Suppose I work over $\mathbb{C}$. Is it known for which locally convex topological vector spaces $V$, we have $\text{Ext}^i(V, \mathbb{C})=\{0\}$ for all $i>0$, working with the type of Ext groups that Wengenroth uses. For instance, is it true when $V$ is Frechet and nuclear?
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The Hahn-Banach theorem implies that $\mathbb C$ is an injective object in the category of complex locally convex topological vector spaces. Therefore, Ext$^i(V,\mathbb C)=0$ for every locally convex space $V$.
answered Jul 20, 2023 at 10:15
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$\begingroup$ I couldn't remember what "the type of Ext groups that Wengenroth uses" are, I am slightly disappointed that they seem to be what I would have naively assumed ;-) $\endgroup$ Commented Jul 20, 2023 at 18:29
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1$\begingroup$ I am sorry to disappoint you. The Ext groups were in fact already considered by Palamodov who was the first to use derived functors in the locally convex theory. $\endgroup$ Commented Jul 20, 2023 at 19:44
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$\begingroup$ Indeed, back in the days when I was trying to study Banach versions, I had looked at some of the general properties of categories which Palamodov calls semi-abelian, but only for the Banach-space categories mathoverflow.net/questions/272537 $\endgroup$ Commented Jul 20, 2023 at 21:11
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$\begingroup$ I was aware that Palamodov defined these groups before Wengenroth! I just didn't know how to refer to them. $\endgroup$ Commented Jul 21, 2023 at 11:24
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$\begingroup$ Sorry, I don't understand your last comment. $\endgroup$ Commented Jul 21, 2023 at 14:22