# Questions tagged [local-rings]

In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.

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### $aX + bY$ is an element of $M^2$ if and only if the line $aX + bY = 0$ is tangent to $W$ at $(0, 0)$

Let $F(X, Y) = Y^2 - X^3 + X \in \mathbb{C}[X, Y]$, and let $a$ and $b$ be constants (elements of $\mathbb{C}$). Write $W = V(F)$, and let $P$ be the point $(0, 0)$ on $W$. Show that $aX + bY$ is an ...
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### local ring of variety over not necessarily algebraically closed field

Let $V$ be an affine variety. The ideal of $V$ is $I(V) = \{f\in \bar K[X] \mid f(P)=0\;\forall P\in V \}$. If $I(V)$ is generated by elements in $K[X]$, the $V$ is said to be defined over $K$ and ...
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### Is there a non-regular depth 1 noetherian local ring with this property?

Let $(R, \mathfrak{m})$ be a non-regular depth 1 noetherian local ring. Then if $x$ is any regular element of $R$ the module $R/(x)$ will have depth 0, and so it has $\mathfrak{m}$ as an associated ...
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### Exact functors in local cohomology

In local cohomology the ideal transform functors with respect to a pair of ideals are defined by $D_{I,J}(-)=\underset{\textbf{a} \in \tilde{w‎‎} ‎(I,J)‎} {\varinjlim}\,\,\text D_{\textbf{a}}(-)$. ...
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### Exponent = index in Brauer group over the quotient field of an Iwasawa algebra

Let $K$ be a field and $B(K)$ its Brauer group. We know that for all $[A]\in B(K)$, $$\exp[A]=\operatorname{ind}[A]$$ when $K$ is the quotient field of a complete DVR or when $K$ is a global field. (...
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### My trial for showing that $K[[x]]$ over a field is a local ring.

Here is the question I want to answer letter $(b)$ in it: A commutative ring $R$ is local if it has a unique maximal ideal $\mathfrak{m}.$ In this case, we say $(R, \mathfrak{m})$ is a local ring. For ...