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I would like to know what is the "correct" algebraic $K$-theory "with proper support". I suppose that the answer should be found in the condensed world, which is mostly inspired the existence of six functors. See Lectures on Condensed Mathematics, Appendix to Lecture VIII.

$\DeclareMathOperator\Tor{Tor}\DeclareMathOperator\Perf{Perf}\DeclareMathOperator\Spec{Spec}$More precisely, given a map $f\colon R\to S$ of finitely generated rings, we have the functor $f_!\colon D(S_\blacksquare)\to D(R_\blacksquare)$ of "direct image with proper support" which preserves small colimits. Suppose that $f$ is of finite $\Tor$-amplitude, then $f_!$ induces a functor $\Perf(S_\blacksquare)\to\Perf(R_\blacksquare)$ between compact objects in $D(S_\blacksquare)$ and $D(R_\blacksquare)$ respectively. The ordinary $K$-theoretic machinery leads to a map $K(\Perf(S_\blacksquare))\to K(\Perf(R_\blacksquare))$ which looks like some kind of "integral along fibers" of $\Spec S\to\Spec R$ for compactly supported cohomologies. However, this does not seem to be satisfactory, since both sides are discrete.

One way to get a condensed structure might be the following: $\Perf(R_\blacksquare)$ (resp. $\Perf(S_\blacksquare)$) should be a "condensed stable symmetric monoidal $\infty$-category". There might be set-theretic issues, but let's ignore them for a moment. Take the maximal groupoid, we get a map of condensed $E_\infty$-monoids and taking the group completion, we get a map of condensed spectra.

Edit: I was mistaken, we should pass to the Waldhausen $S_\bullet$ construction for this, but seemingly this could also be equipped with a condensed structure.

It is not immediately clear whether these spectra are solid. Is there alternative characterizations of these condensed spectra, hopefully without reference to "condensed $\infty$-categories"?

On the other hand, there is a concept of condensed $K$-theory in Lectures on Analytic Geometry, Prop 10.6. However, this does not seem to coincide with the construction above.

Update: As Denis-Charles Cisinski pointed out, the approach sketched above does not work since the corresponding $K$-theory is trivial by Eilenberg's Swindle applied to the infinite product). It is not immediate for me what is a replacement, in view of the compactly supported topological $K$-theory, for example.

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    $\begingroup$ I do not think perfect complexes should be defined as compact objects: in the solid world there are many compact objects. For instance, in the solid derived category of condensed abelian groups, the subcategory of compact objects is (the opposite of) the bounded derived category of abelian groups, the $K$-theory of which is trivial (through Eilenberg's swindle); I do not think that taking the condensed version of Waldhausen construction would solve that (the solidification of zero is null). $\endgroup$ Apr 10, 2021 at 19:55

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Actually, the possibility of defining such a thing was one of my motivations for studying this condensed mathematics in the first place. That said, the story is far from complete.

First, I guess a reasonable definition is the following. For $R\rightarrow S$ a map of finitely generated commutative rings, recall that there is an idempotent commutative algebra object $S_{\infty/R}$ in derived solid $(S,R)$-modules such that modules over $S_{\infty/R}$ are exactly those modules which die on localization to solid $S$-modules. For example, if $S=R[t]$ then $S_{\infty/R}=R((t^{-1}))$, but in general $S_{\infty/R}$ lives in several cohomological degrees. Now the K-theory wth proper support should be the homotopy fiber of

$$K(S)\rightarrow K(S_{\infty/R}).$$

However, this leaves open the question of how $K(S_{\infty/R})$ should be defined. one could do it naively by just taking perfect complexes, but this leads to a messy theory. Peter Scholze and I realized that the correct way to define such K-theory is instead to take the continuous K-theory, in the sense of Efimov, of a certain dualizable category of modules which contains, but is bigger than, the ind-category of perfect complexes. This is the category of nuclear modules briefly described in Analytic.pdf.

This probably defines what is the "correct" object. That said, I view this actual definition as unsatisfactory, because I would like a natural dualizable category of modules whose continuous K-theory directly gives the K-theory with proper supports. I don't know how to do this; the best I've managed so far is one that gives the suspension.

Another thing I'd like to mention is that the reason I was interested in this K-theory with proper supports is that it shows up in the approach I worked out to Artin maps, https://arxiv.org/abs/1703.07842 . If $R$ is a finite type $\mathbb{Z}$-algebra, the source of the Artin map as I define it is the K-theory of the category of $R$-modules in $D^b(LCA)$, the bounded derived category of locally compact abelian groups. If $R$ is a regular $\mathbb{F}_p$-algebra, I conjecture that this K-theory is equivalent to the suspension of the compactly supported K-theory of $R$ over $\mathbb{Z}$ in the above sense, if you interpret $K(S_{\infty/R})$ in the naive way with perfect complexes. By the way, the hypotheses on $R$ are not crucial: one can remove the regularity assumption by working with G-theory, and the $\mathbb{F}_p$-algebra assumption by adding in the appropriate archimedean part at $\infty$ as well. When $R$ is one-dimensional I believe these conjectures have been proved (though with slightly different definitions) by Braunling, https://arxiv.org/abs/1710.10819 .

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  • $\begingroup$ The continuous K-theory in question is discrete and so is the proposed K-theory? I don't know the ring of functions near the boundary in general, but does that matter, in view of the Morita-invariance? By the way, I don't know whether the compactly supported topological K-theory is the K-theory of some category. $\endgroup$
    – Z. M
    Apr 10, 2021 at 22:17
  • $\begingroup$ Right, this K-theory is disrete, but you can promote it to a condensed spectrum by replacing the rings with internal hom from an extr. disc. profinite T to them to define the T-valued points of the condensed K-theory spectrum. About the Morita-invariance question, I don't follow waht you're asking. About compactly supported topological K-theory being the K-theory of a category, it's also not clear to me, but I think it should be. $\endgroup$ Apr 12, 2021 at 10:30
  • $\begingroup$ I am a bit confused by the direction of the map you depicted. If I am not mistaken, the natural map is $S\to S_\infty$, and then I guess that the map should look like $K(S)\to K(S_\infty)$? Maybe I am mistaken, but $D(S_\infty)$ looks like a reflective subcategory of $D((S,\mathbb Z)_\blacksquare)$? $\endgroup$
    – Z. M
    Apr 12, 2021 at 10:43
  • $\begingroup$ Yes, the map is $K(S) \rightarrow K(S_\infty)$, exactly as you say. Where is the confusion? $\endgroup$ Apr 12, 2021 at 10:47
  • $\begingroup$ It's a map of ring objects in solid $\mathbb{Z}$-modules, and everything can be done in that context. $R=\mathbb{Z}$ here. $\endgroup$ Apr 12, 2021 at 10:49

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