List questions from projective-module

Published on 15 Apr 2026 - 0:51

77

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Multiplication map of an $R$-algebra $A$ is projective module iff diagonal is open?

Published on 15 Apr 2026 - 2:09

213

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For $0 \to N' \to N \to M \to 0 $ with $N$ finitely generated and $M$ finitely presented then $N^{'}$ is also finitely generated.

Published on 16 Apr 2026 - 5:45

1.1k

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Projective modules and ring homomorphisms.

Published on 15 Apr 2026 - 10:08

1k

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$\operatorname{Ext}^1(M,R/m)=0$ iff $M$ is projective

Published on 17 Apr 2026 - 15:53

242

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Ext, Tor and ideals (2)

Published on 11 Apr 2026 - 21:36

101

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Prove $\operatorname{pd}_R M\leq n$ iff $\operatorname{Ext}^{n+1}(M,N)=0$

Published on 17 Apr 2026 - 4:28

446

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Is this a valid proof for why all modules over a field are projective?

Published on 13 Apr 2026 - 1:22

496

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$K$ is not projective as $R$ module where $K$ is quotient field of a PID $R$.

Published on 14 Apr 2026 - 16:27

330

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In the proof of every free R-module is projective, where do we use being free?

Published on 13 Apr 2026 - 23:09

795

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Projective modules

Published on 13 Apr 2026 - 22:14

7.8k

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Finitely generated projective modules are locally free

Published on 12 Apr 2026 - 4:54

561

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Projective modules over a semi-local ring

Published on 14 Apr 2026 - 22:40

732

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Projective but not free (exercise from Adkins - Weintraub)

Published on 12 Apr 2026 - 21:15

2.6k

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A module is projective iff it has a projective basis

Published on 16 Apr 2026 - 2:27

List Question

For an exact sequence prove that $B\cong A\oplus C$

Multiplication map of an $R$-algebra $A$ is projective module iff diagonal is open?

For $0 \to N' \to N \to M \to 0 $ with $N$ finitely generated and $M$ finitely presented then $N^{'}$ is also finitely generated.

Projective modules and ring homomorphisms.

$\operatorname{Ext}^1(M,R/m)=0$ iff $M$ is projective

Ext, Tor and ideals (2)

Prove $\operatorname{pd}_R M\leq n$ iff $\operatorname{Ext}^{n+1}(M,N)=0$

Is this a valid proof for why all modules over a field are projective?

$K$ is not projective as $R$ module where $K$ is quotient field of a PID $R$.

In the proof of every free R-module is projective, where do we use being free?

Projective modules

Finitely generated projective modules are locally free

Projective modules over a semi-local ring

Projective but not free (exercise from Adkins - Weintraub)

A module is projective iff it has a projective basis

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