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?

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?