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Let $p:X\to S$ be the unique map from a (locally compact) topological space $X$ to a point. Since $\underline{\hom}(\underline{\mathbb{Z}},-)$ is the identity functor, we have that $\Gamma(X,-)=\hom(\underline{\mathbb{Z}},-)$ and so $$H^i(X,-)=\operatorname{Ext}^i(\underline{\mathbb{Z}},-)=\hom_{\mathsf{D}(X)}(\underline{\mathbb{Z}},-[i]).$$ We can then use the adjunction to write this as $H^i(X,-)=\hom_{\mathsf{D}(S)}(\underline{\mathbb{Z}},p_*- [i])$. In particular, we have that $H^i(X,\underline{\mathbb{Z}})=\hom_{\mathsf{D}(S)}(\underline{\mathbb{Z}},p_*p^*\underline{\mathbb{Z}} [i])$. The same kind of reasoning shows that $H^i_c(X,\underline{\mathbb{Z}})=\hom_{\mathsf{D}(S)}(\underline{\mathbb{Z}},p_!p^*\underline{\mathbb{Z}} [i])$.

Now, most references on Verdier duality would call $\hom_{\mathsf{D}(S)}(\underline{\mathbb{Z}},p_*p^!\underline{\mathbb{Z}} [-i])$ the $i$-th homology of $X$. I get that this is defined precisely in a way that recovers Poincaré duality in its general form. Is there more to it? Why does this deserve to be called a homology group?

LSpice
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Gabriel
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    You might enjoy the related discussion around David Hansen's blog post https://totallydisconnected.wordpress.com/2021/03/08/remarks-on-fargues-scholze-part-2/ – David Ben-Zvi Sep 30 '21 at 16:48
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    .. as well as here: https://mathoverflow.net/questions/277069/what-is-homology-anyway – David Ben-Zvi Sep 30 '21 at 16:59
  • Although it might be obvious, the comments by Scholze and Clausen are also valuable in Hensen's blog post that @DavidBen-Zvi mentioned. – Z. M Sep 30 '21 at 17:00
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    If I understand well, it all boils down to "this is the dual of cohomology in some sense, so it deserves to be called homology". But this is only somewhat true in algebraic topology; it only holds over a field and if homology is finite dimensional. Also, in the case of coefficients in a field, we have less interest in homology (in this context), since we have a duality with compactly supported cohomology. (I can explain this better if anyone wants.) – Gabriel Oct 02 '21 at 09:59
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    @Gabriel For the record, the restriction to fields is unnecessary, since we are working in a derived setting (that is $C^(X)$ is* the dual of $C_(X)$ in the derived category, so the only obstruction is whether $C_(X)$ is perfect (and so in particular equal to its double dual), which is true for example for every finite CW complex – Denis Nardin Oct 03 '21 at 07:20
  • @DenisNardin cool! There's, however, another problem in the context of étale cohomology, I think. In order for the dual on the derived category $\hom_{\mathsf{D}(S)}(-,\mathscr{O}_S)$ to correspond to the dual in a category of modules, $S$ has to be a point. (Since, in this case, the category of sheaves over $S$ is equivalent to the category of modules over $\Gamma(S,\mathscr{O}_S)$.) Of course, there are more general rings $A$ such that $\operatorname{Spec}A$ has only one point, but this restricts the possible base rings a lot. – Gabriel Oct 03 '21 at 09:37
  • @Z.M, re, I see a comment by Scholze. Where is the comment by Clausen? – LSpice Oct 03 '21 at 18:34
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    @LSpice Dustin is Dustin Clausen. – Z. M Oct 03 '21 at 19:54

1 Answers1

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One nice geometric way to view homology is a measurement of your space $X$ given by probing $X$ with other, nicer spaces, for instance, singular homology probes with simplices. One good reason to call this abstract thing homology is that it too admits elements corresponding to nice enough maps from test objects into $X$.

Lets work in a general six functor formalism, but I'm secretly thinking of constructible sheaves on nice spaces. Then we have our unit object $\mathbf{1}$ ($\underline{\mathbb{Z}}$ in our case) and a dualising object $p^!\mathbf{1}:=\omega$, in our case, this is the usual dualising sheaf $p^!\underline{\mathbb{Z}}$.

Then we can define a "smooth" object $Z$ to be one which admits an isomorphism $\gamma:\mathbf{1}_Z\rightarrow \omega_Z[-d_Z]$, for some integer $d_Z$. In the constructible/sheafy setting, manifolds give plenty of examples of "smooth" objects, and $d_Z$ is the dimension.

Then if we have $Z$ a closed subset of $X$, or more generally, if $i_*\cong i_!$ for $i:Z\rightarrow X$, then we have the following map, where all arrows are coming from adjunctions in our formalism, except the second one which says $Z$ is "smooth".

$$\mathbf{1}_X\rightarrow i_*\mathbf{1_Z}\xrightarrow{i_*\gamma} i_*\omega_Z[-d_Z]\xrightarrow{\cong}i_!\omega_Z[-d_Z]\rightarrow\omega_X[-d_Z]$$

So the formalism gives us elements of this group for each "smooth" closed subset $Z$ in $X$, so in my mind, this deserves the label of homology, it admits a direct link to the geometry of nice subsets of your space!

Chris H
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