## 60.6 Identifying local rings versus ind-Zariski

An ind-Zariski ring map $A \to B$ identifies local rings (Lemma 60.4.6). The converse does not hold (Examples, Section 108.43). However, it turns out that there is a kind of structure theorem for ring maps which identify local rings in terms of ind-Zariski ring maps, see Proposition 60.6.6.

Let $A$ be a ring. Let $X = \mathop{\mathrm{Spec}}(A)$. The space of connected components $\pi _0(X)$ is a profinite space by Topology, Lemma 5.23.9 (and Algebra, Lemma 10.26.2).

Lemma 60.6.1. Let $A$ be a ring. Let $X = \mathop{\mathrm{Spec}}(A)$. Let $T \subset \pi _0(X)$ be a closed subset. There exists a surjective ind-Zariski ring map $A \to B$ such that $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ induces a homeomorphism of $\mathop{\mathrm{Spec}}(B)$ with the inverse image of $T$ in $X$.

Proof. Let $Z \subset X$ be the inverse image of $T$. Then $Z$ is the intersection $Z = \bigcap Z_\alpha$ of the open and closed subsets of $X$ containing $Z$, see Topology, Lemma 5.12.12. For each $\alpha$ we have $Z_\alpha = \mathop{\mathrm{Spec}}(A_\alpha )$ where $A \to A_\alpha$ is a local isomorphism (a localization at an idempotent). Setting $B = \mathop{\mathrm{colim}}\nolimits A_\alpha$ proves the lemma. $\square$

Lemma 60.6.2. Let $A$ be a ring and let $X = \mathop{\mathrm{Spec}}(A)$. Let $T$ be a profinite space and let $T \to \pi _0(X)$ be a continuous map. There exists an ind-Zariski ring map $A \to B$ such that with $Y = \mathop{\mathrm{Spec}}(B)$ the diagram

$\xymatrix{ Y \ar[r] \ar[d] & \pi _0(Y) \ar[d] \\ X \ar[r] & \pi _0(X) }$

is cartesian in the category of topological spaces and such that $\pi _0(Y) = T$ as spaces over $\pi _0(X)$.

Proof. Namely, write $T = \mathop{\mathrm{lim}}\nolimits T_ i$ as the limit of an inverse system finite discrete spaces over a directed set (see Topology, Lemma 5.22.2). For each $i$ let $Z_ i = \mathop{\mathrm{Im}}(T \to \pi _0(X) \times T_ i)$. This is a closed subset. Observe that $X \times T_ i$ is the spectrum of $A_ i = \prod _{t \in T_ i} A$ and that $A \to A_ i$ is a local isomorphism. By Lemma 60.6.1 we see that $Z_ i \subset \pi _0(X \times T_ i) = \pi _0(X) \times T_ i$ corresponds to a surjection $A_ i \to B_ i$ which is ind-Zariski such that $\mathop{\mathrm{Spec}}(B_ i) = X \times _{\pi _0(X)} Z_ i$ as subsets of $X \times T_ i$. The transition maps $T_ i \to T_{i'}$ induce maps $Z_ i \to Z_{i'}$ and $X \times _{\pi _0(X)} Z_ i \to X \times _{\pi _0(X)} Z_{i'}$. Hence ring maps $B_{i'} \to B_ i$ (Lemmas 60.3.8 and 60.4.6). Set $B = \mathop{\mathrm{colim}}\nolimits B_ i$. Because $T = \mathop{\mathrm{lim}}\nolimits Z_ i$ we have $X \times _{\pi _0(X)} T = \mathop{\mathrm{lim}}\nolimits X \times _{\pi _0(X)} Z_ i$ and hence $Y = \mathop{\mathrm{Spec}}(B) = \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Spec}}(B_ i)$ fits into the cartesian diagram

$\xymatrix{ Y \ar[r] \ar[d] & T \ar[d] \\ X \ar[r] & \pi _0(X) }$

of topological spaces. By Lemma 60.2.5 we conclude that $T = \pi _0(Y)$. $\square$

Example 60.6.3. Let $k$ be a field. Let $T$ be a profinite topological space. There exists an ind-Zariski ring map $k \to A$ such that $\mathop{\mathrm{Spec}}(A)$ is homeomorphic to $T$. Namely, just apply Lemma 60.6.2 to $T \to \pi _0(\mathop{\mathrm{Spec}}(k)) = \{ *\}$. In fact, in this case we have

$A = \mathop{\mathrm{colim}}\nolimits \text{Map}(T_ i, k)$

whenever we write $T = \mathop{\mathrm{lim}}\nolimits T_ i$ as a filtered limit with each $T_ i$ finite.

Lemma 60.6.4. Let $A \to B$ be ring map such that

1. $A \to B$ identifies local rings,

2. the topological spaces $\mathop{\mathrm{Spec}}(B)$, $\mathop{\mathrm{Spec}}(A)$ are w-local,

3. $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is w-local, and

4. $\pi _0(\mathop{\mathrm{Spec}}(B)) \to \pi _0(\mathop{\mathrm{Spec}}(A))$ is bijective.

