# The Stacks Project

## Tag 00F6

Lemma 10.28.1. Let $U \subset \mathop{\rm Spec}(R)$ be open. The following are equivalent:

1. $U$ is retrocompact in $\mathop{\rm Spec}(R)$,
2. $U$ is quasi-compact,
3. $U$ is a finite union of standard opens, and
4. there exists a finitely generated ideal $I \subset R$ such that $X \setminus V(I) = U$.

Proof. We have (1) $\Rightarrow$ (2) because $\mathop{\rm Spec}(R)$ is quasi-compact, see Lemma 10.16.10. We have (2) $\Rightarrow$ (3) because standard opens form a basis for the topology. Proof of (3) $\Rightarrow$ (1). Let $U = \bigcup_{i = 1\ldots n} D(f_i)$. To show that $U$ is retrocompact in $\mathop{\rm Spec}(R)$ it suffices to show that $U \cap V$ is quasi-compact for any quasi-compact open $V$ of $\mathop{\rm Spec}(R)$. Write $V = \bigcup_{j = 1\ldots m} D(g_j)$ which is possible by (2) $\Rightarrow$ (3). Each standard open is homeomorphic to the spectrum of a ring and hence quasi-compact, see Lemmas 10.16.6 and 10.16.10. Thus $U \cap V = (\bigcup_{i = 1\ldots n} D(f_i)) \cap (\bigcup_{j = 1\ldots m} D(g_j)) = \bigcup_{i, j} D(f_i g_j)$ is a finite union of quasi-compact opens hence quasi-compact. To finish the proof note that (4) is equivalent to (3) by Lemma 10.16.2. $\square$

The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 5079–5090 (see updates for more information).

\begin{lemma}
\label{lemma-qc-open}
Let $U \subset \Spec(R)$ be open. The following
are equivalent:
\begin{enumerate}
\item $U$ is retrocompact in $\Spec(R)$,
\item $U$ is quasi-compact,
\item $U$ is a finite union of standard opens, and
\item there exists a finitely generated ideal $I \subset R$ such
that $X \setminus V(I) = U$.
\end{enumerate}
\end{lemma}

\begin{proof}
We have (1) $\Rightarrow$ (2) because $\Spec(R)$ is quasi-compact, see
Lemma \ref{lemma-quasi-compact}. We have (2) $\Rightarrow$ (3) because
standard opens form a basis for the topology. Proof of (3) $\Rightarrow$ (1).
Let $U = \bigcup_{i = 1\ldots n} D(f_i)$. To show that $U$ is retrocompact
in $\Spec(R)$ it suffices to show that $U \cap V$ is quasi-compact for any
quasi-compact open $V$ of $\Spec(R)$. Write
$V = \bigcup_{j = 1\ldots m} D(g_j)$ which is possible by (2) $\Rightarrow$
(3). Each standard open is homeomorphic to the spectrum of a ring and hence
quasi-compact, see Lemmas \ref{lemma-standard-open} and
\ref{lemma-quasi-compact}. Thus
$U \cap V = (\bigcup_{i = 1\ldots n} D(f_i)) \cap (\bigcup_{j = 1\ldots m} D(g_j)) = \bigcup_{i, j} D(f_i g_j)$ is a finite union of quasi-compact opens
hence quasi-compact. To finish the proof note
that (4) is equivalent to (3) by
Lemma \ref{lemma-Zariski-topology}.
\end{proof}

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