References for the "nerve of an algebraic variety"

Question

Let's do algebraic geometry over an arbitrary base ring $k$.

I've frequently seen a definition of the algebraic $n$-simplex, as follows: $$\Delta^n = \operatorname{Spec}\left(k[x_0,\ldots,x_n]/(\Sigma x_i-1)\right).$$ This is just a copy of $\mathbb{A}^n$, conveniently endowed with the barycentric coordinate system. Of course, the topological $n$-simplex, whose definition this imitates, is the subset of points where $x_i\geq 0$ for all $i$. As $n$ varies, the algebraic $n$-simplices fit together to form a cosimplicial algebraic variety, in exactly the same way as the topological $n$-simplices do.

Here and there, I've even seen people describe what possibly ought to be called the nerve of a variety $X$: the simplicial set $N(X)$ whose set of $n$-simplices $N(X)_n$ is the set of maps of varieties $\Delta^n\rightarrow X$. I've even heard it suggested that, as one might hope, the topology of this simplicial set is relevant to the study of the usual cohomological invariants of the variety.

Can anyone suggest a reasonably comprehensive reference which makes that relevance precise? Is there even a universally accepted name for the construction?

Answer 1

The simplicial set itself does not give much (most varieties don't have very many maps from affine spaces), but Suslin introduced something along these lines, using maps from algebraic simplices to symmetric powers of $X$ as algebraic-geometry versions of singular chains. Here is a paper by Suslin and Voevodsky: http://www.math.uiuc.edu/K-theory/0032/

Answer 2

One way to understand your question is in the framework of $\mathbf{A}^1$-homotopy theory. This is because your nerve functor is better understood when defined on a cocomplete category like the category of motivic spaces $\mathrm{Spc}(k)$. The latter is roughly speaking the free $\infty$-cocompletion of the category $\mathrm{Sm}(k)$ of smooth $k$-schemes, localized with respect to Nisnevich covers and $\mathbf{A}^1$-equivalences. (So its objects are $\infty$-presheaves, and it is presented by a model structure on the category of simplicial presheaves on $\mathrm{Sm}(k)$.)

The nerve functor $N : \mathrm{Spc}(k) \to \mathrm{sSet}$ is just the restriction of scalars functor induced from $\Delta^\bullet_k : \mathbf{\Delta} \to \mathrm{Spc}(k)$. Since $\mathrm{Spc}(k)$ is cocomplete, by abstract nonsense (as tetrapharmakon mentioned in his answer) this admits a left adjoint which is the (motivic) geometric realization functor $|-|_k : \mathrm{sSet} \to \mathrm{Spc}(k)$. Certainly these constructions have been studied in this setting, in various papers of Voevodsky for example, though I am not sure if he ever used the term "nerve".

There is another functor $\mathrm{Sing}_* : \mathrm{Spc}_k \to \mathrm{Spc}_k$ which is given by the formula $$ \mathrm{Sing}_n(X)(U) = \mathrm{Hom}_{\mathrm{Spc}(k)}(U \times \Delta^n_k, X) $$ for a smooth scheme $U$. This seems to be a more refined version of the construction of Suslin-Voevodsky referenced by Tom Goodwillie, and I believe that these simplicial sets are more directly analogous to the nerve or singular simplicial complex in topology than the above nerve $N$. When one takes for $X$ the motivic Eilenberg-Mac Lane spaces, the induced functor $$\mathrm{Sing}_*(K(\mathbf{Z}(n), 2n)) : \mathrm{Sm}(k)^{\mathrm{op}} \to \mathrm{sSet} $$ is particularly interesting. For a "good" smooth scheme $U$, the simplicial set it gives has its homotopy groups identified with the motivic cohomology of $U$. So certainly the topology of this simplicial set is relevant to the study of cohomological invariants.

Surely there are many more interesting things to be said, but I am not an expert so I will refer to papers of Voevodsky. His ICM talk may be a good place to start.

Edit: The fact that $\mathrm{Sing}_*$ should be viewed as a refined version of the nerve $N : \mathrm{sSet} \to \mathrm{Spc}(k)$ can be made more precise by the following observation: $\mathrm{Spc}(k)$ is presented by a simplicial model category, and hence is tensored and powered over $\mathrm{sSet}$. This means in particular that there are adjoint pairs $(- \otimes \Delta^n, \underline{\mathrm{Hom}}(\Delta^n, -))$ for each $n$. For $X \in \mathrm{Spc}(k)$ each $\underline{\mathrm{Hom}}(\Delta^n, X)$ is a actually a presheaf (and not a set, like $N(X)_n = \mathrm{Hom}(\Delta^n, X)$). And the point is that $\mathrm{Sing}_n$ is by definition precisely the power functor $\underline{\mathrm{Hom}}(\Delta^n, -)$.

Edit 2: Since you asked for a reference, I just noticed that there is a good discussion about the functor $\mathrm{Sing}_*$ in section 2.3.2 of

• Fabien Morel, Vladimir Voevodsky, A^1-homotopy theory of schemes, 1998, web.

Answer 3

There is, indeed, something general under the carpet. I would have preferred a comment but I needed some more space.

1. Whenever you are able to to produce a functor $f:\Delta\to Something$ and "something" is a cocomplete category, then the functor you define ($X$ goes to the simplicial set $Map(\Delta[\bullet], X)$) has a left adjoint, obtained as the left Yoneda extension of your original $f$.

2. This mimicks (and in fact generalizes) the classical topological realization of a simplicial set, starting from how this should be defined on simplices, and then extending by cocontinuity, using that every presheaf is a colimit of representables

3. This has plenty of declinations in algebraic topology and category theory, so I'm not surprised it can be adapted to algebro-geometric settings! See in particular these discussions.