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"](https://mathoverflow.net/questions/166129/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?



Computing naive algebraic singular homology
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.](http://www.math.uwo.ca/~jardine/cv/jardine-pub.html) You can also check [Jardine's thesis.](http://www.math.uwo.ca/~jardine/papers/preprints/Jardine-thesis.pdf) 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.](http://arxiv.org/abs/1111.4664) 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,](http://arxiv.org/pdf/1207.2364v1.pdf) 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](http://www.mathematik.uni-muenchen.de/~morel/Prepublications/A1TopologyLNM.pdf) and [in this paper.](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=7606992)
* 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](https://mathoverflow.net/questions/163250/) and some results related to computation of $H\_3(BSL\_2)(k)$ can be found [in my paper with Kevin Hutchinson](http://arxiv.org/abs/1307.3069).


