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Results tagged with ac.commutative-algebra
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user 5498
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
3
votes
Under the condition specified below, is $\mathcal{O}_X(X-V(I))=R$, where $X=\mathrm{Spec}R$?
Yes, it is true. This is a special case of [EGA IV$_2$, 5.10.5] (see also 5.9.9 if needed) combined with Serre's criterion for normality (see http://stacks.math.columbia.edu/tag/031S). The latter tell …
- 5,009
2
votes
irreducibility of general fiber
It seems to me that what you may be looking for is [EGA IV$_3$, 9.7.8], which says in particular that if $S$ is an irreducible scheme with function field $K$ and $X$ is an $S$-scheme of finite present …
- 5,009
2
votes
Accepted
Iwasawa algebra
For such an $M$, set $\lambda^{\prime}(M) := \dim_{\mathbf{F}_p} M/pM - \dim_{\mathbf{F}_p} M[p]$; since $\mu(M) = 0$, the dimensions are actually finite---this can be extracted from the argument belo …
- 5,009
1
vote
Metrizability of $\mathfrak{a}$-adic topology
No, it is not true in general. Take $\mathfrak{a} = A$ and $A$ nonzero to get that $A$ equipped with the $\mathfrak{a}$-adic topology is not Hausdorff and hence not a metric space (since $|A| > 1$). M …
- 5,009
3
votes
1
answer
930
views
Module of Kahler differentials of rings of integers of number fields
How does one prove that if $L/K$ is an extension of number fields with rings of integers $B/A$, then the module of Kahler differentials $\Omega^1_{B/A}$ can be generated by one element as a $B$-module …
- 5,009
26
votes
Accepted
Does completion commute with localization?
It is true. $(\widehat{A}, \widehat{\mathfrak{m}})$ is a Noetherian local ring so your left hand side could be simplified replacing it by $\widehat{A}$. Now let's use the definitions: $\widehat{A} = \ …
- 5,009
19
votes
Accepted
Torsors and the fpqc topology
One way to define an fppf (or etale, or...) torsor is to require that $G \times_S P \rightarrow P \times_S P$ given by $(g, p) \mapsto (gp, p)$ is an isomorphism and that $P$ has a section fppf (resp. …
- 5,009