nLab étale (infinity,1)-site

Idea

The étale (∞,1)-site is an (∞,1)-site whose (∞,1)-topos encodes the derived geometry version of the geometry encoded by the topos over the étale site.

Its underlying (∞,1)-category is the opposite (∞,1)-category $sCAlg_k^{op}$ of commutative simplicial algebras over a commutative ring $k$, whose covering families are essentially those which under decategorification become coverings in the étale site of ordinary $k$-algebras.

Definition

Let $k$ be a commutative ring. Let $T$ be the Lawvere theory of commutative associative algebras over $k$.

Definition

Let

$$\infty CAlg_k := T Alg_\infty$$

be the (∞,1)-category of ∞-algebras over $T$ regarded as an (∞,1)-algebraic theory.

Proposition

Let $sCAlg_k = (T Alg)^{\Delta^{op}}$ be the sSet-enriched category of simplicial commutative associative k-algebras equipped with the standard model structure on simplicial T-algebras. Write $sCAlg_k^\circ$ for the $(\infty,1)$-category presentable (∞,1)-category. Then we have an equivalence of (∞,1)-categories

$$\infty CAlg_k \simeq (sCAlg_k)^\circ \,.$$

This is a special case of the general statement discussed at (∞,1)-algebraic theory. See also (Lurie, remark 4.1.2).

Notation

For $X \in \infty CAlg_k^{op}$ we write $\mathcal{O}(X)$ for the corresponding object in $\infty CAlg_k$ and conversely for $A \in \infty CAlg_k$ we write $Spec A$ for the corresponding object in $\infty CAlg_k^{op}$.

So $Spec \mathcal{O} X = X$ and $\mathcal{O} Spec A = A$, by definition of notation.

Notice from the discussion at model structure on simplicial algebras the homotopy group functor

$$\pi_* : sCAlg_k \to CAlg_k \,.$$

Definition

A morphism $Spec A \to Spec B$ in $\infty CAlg_k^{op}$ is an étale morphism if

1. The underlying morphism $Spec \pi_0(A) \to Spec \pi_0(B)$ is an étale morphism of schemes;

2. for each $i \in \mathbb{N}$ the canonical morphism

   $$\pi_i(A) \otimes_{\pi_0(A)} \pi_0(B) \to \pi_i(B)$$

   is an isomorphism.

Definition

The étale $(\infty,1)$-site is the (∞,1)-site whose underlying $(\infty,1)$-category is the opposite (∞,1)-category $\infty CAlg_k^{op}$ and whose covering famlies $\{Spec A_i \to Spec B\}_{i \in I}$ are those collections of morphisms such that

1. every $Spec A_i \to Spec B$ is an étale morphism

2. there is a finite subset $J \subset I$ such that the underlying decategorified family $\{Spec \pi_0(A_j) \to Spec \pi_0(B)\}_{j \in J}$ is a covering family in the 1-étale site.

This appears as (ToënVezzosi, def. 2.2.2.12) and as (Lurie, def. 4.3.3; def. 4.3.13).

Properties

Derived étale geometry

The following definition and theorem show how the étale $(\infty,1)$-site arises naturally from the étale 1-site, and naturally encodes the derived geometry induced by the étale site.

Definition

(étale pregeometry)

Let $\mathcal{T}_{et}$ be the 1-étale site regarded as a pregeometry (for structured (∞,1)-toposes) as follows.

• the underlying (∞,1)-category is the 1-category

  $$(CAlg_k^{sm})^{op} \hookrightarrow CAlg_k ^{op} \,,$$

  which is the full subcategory of $CAlg_k$ on those objects $A \in CAlg_k$ for which there exists an étale morphism $k[x^1, \cdots, x^n] \to A$ from the polynomial algebra in $n$ generators for some $n \in \mathbb{N}$;

• the admissible morphisms in the pregeometry are the étale morphisms;

• a collection of admissible morphisms is a covering family if it is so as a family of morphisms in the étale site.

This is (Lurie, def. 4.3.1).

Definition

(étale geometry)

Let $\mathcal{G}_{et}$ be the geometry (for structured (∞,1)-toposes) given by

• the underlying (∞,1)-site is the étale $(\infty,1)$-site;

• the admissible morphisms are the étale morphisms.

This is (Lurie, def. 4.3.13).

Theorem

The geometry generated by the étale pregeometry $\mathcal{T}_{et}$ is the étale geometry $\mathcal{G}_{et}$.

This is (Lurie, prop. 4.3.15).

Related concepts

• étale morphism, étale site, étale cohomology

• étale morphism of E-∞ rings

• étale $(\infty,1)$-site

References

In its presentation as a model site the étale $(\infty,1)$-site is given in definition 2.2.2.12 of

• Bertrand Toën, Gabriele Vezzosi, Homotopical Algebraic Geometry II: geometric stacks and applications (arXiv)

  .

A discussion in the context of structured (∞,1)-toposes is

• Jacob Lurie, section 4.3 of Structured Spaces

• Jacob Lurie, section 3 and 4 of, Descent theorems (pdf)

See also

• Benjamin Antieau, David Gepner, Brauer groups and étale cohomology in derived algebraic geometry (arXiv:1210.0290)