Geometries

7.2. Geometries

Classical algebraic geometry is built from affine schemes, basic open immersions, and covers by basic opens. Lurie's notion of a geometry abstracts exactly this pattern; see [Lurie 2011]:

For the applications below, we will use the following slightly site-theoretic formulation.

7.2.1. Zariski geometries

The basic example is the classical Zariski geometry. Its underlying category is

where \(\CRing^{\mathrm{fp}}\) denotes the category of finitely presented ordinary commutative rings. Since \(\GZar^{\mathrm{cl}}\) is the opposite of a category of rings, we will describe its morphisms in terms of the corresponding maps of rings in the opposite direction.

The admissible morphisms in \(\GZar^{\mathrm{cl}}\) correspond to principal localizations

Geometrically, this is the inclusion of the principal open

A finite family

generates a covering sieve precisely when the elements \(f_i\) generate the unit ideal in \(R\).

Proof

Finite limits in \(\GZar^{\mathrm{cl}}\) are finite colimits of finitely presented rings. The topology is generated by admissible covers by definition. Base change of admissible morphisms follows from the formula

\[S \otimes_R R[f^{-1}] \cong S[\varphi(f)^{-1}]\]

for a map \(\varphi\colon R \to S\).

For left cancellability, suppose that a principal localization \(R \to R[g^{-1}]\) factors through another principal localization \(R \to R[f^{-1}]\). Then \(f\) is already invertible in \(R[g^{-1}]\), and hence

\[R[g^{-1}] \cong R[g^{-1}][f^{-1}] \cong R[f^{-1}][g^{-1}].\]

It follows that the induced map \(R[f^{-1}] \to R[g^{-1}]\) is again a principal localization.

It remains to check closure under retracts. Consider a retract diagram

in \(\CRing^{\mathrm{fp}}\), where \(\ell\) is localization at \(f\). For every ring \(C\), the restriction map

\[\Hom_{\CRing}(B,C) \longrightarrow \Hom_{\CRing}(A,C)\]

is a retract of

\[\Hom_{\CRing}(R[f^{-1}],C) \longrightarrow \Hom_{\CRing}(R,C).\]

The latter map is a monomorphism whose image consists of the maps that send \(f\) to a unit. Consequently, restriction along \(\varphi\) is also a monomorphism. If \(a:=\rho(f)\in A\), the retract identities show that a map \(A\to C\) extends across \(\varphi\) if and only if it sends \(a\) to a unit. The universal property of localization therefore gives an isomorphism \(B\cong A[a^{-1}]\) under \(A\).

There is also a spectral Zariski geometry

where \(\CAlg^{\omega}\) denotes the category of compact commutative \(\Einf\)-rings. The admissible morphisms are again principal localizations

and a finite family \(\{R \to R[f_i^{-1}]\}_{i \in I}\) is a cover precisely when the elements \(f_i\) generate the unit ideal in \(\pi_0(R)\). The verification is the same as in the classical case, using the corresponding localization formula for commutative \(\Einf\)-rings.

7.2.2. Other examples

Here are some other geometries obtained by changing the category of affine test objects, the admissible morphisms, or the topology:

• Derived Zariski geometry: the analogous Zariski geometry based on compact animated rings.

• Classical étale geometry: take \(C = (\CRing^{\mathrm{fp}})\catop\), take \(\tau\) to be the étale topology, and let \(C^{\ad}\) be the class of morphisms corresponding to étale maps of rings.

• Derived étale geometry: the analogous version based on compact animated rings.

• Spectral étale geometry: replace ordinary rings by compact commutative \(\Einf\)-rings and use the spectral étale topology.

• Differential geometry: use finitely presented \(C^{\infty}\)-rings, with admissible maps encoding open immersions.

7.2.3. Structured topoi

Let \(\Gg = (C,C^{\ad},\tau)\) be a geometry and let \(T\) be a topos. A \(\Gg\)-structure on \(T\) is a left exact functor

such that for every admissible cover \(\{U_i \to X\}_{i \in I}\), the induced map

is an effective epimorphism in \(T\). We denote by \(\Str_{\Gg}(T)\) the full subcategory of \(\Fun^{\lex}(C,T)\) spanned by the \(\Gg\)-structures.

In the classical Zariski case, the equivalence

identifies a left exact functor \(\GZar^{\mathrm{cl}} \to \An\) with an ordinary commutative ring \(A\). Indeed, every ordinary ring is a filtered colimit of finitely presented rings, and filtered colimits of discrete animae are discrete. Under this identification, the functor associated to \(A\) sends a finitely presented ring \(S\) to

The Zariski cover condition says that, whenever \(f_1,\dots,f_n \in S\) generate the unit ideal, the map

is surjective. In concrete terms: for every map \(\varphi\colon S \to A\), at least one of the elements \(\varphi(f_i) \in A\) must be invertible.

This condition is equivalent to \(A\) being a local ring. Indeed, if \(A\) is local, then not all of the elements \(\varphi(f_i)\) can lie in the maximal ideal, so one must be invertible. Conversely, suppose that the displayed condition holds. If \(a+b=1\), apply it to the map

and to the cover by \(D(x)\) and \(D(y)\). It follows that either \(a\) or \(b\) is invertible. Replacing \((a,b)\) by \((-a,a+b)\) shows that the sum of two nonunits is again a nonunit. Since the nonunits are also closed under multiplication by arbitrary elements, they form an ideal, necessarily the unique maximal ideal. All in all, we see that

In the spectral Zariski case, the same argument starts from

and identifies a left exact functor \(\GZar^{\mathrm{sp}} \to \An\) with a commutative ring spectrum \(R\). The cover condition says that if \(f_1,\dots,f_n \in \pi_0(S)\) generate the unit ideal, then for every map \(\varphi\colon S \to R\) at least one of the elements \(\varphi(f_i) \in \pi_0(R)\) is invertible. Equivalently, \(\pi_0(R)\) is a local ring. Thus

The upshot of this discussion is that Zariski structures provide a point-free version of the usual locality condition on stalks. Left exact functors from the Zariski geometry to a topos \(T\) correspond to internal commutative ring objects of \(T\). Pulling such a functor back along a point \(x^*\colon T \to \An\) produces the corresponding stalk. If \(T\) admits enough points, then these inverse-image functors jointly detect effective epimorphisms, so the admissible-cover condition is equivalent to requiring every stalk to be a local ring. If \(T\) does not admit enough points, the definition in terms of admissible covers remains meaningful and supplies the correct point-free replacement.

7.2.4. Local morphisms of structures

Let \(\Oo,\Oo' \in \Str_{\Gg}(T)\) be two \(\Gg\)-structures. A natural transformation

is called local if for every admissible morphism \(U \to X\) in \(C\), the square

is a pullback square in \(T\).

For the classical Zariski geometry and \(T=\An\), this recovers the usual notion of a local homomorphism. A map of Zariski structures corresponds to a map of local rings \(A \to B\). Testing locality on an admissible map \(S \to S[f^{-1}]\) says that the square

is a pullback. In other words, an element of \(A\) is invertible if and only if its image in \(B\) is invertible. For local rings this is equivalent to the classical condition that the preimage of the maximal ideal of \(B\) is the maximal ideal of \(A\). The spectral case is identical after passing to \(\pi_0\): a map \(R \to R'\) of local commutative \(\Einf\)-rings is local precisely when \(\pi_0(R) \to \pi_0(R')\) is local in the classical sense.