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An affine scheme is quasi-compact. In fact, a scheme is quasi-compact if and only if it is a finite union of open affine subschemes.
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The composition of quasi-compact morphisms is quasi-compact. The base change of a quasi-compact morphism is quasi-compact.
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is quasi-compact, then the pre-image of a compact open subscheme (e.g., open affine subscheme) under
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gives a necessary and sufficient condition for a quasi-compact scheme to be affine.
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A morphism from a quasi-compact scheme to an affine scheme is quasi-compact.
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Schwede, Karl (2005), "Gluing schemes and a scheme without closed points",
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is closed if and only if it is stable under specialization.
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A quasi-compact scheme has at least one closed point.
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Recent progress in arithmetic and algebraic geometry
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497:: CS1 maint: DOI inactive as of April 2024 (
386:needs attention from an expert in Mathematics
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542:When is an irreducible scheme quasi-compact?
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207:{\displaystyle X=\operatorname {Spec} A}
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140:{\displaystyle f^{-1}(V_{i})}
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