Homology of the étale homotopy type

$$\DeclareMathOperator\Et{Et}$$Let $$X$$ be a scheme and denote by $$\Et(X)$$ the associated étale homotopy type. Then by the work of Artin–Mazur, we know that for an abelian group $$A$$, we have $$H^n(\Et(X),A)=H^n_{\text{ét}}(X,A)$$ and
$$\pi^1(\Et(X))\cong \pi^1_{\text{alg}}(X).$$ Therefore I wonder what $$H_n(\Et(X),A)$$ is? Naïvely I would hope that it is $$H_{n,\text{ét}}(X,A)$$, however, it seems that no survey or reference mentions this.

• What's your definition of étale homology? Aug 3, 2020 at 8:23
• In the case that $A=\mathbb{Q}_p$ (which is the case I'm most interested in), I know it as the dual of the étale cohomology. Aug 3, 2020 at 17:33
• In that case, doesn't the theorem basically follow from the theorem for cohomology by taking duals? Aug 4, 2020 at 5:10
• It does absolutely, but I'm wondering if this agrees with more general coefficients. The way I've heard about constucting the étale homology is to consider étale cosheaves and their derived functor. Aug 4, 2020 at 13:59

Work on the big pro-etale site on all schemes, which maps to the pro-etale site of a point, $$\pi: \mathrm{Sch}_{\mathrm{proet}}\to \ast_{\mathrm{proet}}$$. Also, let's consider (hypercomplete) sheaves of anima. Then $$\pi^\ast$$ has a left adjoint $$\pi_\natural$$, and (for our given scheme $$X$$) $$\pi_\natural(X)$$ is a condensed anima that "is" the etale homotopy type of $$X$$. Concretely, if $$X_\bullet\to X$$ is a proetale hypercover by w-contractible $$X_\bullet$$, then $$\pi_\natural(X)$$ is represented by the simplicial extremally disconnected profinite set $$\pi_0(X_\bullet)$$. By adjunction, there is a natural map $$X\to \pi^\ast \pi_\natural (X)$$.

Now giving a $$\mathbb Q_\ell$$-local system $$\mathbb L$$ (same for other coefficient rings) is the same as giving a map $$X\to \pi^\ast B\mathrm{GL}_n(\mathbb Q_\ell)$$, i.e. equivalently a map $$\pi_\natural(X)\to B\mathrm{GL}_n(\mathbb Q_\ell)$$, i.e. a $$\mathbb Q_\ell$$-local system on the condensed anima $$\pi_\natural(X)$$.

Now what is homology of $$X$$ with coefficients in $$\mathbb L$$? One definition uses the formalism of solid $$\mathbb Q_\ell$$-sheaves $$D_\blacksquare(X,\mathbb Q_\ell)$$ (no reference for them in the case of schemes yet, sorry!). In that setting, pullback $$D_\blacksquare(\ast,\mathbb Q_\ell)\to D_\blacksquare(X,\mathbb Q_\ell)$$ has a left adjoint, which takes $$\mathbb L$$ to the homology of $$X$$ with coefficients in $$\mathbb L$$. Here $$D_\blacksquare(\ast,\mathbb Q_\ell)$$ is the "usual" derived category of solid $$\mathbb Q_\ell$$-modules.

Working with the condensed anima $$\pi_\natural(X)$$ instead, one can apply a similar procedure, and the two answers will agree, so indeed the homology of $$X$$ with coefficients in $$\mathbb L$$ agrees with the homology of $$\pi_\natural(X)$$ with coefficients in $$\mathbb L$$.

In practice, if $$X$$ is sufficiently nice, then these homology groups will be finite-dimensional over $$\mathbb Q_\ell$$ and dual to cohomology, so this would indeed follow from the usual statements about cohomology. For general $$X$$, this statement about homology is however slightly finer (as cohomology is always the dual of homology, but not conversely).

Edit: Here is a way to phrase the answer so that it does not involve a reference to $$D_\blacksquare(X,\mathbb Q_\ell)$$. One can instead use the full derived category of pro-etale $$\mathbb Q_\ell$$-sheaves $$D(X_{\mathrm{proet}},\mathbb Q_\ell)$$; then, just like for sheaves of anima, the pullback along $$f: X_{\mathrm{proet}}\to \ast_{\mathrm{proet}}$$ has a left adjoint $$f_\natural$$, and $$f_\natural \mathbb L\in D(\ast_{\mathrm{proet}},\mathbb Q_\ell)$$ is a complex of condensed $$\mathbb Q_\ell$$-vector spaces that can be considered as the homology of $$\mathbb L$$. Again, a similar construction can be done for $$\pi_\natural(X)$$, and these two notions of homology agree.

On the other hand, I expect that it is extremely difficult to compute this notion of homology even for $$X=\mathbb P^1_k$$ for $$k$$ an algebraically closed field. However, passing to the solidification, one can compute it in practice, and I guess it usually agrees with the homology of the Artin--Mazur pro-(homotopy type).

• I do like very much everything you do on analytic rings and condensed mathematics, but I have trouble to see why solid modules are necessary here. I mean that the derived category of pro-étale l-adic sheaves, as documented in your joint paper with Bhatt, is sufficient to make sense of the left adjoint of the pullback functor: we can simply apply the $l$-adic version of $\pi_\sharp$ to $\mathbb{L}$ to get homology with coefficients. Could you explain what to expect from the consideration of solid modules everywhere? Mar 17, 2021 at 12:30
• As long as $X$ is nice, you are right, the left adjoint exists on the usual level, but not for general $X$. If $X=\mathrm{Spec} \mathbb Q$ for example, $H^1(X,\mathbb F_2)$ is infinite, and dually $H_1(X,\mathbb F_2)$ ought to be an infinite product of $\mathbb F_2$'s. So you need some solid formalism, I think. Mar 17, 2021 at 12:35
• On the other hand, I wouldn't be completely sure how to define homology of the pro-etale homotopy type with coefficients. I guess usually one would use coefficients that come from stage in the pro-limit, and then take the projective limit of the homologies? I think this implicit passage to the projective limit is what's mirrored here by the use of solid modules, specifically $\mathbb Z[S]^\blacksquare = \varprojlim_i \mathbb Z[S_i]$ for profinite $S=\varprojlim_i S_i$. Mar 17, 2021 at 12:37
• I should maybe mention that one advantage of using the condensed anima $\pi_\natural(X)$ in place of the Artin--Mazur pro-homotopy type is that $\pi_\natural(X)$ can see general $\mathbb Q_\ell$-local systems (as in the argument above), which I think is not the case for the latter (whose $\pi_1$ is given by the SGA3 $\pi_1$ which is too small for say nodal curves of positive genus). But then to define the homology of a condensed anima, I think I really need the solid formalism. Mar 17, 2021 at 12:47
• Right -- this is the left adjoint I was referring to in the beginning of my comments. But if you take the one on general pro-etale $\mathbb Q_\ell$ (or even $\mathbb F_\ell$)-sheaves, it gets more complicated. So you have three options: Restrict to nice $X$ and nice coefficients, and get a good left adjoint; take general $X$ and solid sheaves, and get a good left adjoint; take general $X$ and all pro-etale sheaves, and get a weird left adjoint. Mar 17, 2021 at 13:36