nLab scheme as a locally affine structured (infinity,1)-topos

This is a sub-entry of scheme and derived scheme.

This entry discusses a higher-categorical perspective on that standard notion.

On p. 37 of

• Jacob Lurie, Structured Spaces (StSp),

it is argued that the standard definition of scheme with its insistance on an underlying topological space is slightly misleading:

it is in some sense coincidental that $Spec A$ is described by a topological space. What arises more canonically is the lattice of open subsets of $Spec A$, which is generated by basic open sets of the form $U_f$. This lattice naturally forms a locale, or 0-topos. It happens that this locale has enough points, and can therefore be described as the lattice of open subsets of a topological space. However, there are various reasons we might want to disregard this fact:

a) The existence of enough points for $Spec A$ is equivalent to the assertion that every nonzero commutative ring contains a prime ideal, and the proof of this assertion requires the axiom of choice.

b) In relative situations, the relevant construction may well fail to admit enough points, even if the axiom of choice is assumed. However, the underlying locale (and its associated sheaf theory) are still well-behaved.

c) If we wish to replace the Zariski topology by some other topology (such as the étale topology), then we are forced to work with $Spec A$ as a topos rather than simply as a topological space: the category of étale sheaves on $Spec A$ is not generated by subobjects of the final object.

When we study derived algebraic geometry, we will want to study sheaves on $Spec A$ of a higher categorical nature, such as sheaves of spaces or sheaves of spectra. For these purposes it will be most convenient to regard $Spec A$ as an ∞-topos, rather than as a topological space.

From that perspective then a scheme is realized as a structured (∞,1)-topos with respect to the Zariski geometry (for structured (∞,1)-toposes) $\mathcal{G}_{Zar}$. See the examples at derived scheme for more details on this.

Proposition (StSp, 4.2.9)

Let $Sch_{\leq 0}^{\leq 0}(\mathcal{G}_{Zar}) \subset Sch(\mathcal{G}_{Zar})$ be the full subcategory of all derived schemes on the 0-truncated and 0-localic ones. This is canonically equivalent to the ordinary category $Sch(k)$ of relative schemes over $k$:

$$Sch_{\leq 0}^{\leq 0}(\mathcal{G}_{Zar}(k)) \simeq Sch(k) \,.$$

category: algebraic geometry