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Let $C$ be an abelian category. Suppose that $(N_i)_{i\in I}$ is an inverse system of objects in $C$. Under which conditions does the hypothesis that $$\operatorname{Ext}_C^1(M,N_i)=0\quad\forall i\in I\tag{1}$$ imply $$\operatorname{Ext}_C^1\left(M,\lim\limits_\longleftarrow N_i\right)=0?\tag{2}$$

The functor $\operatorname{Ext}^1_C$ is the Yoneda $\operatorname{Ext}^1$-functor. So we don't have to worry whether $C$ has enough injectives/projectives. However, I would be satisfied if there are some results when $C$ is the category of (unitary) left modules over a ring $R$ with unity.

In the case that $C$ is the category of (unitary) left modules over a ring $R$ with unity, every object in $C$ is an inverse limit of injective modules due to this paper. Therefore, an object $M\in C$ satisfies $$\operatorname{Ext}_C^1\left(M,\lim\limits_\longleftarrow N_i\right)=\lim\limits_\longleftarrow \operatorname{Ext}_C^1(M,N_i)\tag{3}$$ for any inverse system $(N_i)_{i\in I}$ in $C$ iff $\operatorname{Ext}^1_C(M,N)=0$ for all $N\in C$ iff $M$ is projective.

However, I'm not imposing that $(3)$ should be true. I only suppose that $(1)$ is true, and I'd like to know when $(2)$ is also true. Counterexamples in which $(1)$ is true but $(2)$ is false will also be very helpful. Thank you in advance.


An answer to the dual problem below will also be greatly appreciated. If there are cases where $(1')$ is true but $(2')$ is not, I would also like to see examples.

Let $C$ be an abelian category. Suppose that $(N_i)_{i\in I}$ is an directed system of objects in $C$. Under which conditions does the hypothesis that $$\operatorname{Ext}_C^1(N_i,M)=0\quad\forall i\in I\tag{$1'$}$$ imply $$\operatorname{Ext}_C^1\left(\lim\limits_\longrightarrow N_i,M\right)=0?\tag{$2'$}$$

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    $\begingroup$ In the case where $I=\mathbb N$ and over $R-\mathbf{Mod}$, under niceness assumptions on your inverse system, you have the Milnor exact sequence : $0\to \varprojlim^1_n \mathrm{Ext}^2_R(M,N_i) \to \mathrm{Ext}^1_R(M,\varprojlim_i N_i) \to \varprojlim_i \mathrm{Ext}^1_R(M,N_i)\to 0$; e.g. if your system satisfies the Mittag-Leffler condition; so the conditions might involve conditions on your system $(N_i)$, as well as on $\mathrm{Ext}^2_R$ $\endgroup$ Aug 14, 2020 at 8:31
  • $\begingroup$ @MaximeRamzi Thank you. Do you have a reference for this? I have seen something similar in Weibel's book, but it has the assumption $R=\Bbb Z$, and not for a general $R$. $\endgroup$
    – Tanimura
    Aug 14, 2020 at 10:13
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    $\begingroup$ Sorry, actually it should be $\hom$, not $\mathrm{Ext}^2_R$, I messed up between homological and cohomological grading. For a reference, I'm not exactly sure. The point is that under these conditions, you $\varprojlim_i N_i$ is an actual homotopy limit, and so (with everything derived) $Hom(M,\varprojlim_i N_i)\simeq \varprojlim_i Hom(M,N_i)$. Then you can replace this inverse system $(Hom(M,N_i))$ with a good inverse system (which doesn't change its homology) and apply the Milnor sequence as seen here (ncatlab.org/nlab/show/…) $\endgroup$ Aug 14, 2020 at 10:23
  • $\begingroup$ You just have to be careful (unlike I was) about homological vs cohomological grading : If you want $H_q$ (with their notation) to be $\mathrm{Ext}^1$, you will have $H_{q+1}$ being $\hom$ (not $\mathrm{Ext}^2$) $\endgroup$ Aug 14, 2020 at 10:24

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The standard result in this direction is the dual Eklof lemma (for your first problem) or the Eklof lemma (for your dual problem). Any version of the Eklof lemma presumes that your direct/inverse system is indexed by a well-ordered set. For an inverse system, it should be a smooth chain of epimorphisms with the kernels which also satisfy the $\operatorname{Ext}^1_C(M,-)$ vanishing. For a direct system, it should be a smooth chain of monomorphisms with the cokernels which also satisfy the $\operatorname{Ext}^1_C(-,M)$ vanishing. Then the answer to your question is positive.

References:

  1. P.C. Eklof, J. Trlifaj, "How to make Ext vanish", Bull. London Math. Soc. 33, #1, p. 41-51, 2001, https://doi.org/10.1112/blms/33.1.41 . Lemma 1 is the Eklof lemma (for a direct system), Proposition 18 is the dual Eklof lemma (for an inverse system). This is a paper about modules over associative rings.

  2. L. Positselski, J. Rosický, "Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories", Journ. of Algebra 483, p. 83-128, 2017, https://doi.org/10.1016/j.jalgebra.2017.03.029 , https://arxiv.org/abs/1512.08119 . Lemma 4.5 is the Eklof lemma for abelian categories. This is for direct systems, but you can pass to the inverse systems by inverting the arrows.

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