Let $(S,\mathcal O_S)$ be a scheme. What's the definition of $\mathcal O_S$-algebra?
2026-03-26 00:53:32.1774486412
What's the meaning of $\mathcal O_S$-algebra?
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It's a sheaf $\mathcal{A}$ of $\mathcal{O}_S$-modules on $S$ which is also a sheaf of rings, such that the $\mathcal{O}_S$-module map $\mathcal{O}_S \to \mathcal{A}$ induced by $1 \mapsto 1$ is a ring map.
(Equivalently, a sheaf of rings $\mathcal{A}$ on $S$ together with a ring homomorphism $\mathcal{O}_S \to \mathcal{A}$.)