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Results tagged with ac.commutative-algebra
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user 5480
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
16
votes
Accepted
maximal ideals of $k[x_1,x_2,...]$
If $|k| > |I|$ then the usual cheap proof of Nullstellensatz still works: let $K$ be a residue field. Then $\dim_k K \le \dim_kR = |I|$, but if $t\in K$ is transcendental over $k$, the elements $1/(t- …
- 2,441
7
votes
Accepted
Abelian varieties over local fields
$V$ is an irreducible $\mathbb{F}_p$-representation of $I$. As $P$ is a pro-$p$ group, $V^P\ne 0$, and $P$ is normal in $I$ so $V^P$ is stable under $I$. Therefore $V^P=V$. You might like to look at T …
- 2,441
15
votes
Accepted
Explicit elements of $K((x))((y)) \setminus K((x,y))$
Suppose $R$ is a domain with field of fractions $F$. Let $f\in F[[y]]$ and suppose that $f\in Frac(R[[y]])$. Then $f=h/g$ with $g,h\in R[[y]]]$ and we may assume that $g=b_0+b_1y+\dots$ with $b_0\ne0$ …
- 2,441
12
votes
Two questions about finiteness of ideal classes in abstract number rings
Here is an answer to your "question 0" - an example of an "exotic" number ring. (This should be a comment but it is too long.)
Construct a sequence of number rings $\mathbb{Z}=R_0\subset R_1\subset …
- 2,441