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.

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

Revised on March 6, 2013 19:28:07
by Zoran Škoda
(161.53.130.104)