Then $A \to B$ is an isomorphism

Proof. Let $X_0 \subset X = \mathop{\mathrm{Spec}}(A)$ and $Y_0 \subset Y = \mathop{\mathrm{Spec}}(B)$ be the sets of closed points. By assumption $Y_0$ maps into $X_0$ and the induced map $Y_0 \to X_0$ is a bijection. As a space $\mathop{\mathrm{Spec}}(A)$ is the disjoint union of the spectra of the local rings of $A$ at closed points. Similarly for $B$. Hence $X \to Y$ is a bijection. Since $A \to B$ is flat we have going down (Algebra, Lemma 10.39.19). Thus Algebra, Lemma 10.41.11 shows for any prime $\mathfrak q \subset B$ lying over $\mathfrak p \subset A$ we have $B_\mathfrak q = B_\mathfrak p$. Since $B_\mathfrak q = A_\mathfrak p$ by assumption, we see that $A_\mathfrak p = B_\mathfrak p$ for all primes $\mathfrak p$ of $A$. Thus $A = B$ by Algebra, Lemma 10.23.1. $\square$

Lemma 60.6.5. Let $A \to B$ be ring map such that

1. $A \to B$ identifies local rings,

2. the topological spaces $\mathop{\mathrm{Spec}}(B)$, $\mathop{\mathrm{Spec}}(A)$ are w-local, and

3. $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is w-local.

Then $A \to B$ is ind-Zariski.

Proof. Set $X = \mathop{\mathrm{Spec}}(A)$ and $Y = \mathop{\mathrm{Spec}}(B)$. Let $X_0 \subset X$ and $Y_0 \subset Y$ be the set of closed points. Let $A \to A'$ be the ind-Zariski morphism of affine schemes such that with $X' = \mathop{\mathrm{Spec}}(A')$ the diagram

$\xymatrix{ X' \ar[r] \ar[d] & \pi _0(X') \ar[d] \\ X \ar[r] & \pi _0(X) }$

is cartesian in the category of topological spaces and such that $\pi _0(X') = \pi _0(Y)$ as spaces over $\pi _0(X)$, see Lemma 60.6.2. By Lemma 60.2.5 we see that $X'$ is w-local and the set of closed points $X'_0 \subset X'$ is the inverse image of $X_0$.

We obtain a continuous map $Y \to X'$ of underlying topological spaces over $X$ identifying $\pi _0(Y)$ with $\pi _0(X')$. By Lemma 60.3.8 (and Lemma 60.4.6) this is corresponds to a morphism of affine schemes $Y \to X'$ over $X$. Since $Y \to X$ maps $Y_0$ into $X_0$ we see that $Y \to X'$ maps $Y_0$ into $X'_0$, i.e., $Y \to X'$ is w-local. By Lemma 60.6.4 we see that $Y \cong X'$ and we win. $\square$

The following proposition is a warm up for the type of result we will prove later.

Proposition 60.6.6. Let $A \to B$ be a ring map which identifies local rings. Then there exists a faithfully flat, ind-Zariski ring map $B \to B'$ such that $A \to B'$ is ind-Zariski.

Proof. Let $A \to A_ w$, resp. $B \to B_ w$ be the faithfully flat, ind-Zariski ring map constructed in Lemma 60.5.3 for $A$, resp. $B$. Since $\mathop{\mathrm{Spec}}(B_ w)$ is w-local, there exists a unique factorization $A \to A_ w \to B_ w$ such that $\mathop{\mathrm{Spec}}(B_ w) \to \mathop{\mathrm{Spec}}(A_ w)$ is w-local by Lemma 60.5.5. Note that $A_ w \to B_ w$ identifies local rings, see Lemma 60.3.4. By Lemma 60.6.5 this means $A_ w \to B_ w$ is ind-Zariski. Since $B \to B_ w$ is faithfully flat, ind-Zariski (Lemma 60.5.3) and the composition $A \to B \to B_ w$ is ind-Zariski (Lemma 60.4.3) the proposition is proved. $\square$

The proposition above allows us to characterize the affine, weakly contractible objects in the pro-Zariski site of an affine scheme.

Lemma 60.6.7. Let $A$ be a ring. The following are equivalent

1. every faithfully flat ring map $A \to B$ identifying local rings has a section,

2. every faithfully flat ind-Zariski ring map $A \to B$ has a section, and

3. $A$ satisfies

1. $\mathop{\mathrm{Spec}}(A)$ is w-local, and

2. $\pi _0(\mathop{\mathrm{Spec}}(A))$ is extremally disconnected.

Proof. The equivalence of (1) and (2) follows immediately from Proposition 60.6.6.

