Definition 61.4.1. A ring map $A \to B$ is said to be ind-Zariski if $B$ can be written as a filtered colimit $B = \mathop{\mathrm{colim}}\nolimits B_ i$ with each $A \to B_ i$ a local isomorphism.
61.4 Ind-Zariski algebra
We start with a definition; please see Remark 61.6.9 for a comparison with the corresponding definition of the article [BS].
An example of an Ind-Zariski map is a localization $A \to S^{-1}A$, see Algebra, Lemma 10.9.9. The category of ind-Zariski algebras is closed under several natural operations.
Lemma 61.4.2. Let $A \to B$ and $A \to A'$ be ring maps. Let $B' = B \otimes _ A A'$ be the base change of $B$. If $A \to B$ is ind-Zariski, then $A' \to B'$ is ind-Zariski.
Proof. Omitted. $\square$
Lemma 61.4.3. Let $A \to B$ and $B \to C$ be ring maps. If $A \to B$ and $B \to C$ are ind-Zariski, then $A \to C$ is ind-Zariski.
Proof. Omitted. $\square$
Lemma 61.4.4. Let $A$ be a ring. Let $B \to C$ be an $A$-algebra homomorphism. If $A \to B$ and $A \to C$ are ind-Zariski, then $B \to C$ is ind-Zariski.
Proof. Omitted. $\square$
Lemma 61.4.5. A filtered colimit of ind-Zariski $A$-algebras is ind-Zariski over $A$.
Proof. Omitted. $\square$
Lemma 61.4.6. Let $A \to B$ be ind-Zariski. Then $A \to B$ identifies local rings,
Proof. Omitted. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)