# Questions tagged [commutative-algebra]

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

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### $p$-adic and mod $p$ relation on tensor

Let $G$ an abelian group. $\Bbb Z_{(p)}$ is the $p$-adic integers. What can one say about $\Bbb Z/p \otimes G$ given that $\Bbb Z_{(p)} \otimes G \not= 0$? Is it possible to conclude whether the ...
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### On an isomorphism of rings

Let $k$ be a field, it's claimed in algebraic geometry textbook that $k[v^2, v^3]\cong k[t, u]/(t^2-u^3)$ via $v^2\mapsto u, v^3\mapsto t$. But I can't show it's well-defined, since an integer can ...
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### Proof of Quillen's patching theorem

The following are from Lam's book Serre's problem on projective modules, on page 163 and 164. For any ring $A$, the notation $m \in \mathscr{R}^A(A[t_1, \dots, t_n])$ means that there exists a $A-$ ...
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### Showing surjectivity of the trace map $B\to A$ from faithful flatness

I've come across the claim if $A\hookrightarrow B$ is a finite etale extension of rings (commutative w/ $1$) with $A$ Noetherian then the trace map $\operatorname{Tr}_{B/A}:B\to A$ is surjective and ...
I am just trying to cross check my answer as it slightly differs from https://math.berkeley.edu/~reb/courses/256A/1.5.pdf to be sure of any mistake I am making. Here $k$ is an algebraically closed ...