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Results tagged with flatness
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user 40297
14
votes
Accepted
Flatness from constancy of dimension of fibers
Yes if $R$ is reduced (no nilpotent elements). This is (for instance) Lemma 1 p. 51 in Mumford's Abelian Varieties (2nd edition). No in general: just take $R=k[\varepsilon ]/(\varepsilon ^2)$, $M=k$. …
- 38k
7
votes
Is the zero locus of a global section flat?
No. Take $X=\mathbb{P}^1\times \mathbb{P}^1$, $Y=\mathbb{P}^1$, $f$ the first projection, $\mathcal{L}=\mathcal{O}_{\mathbb{P}^1}(1)\boxtimes \mathcal{O}_{\mathbb{P}^1}(1)$, $s=X\otimes X'$, where $(X …
- 38k