, which contains methods of constructing resolutions in $\Ab^{\kappa^{\op}}$, but the resolutions are not of finite length.




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Another way of saying that $\varprojlim^2\_{n<\omega} = 0$ in abelian groups is that $\varprojlim^1\_{n<\omega}$ is right exact. So a closely related question is:


**Question 2:** What is an example of an epimorphism $(X\_\alpha \to Y\_\alpha)\_{\alpha<\kappa}$ of inverse systems of abelian groups such that the induced map $\varprojlim^1\_{\alpha<\kappa} X\_\alpha \to \varprojlim^1\_{\alpha<\kappa} Y\_\alpha$ is not an epimorphism?


And by the way,


**Question 3:** What is the global dimension of the category $\Ab^{\kappa^{\op}}$ of $\kappa$-indexed inverse systems of abelian groups, for a given regular cardinal $\kappa$? How about $\Vect^{\kappa^{\op}}$, where $\Vect$ is the category of vector spaces over your favorite field?



Example of an uncountable sequence of abelian groups with nonvanishing $\varprojlim^2$?
A great survey on this and some related topics is Osofsky's "[The subscript of $\aleph\_n$, projective dimension, and the vanishing of $\varprojlim^{(n)}$](https://projecteuclid.org/euclid.bams/1183535284)." As far as I am aware, this 1974 paper still describes the state of the art on the matter.


Osofsky (exposing material from Mitchell's book on rings with many objects) gives the following example of a sequence such as you want. Fix any ring $R$ and let $\Delta R$ be the constant functor at $R$ indexed by $\aleph^{\mathrm{op}}\_1$. Then $\Delta R$ can be shown to have homological dimension $2$ via a technical argument involving its nice ordered basis (see section 5 of Osofsky.) Now considering any projective resolution
$$\cdots\to P\_2\to P\_1\to P\_0\to \Delta R\to 0,$$
if $K$ is the kernel of $P\_1\to P\_0$ then the length-2 extension
$$0\to K \to P\_1\to P\_0\to \Delta R\to 0$$
must be nontrivial, or else the kernel $K'$ of $P\_0\to \Delta R$ would be a summand of $P\_1$, making $\Delta R$ of projective dimension at most $1$.


That is, $\mathrm{Ext}^2(\Delta R, K)=\varprojlim^{(2)}\_{\aleph\_1^{\mathrm{op}}} K$ is nontrivial whenever $K$ is the kernel of the map between the first two projectives resolving $\Delta R$, for any ring $R$. The same arguments go through when $\aleph\_1$ is replaced by $\aleph\_n$ to get nontrivial $n$th derived functors of lim. So the answer to your question 3 is "$n+1$, if $\kappa=\aleph\_n$, and $\infty$, if it's larger." These holds, passing to cofinalities, even when $\kappa$ is replaced with an arbitrary directed set, and I think even an arbitrary filtered category.