Assume (3)(a) and (3)(b). Let $A \to B$ be faithfully flat and ind-Zariski. We will use without further mention the fact that a flat map $A \to B$ is faithfully flat if and only if every closed point of $\mathop{\mathrm{Spec}}(A)$ is in the image of $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ We will show that $A \to B$ has a section.

Let $I \subset A$ be an ideal such that $V(I) \subset \mathop{\mathrm{Spec}}(A)$ is the set of closed points of $\mathop{\mathrm{Spec}}(A)$. We may replace $B$ by the ring $C$ constructed in Lemma 60.5.8 for $A \to B$ and $I \subset A$. Thus we may assume $\mathop{\mathrm{Spec}}(B)$ is w-local such that the set of closed points of $\mathop{\mathrm{Spec}}(B)$ is $V(IB)$.

Assume $\mathop{\mathrm{Spec}}(B)$ is w-local and the set of closed points of $\mathop{\mathrm{Spec}}(B)$ is $V(IB)$. Choose a continuous section to the surjective continuous map $V(IB) \to V(I)$. This is possible as $V(I) \cong \pi _0(\mathop{\mathrm{Spec}}(A))$ is extremally disconnected, see Topology, Proposition 5.26.6. The image is a closed subspace $T \subset \pi _0(\mathop{\mathrm{Spec}}(B)) \cong V(JB)$ mapping homeomorphically onto $\pi _0(A)$. Replacing $B$ by the ind-Zariski quotient ring constructed in Lemma 60.6.1 we see that we may assume $\pi _0(\mathop{\mathrm{Spec}}(B)) \to \pi _0(\mathop{\mathrm{Spec}}(A))$ is bijective. At this point $A \to B$ is an isomorphism by Lemma 60.6.4.

Assume (1) or equivalently (2). Let $A \to A_ w$ be the ring map constructed in Lemma 60.5.3. By (1) there is a section $A_ w \to A$. Thus $\mathop{\mathrm{Spec}}(A)$ is homeomorphic to a closed subset of $\mathop{\mathrm{Spec}}(A_ w)$. By Lemma 60.2.4 we see (3)(a) holds. Finally, let $T \to \pi _0(A)$ be a surjective map with $T$ an extremally disconnected, quasi-compact, Hausdorff topological space (Topology, Lemma 5.26.9). Choose $A \to B$ as in Lemma 60.6.2 adapted to $T \to \pi _0(\mathop{\mathrm{Spec}}(A))$. By (1) there is a section $B \to A$. Thus we see that $T = \pi _0(\mathop{\mathrm{Spec}}(B)) \to \pi _0(\mathop{\mathrm{Spec}}(A))$ has a section. A formal categorical argument, using Topology, Proposition 5.26.6, implies that $\pi _0(\mathop{\mathrm{Spec}}(A))$ is extremally disconnected. $\square$

Lemma 60.6.8. Let $A$ be a ring. There exists a faithfully flat, ind-Zariski ring map $A \to B$ such that $B$ satisfies the equivalent conditions of Lemma 60.6.7.

Proof. We first apply Lemma 60.5.3 to see that we may assume that $\mathop{\mathrm{Spec}}(A)$ is w-local. Choose an extremally disconnected space $T$ and a surjective continuous map $T \to \pi _0(\mathop{\mathrm{Spec}}(A))$, see Topology, Lemma 5.26.9. Note that $T$ is profinite. Apply Lemma 60.6.2 to find an ind-Zariski ring map $A \to B$ such that $\pi _0(\mathop{\mathrm{Spec}}(B)) \to \pi _0(\mathop{\mathrm{Spec}}(A))$ realizes $T \to \pi _0(\mathop{\mathrm{Spec}}(A))$ and such that

$\xymatrix{ \mathop{\mathrm{Spec}}(B) \ar[r] \ar[d] & \pi _0(\mathop{\mathrm{Spec}}(B)) \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] & \pi _0(\mathop{\mathrm{Spec}}(A)) }$

is cartesian in the category of topological spaces. Note that $\mathop{\mathrm{Spec}}(B)$ is w-local, that $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is w-local, and that the set of closed points of $\mathop{\mathrm{Spec}}(B)$ is the inverse image of the set of closed points of $\mathop{\mathrm{Spec}}(A)$, see Lemma 60.2.5. Thus condition (3) of Lemma 60.6.7 holds for $B$. $\square$

Remark 60.6.9. In each of Lemmas 60.6.1, 60.6.2, Proposition 60.6.6, and Lemma 60.6.8 we find an ind-Zariski ring map with some properties. In the paper [BS] the authors use the notion of an ind-(Zariski localization) which is a filtered colimit of finite products of principal localizations. It is possible to replace ind-Zariski by ind-(Zariski localization) in each of the results listed above. However, we do not need this and the notion of an ind-Zariski homomorphism of rings as defined here has slightly better formal properties. Moreover, the notion of an ind-Zariski ring map is the natural analogue of the notion of an ind-étale ring map defined in the next section.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 097B. Beware of the difference between the letter 'O' and the digit '0'.