# Questions tagged [commutative-algebra]

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

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### Find (a generating set for) $\mathbb{Q}[x]\cap I$ where $I=\langle x^2-y,y^2-x,x^5-x^2\rangle$ (generate gröbner basis).

Consider the polynomial ring $\mathbb{Q}[x,y]$ and the ideal $I=\langle x^2-y,y^2-x,x^5-x^2\rangle$ in $\mathbb{Q}[x,y]$. $G=(x^2-y,y^2-x)$ is a (reduced) gröbner basis for $I$ wrt. graded ...
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### Unsure about proof for Vakil 5.4.M

I've been having some trouble proving the following in Vakil's FOAG recently: Suppose A is a $k$-algebra, and $l/k$ is a finite extension of fields. (Most likely your proof will not use finiteness; ...
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### Transition functions for $\mathcal{O}(1)$ on $\mathbb{P}^1$

I am trying to understand the transition functions for $\mathcal{O}(1)$ on $\mathbb{P}^1$. I've been stuck on this while reading these notes (Proposition 1.8, page 3) on divisors and invertible ...
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### Is a normal domain whose prime ideals are totally ordeded a valuation ring?

Recall one of the definition of a valuation ring is a domain whose ideals are totally ordered. (Then it will be a normal domain.) But if we restrict to all prime ideals the reverse is not true. The ...
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