# Questions tagged [commutative-algebra]

Questions about commutative rings, their ideals, and their modules.

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### Sharp's Exercise 16.37

I am reading Sharp's book "Steps in Commutative Algebra". But I really had difficulty in the exercise below. So can you give me a hint about the exercise below? Exercise 16.37 Let $R$ denote the ...
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### Factorisation of a homogeneous polynomial over a domain.

Fulton's book on Algebraic Curve asks a question that, over a domain R, a factor of a form (by which he means a homogeneous polynomial) in indeterminates $x_1,.., x_n$ is also a form. For one ...
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### Separated morphisms are stable under base change

Suppose that the map $f$ in the following diagram is a separated morphism (i.e. $\Delta_{X/S}:X\rightarrow X\times_{S}X$ is a closed immersion). I want to prove that $p_{2}$ is also a separated ...
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### Bijection between $\text{Spec}(A\otimes_R B)$ and $\text{Spec}(A)\times_{\text{Spec}(R)}\text{Spec}(B)$ for $R$-algebras A and B.
I'm looking for a bijection between $\text{Spec}(A\otimes_R B)$ and $\text{Spec}(A)\times_{\text{Spec}(R)}\text{Spec}(B)$ for $R$-algebras $A$ and $B$. Where \$\text{Spec}(A)\times_{\text{Spec}(R)}\...