$\DeclareMathOperator{\op}{\mathrm{op}}\DeclareMathOperator{\Ab}{\mathsf{Ab}}\DeclareMathOperator{\Vect}{\mathsf{Vect}}$**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.

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 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, appendix A, which contains methods of constructing resolutions in $\Ab^{\kappa^{\op}}$, but the resolutions are not of finite length.

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?