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I'm not sure if this counts as research level, since it might just be an expression of my ignorance. But anyway.

Let $\Delta^n_A = Spec(A[X_0, \dots, X_n]/\sum X_i - 1)$ denote the standard algebraic simplex over $A.$ These assemble into the standard cosimplicial affine scheme $\Delta_A^\bullet.$ Hence, given any scheme $X,$ one may define a simplical set $Sing_\bullet(X)(A) = Hom(\Delta^\bullet_A, X)$ and then, taking the associated chain complex, we may define the naive algebraic homology groups as

$H_*(X)(A) = h_* \mathbb{Z}Sing_\bullet(X)(A),$

where we do the usual thing of taking the free simplicial abelian group on $Sing_\bullet$ and alternating sums of face maps as differentials.

For more on the construction, compare: References for the "nerve of an algebraic variety"

As is pointed out in that answer, this particular construction is not in general well-behaved. (The good version is to use morphisms into infinite symmetric products, which then computes singular homology.) However: can we compute these naive groups for any spaces at all? What I have come up with:

  • $H_0(\mathbb{G}_m)(A) = A^\times,$ $H_n(\mathbb{G}_m) = 0$ for $n>1$ because $\mathbb{G}_m$ is A^1-invariant.

  • $H_0(\mathbb{A}^n)(A) = \mathbb{Z}$ and the higher groups vanish - because $\mathbb{A}^n$ are algebraically contractible [I haven't checked this in detail.]

  • $H_0(\mathbb{P}^1)(k) = \mathbb{Z}$ for any field $k$

But these observations are all essentially trivial. Using work of cazanave on naive homotopy classes of endomorphisms of $\mathbb{P}^1,$ one may compute $H_0(\mathbb{P}^1)(k[X]),$ but this is already fairly non-trivial.

Perhaps some concrete questions: Can anyone compute $H_n(\mathbb{P^1})(k)$ or $H_n(PGL_2)(k),$ for some $k$ or $n>0$? Are these groups zero for $n$ sufficiently large?

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  • $\begingroup$ In your remark on the work of Cazanave, you probably mean to compute $H_0(\mathbb{P}^1)(R)$ where $R$ is the coordinate ring of the affine variety $GL_2/(\mathbb{G}_m\times\mathbb{G}_m)$? $\endgroup$ – Matthias Wendt Dec 4 '14 at 17:20
  • $\begingroup$ If I'm not mistaken, we have $Hom(A^1 \times \Delta^n, P^1) = Hom(P^1 \times \Delta^n, P^1)$ and so $H_0(P^1)(k[t])$ computes unpointed naive homotopy classes of endomorphisms. $\endgroup$ – Tom Bachmann Dec 8 '14 at 7:01
  • $\begingroup$ By construction, $H_\ast(X)(A)$ is $\mathbb{A}^1$-invariant. It is possible to write down explicit contractions (if I remember correctly, the paper of Morel and Voevodsky is one place where this is written down). So $H_0(\mathbb{P}^1)(k[T])\cong\mathbb{Z}$ again. The identification of evaluation of the singular resolution over $\mathbb{A}^1$ and $\mathbb{P}^1$ does not work. There is a natural restriction map. This takes the identity of $\mathbb{P}^1$ to the natural inclusion $\mathbb{A}^1\to\mathbb{P}^1$. The former is not nullhomotopic, but the latter is (via the obvious homotopy). $\endgroup$ – Matthias Wendt Dec 8 '14 at 10:10
  • $\begingroup$ Continued: another way to see the problem is translating the claim into classical topology. We can compactify $\mathbb{R}$ by $S^1$, but $Hom(\mathbb{R}\times\Delta^\bullet,S^1)$ is surely not equivalent to $Hom(S^1\times\Delta^\bullet,S^1)$. $\endgroup$ – Matthias Wendt Dec 8 '14 at 10:13
  • $\begingroup$ To elaborate on my first comment: the natural projection $GL_2(\mathbb{G}_m\times\mathbb{G}_m)\to\mathbb{P}^1$ is an $\mathbb{A}^1$-weak equivalence so that homotopy classes of maps from this affine variety to $\mathbb{P}^1$ are homotopy classes of endomorphisms of $\mathbb{P}^1$. $\endgroup$ – Matthias Wendt Dec 8 '14 at 10:17
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There has not been so much research on $\operatorname{Sing}(X)(A)$, and most of it concentrated on its homotopy. Nevertheless, some things are known:

  • Among the first results are those by Jardine in a series of papers at the beginning of the 80s, see the first five entries on his list of publications. You can also check Jardine's thesis. From his results you get that $H_0(G)(k)\cong\mathbb{Z}$ for a group $G$ whose $k$-points are generated by unipotent elements. He also proved $H_1(SL_n)(k)\cong K_2(k)$ for $n\geq 3$ and $H_1(Sp_{2n})(k)\cong K_2(k)$ for $k$ algebraically closed.

  • Very classical results of Rector and Gersten identify the homotopy of $\operatorname{Sing}(GL)(R)$ with algebraic K-theory if $R$ is a regular ring (basically a consequence of homotopy invariance for algebraic K-theory). You can read about this in Chapter 4 of Jardine's thesis.

  • The results of Jardine are now know to hold more generally. For $G$ isotropic reductive and $R$ a regular ring, $H_0(G)(R)\cong\mathbb{Z}[G(R)/E(R)]$ by unstable homotopy for $K_1$-functors, a result of Stavrova. This relates $H_0$ with Whitehead groups. For $G$ isotropic reductive and $k$ a field, $H_1(G)(k)\cong H_2(G(k),\mathbb{Z})$ is proved in my paper with Konrad Voelkel, as a consequence of homotopy invariance in one variable for homology of isotropic reductive groups.

  • The identification $H_1(G)(k)\cong H_2(G(k),\mathbb{Z})$ can also be derived from an $\mathbb{A}^1$-homotopy point of view. This is explained in Morel's "$\mathbb{A}^1$-algebraic topology over a field", Section 6.2 and in this paper.

  • Another result which can be deduced from the results in Morel's book: the spaces $\mathbb{A}^n\setminus 0$ have the affine Brown-Gersten property so that the $\mathbb{A}^1$-$(n-2)$-connectivity implies $H_i(\mathbb{A}^n\setminus 0)(k)=0$ for $1\leq i\leq n-2$ and $H_{n-1}(\mathbb{A}^n\setminus 0)(k)\cong K^{MW}_n(k)$.

  • If we allow $X$ to be more general than a scheme, we can also say something about classifying spaces of algebraic groups. The very classical results mentioned above can be reformulated as follows: homotopy invariance of algebraic K-theory implies that for $R$ a regular ring $H_n(BGL)(R)\cong H_n(BGL(R),\mathbb{Z})$. On the left is the naive algebraic singular homology of the infinite Grassmannian, and on the right is group homology of the infinity general linear group over $R$, viewed as discrete group. It is a subject of current investigations if an isomorphism such as this holds for linear algebraic groups (related to weak homotopy invariance). Weak homotopy invariance and its current status are discussed in my MO-answer here and some results related to computation of $H_3(BSL_2)(k)$ can be found in my paper with Kevin Hutchinson.

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