List questions from flatness

352

Views

Local flatness criterion: A morphism $X \to Y$ of schemes over $S$ is flat if and only if it is flat on all fibers $X_s \to Y_s$, $s \in S$?

Published on 01 Nov 2021 - 14:22

182

Views

How to prove that $S^{-1} A$ is a flat A-module

Published on 12 Dec 2021 - 13:57

154

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Support of module and faithfully flat base change

Published on 25 Dec 2021 - 12:03

99

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Exercise 4.3, Atiyah Macdonald

Published on 29 Dec 2021 - 8:47

74

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Confusion about a definition in Atiyah-Macdonald

Published on 01 Jan 2022 - 8:43

56

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About flatness of modules / algebras

Published on 01 Jan 2022 - 9:27

63

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For number field $K$, is maximal order $O_K$ flat over the order $O$?

Published on 12 Jan 2022 - 8:07

237

Views

Let $(R, P)$ be a local ring. If $M$ is a finitely presented module, then $M$ is flat iff $\mathrm{Tor}_{1}^R(M,R/P)=0$.

Published on 15 Jan 2022 - 8:10

43

Views

Is the natural projection of flag varieties a flat morphism?

Published on 26 Mar 2026 - 1:14

50

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A statement for flat modules analogous to the Baer's Criterion

Published on 16 Jan 2022 - 23:09

121

Views

Right exactness of tensor product in a long exact sequence

Published on 05 Feb 2022 - 21:18

87

Views

A characterisation of flatness

Published on 08 Feb 2022 - 14:35

139

Views

Change of rings of scalars and projecivity?

Published on 25 Mar 2026 - 6:04

119

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Faithful flatness and split-exactness

Published on 16 Feb 2022 - 21:58

97

Views

When every finitely generated submodule of an R-module is cyclic?

Published on 17 Feb 2022 - 18:29

List Question

Local flatness criterion: A morphism $X \to Y$ of schemes over $S$ is flat if and only if it is flat on all fibers $X_s \to Y_s$, $s \in S$?

How to prove that $S^{-1} A$ is a flat A-module

Support of module and faithfully flat base change

Exercise 4.3, Atiyah Macdonald

Confusion about a definition in Atiyah-Macdonald

About flatness of modules / algebras

For number field $K$, is maximal order $O_K$ flat over the order $O$?

Let $(R, P)$ be a local ring. If $M$ is a finitely presented module, then $M$ is flat iff $\mathrm{Tor}_{1}^R(M,R/P)=0$.

Is the natural projection of flag varieties a flat morphism?

A statement for flat modules analogous to the Baer's Criterion

Right exactness of tensor product in a long exact sequence

A characterisation of flatness

Change of rings of scalars and projecivity?

Faithful flatness and split-exactness

When every finitely generated submodule of an R-module is cyclic?

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