Let $k$ be a non-Archimedean field. Does there exist a spherical completion $K$ of $k$ such that for any $k$-Banach space $X$, the natural map $X \to X \widehat{\otimes}K$ is an isometric embedding? If so I might also ask if it is possible that an exact admissible sequence $X \to Y \to Z$ of bounded morphisms of $k$-Banach spaces leads to an exact admissible sequence $X \widehat{\otimes}K \to Y\widehat{\otimes}K \to Z\widehat{\otimes}K$ (or even conversely also)? Note: if we only demanded that $K$ is non-trivially valued then such a fact appears in Berkovich's book.
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$\begingroup$ Just to be sure: you do not assume that the field $k$ is complete, do you? $\endgroup$– Jérôme PoineauCommented Oct 7, 2013 at 8:32
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$\begingroup$ Thinking a little further, I do not see how you could use anything from Berkovich's book if your base field is not complete. Then your statements would be classical I believe. Could you clarify this point please? (And maybe also provide a more precise reference to Berkovich.) $\endgroup$– Jérôme PoineauCommented Oct 7, 2013 at 8:42
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$\begingroup$ Dear Jerome, Yes, my base field is complete. I forgot to mention that. $\endgroup$– Oren Ben-BassatCommented Oct 9, 2013 at 18:26
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$\begingroup$ I was talking about Proposition 2.1.2 from Berkovich's book. He shows that you can pass to a situation where the field is non-trivially valued. I am looking for a similar way to pass to the situation where the field is spherically complete. $\endgroup$– Oren Ben-BassatCommented Oct 9, 2013 at 18:29
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$\begingroup$ Dear Oren, OK, thanks. I have other questions then :) I have to say I am a little puzzled by what you asked, since I thought that what you want holds for arbitrary extensions $K/k$. For example, the second part should follow from Gruson's article "Théorie de Fredholm $p$-adique" (numdam.org/numdam-bin/fitem?id=BSMF_1966__94__67_0). Am I missing something here? Do you have examples of extensions where you statements fail? $\endgroup$– Jérôme PoineauCommented Oct 10, 2013 at 6:44
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1 Answer
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Let me try to give an answer. In fact, I think that the statements you want always hold, i.e. for any complete valued field extension $K/k$. Which makes wonder whether I have missed something obvious... Anyway, writing things up clearly should help us start a discussion, so I will give it a try.
As regards the first question about the isometric embedding $X \to X\hat{\otimes}_k K$, it can be deduced from lemma 3.1 (with $A=k$, $B=K$ and $C=X$) from my paper "Les espaces de Berkovich sont angéliques" in Bulletin de la SMF 141 (2), p. 267-297 (2013) (http://smf4.emath.fr/Publications/Bulletin/141/html/smf_bull_141_267-297.php). It eventually relies on arguments from BGR about the existence of $\alpha$-cartesian bases for finite-dimensional spaces over complete valued fields.
As I said in the comments above, the second question about the admissible exact sequence is dealt with by Gruson in "Théorie de Fredholm p-adique" Bulletin de la SMF 94, p. 67-95 (1966) (www.numdam.org/numdam-bin/fitem?id=BSMF_1966_94_67_0).
answered Oct 14, 2013 at 14:00
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$\begingroup$ Dear Jerome, thanks very much. I still will probably want to speak about it with you more, but because it will be slow, I approved this answer in the meantime! $\endgroup$ Commented Oct 14, 2013 at 16:18