# Questions tagged [flatness]

An $A$-module $M$ is flat when the right-exact _$\otimes_AM$ functor becomes left exact (and therefore exact). This applies to $A$-algebras as the latter are $A$-module, saying that $B$ flat over $A$, or that $A\to B$ is flat. The notion passes to morphisms of schemes: a morphism of schemes $f: X\to Y$ is flat if all the induces stalks local morphisms are flat. This passes also to sheaves, etc...

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### Is the quotient of an infinite product of fields by the direct sum injective?

Let $k$ be any field and consider $R=\prod_{\mathbb{N}}k$ the product of copies of $k$ indexed over the natural numbers. The direct sum $\sum_{\mathbb{N}}k=I$ is an ideal of $R$, hence we can form the ...
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### A Criterion for Faithfully Flatness via Intersection Property

Let $R \subset S$ be an injective map of commutative rings, $M \subset N$ an inclusion of $R$-modules. We identify $M, N$ respectively as $M \otimes_R 1_S \subset N \otimes_R 1_S \subset N \otimes S$...
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### [Cor 5.3.24 in Qing Liu]If $X$ is a projective scheme over a DVR $A$, then any coherent sheaf of ideal of $\mathcal{O}_X$ is flat over $A$?

The problem comes from the proof of Corollary 5.3.24 in Qing Liu's book Algebraic Geometry and Arithmetic curve. Let $\DeclareMathOperator{\Spec}{Spec}S=\Spec A$ be the spectrum a discrete valuation ...
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