# Tagged Questions

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

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### Existence of homogeneous non-unit non-zero divisor in a particular graded ring.

Let $R$ be a finitely generated $k$-algebra of dimension greater than $1$, let $Q$ be any maximal ideal of $R$. It is claimed by my lecturer that one can find a homogeneous, non-unit, non-zero divisor ...
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### Exact sequence of graded modules and localization

I know that a sequence of modules is exact iff the localization at each prime ideal is exact What happens in the case we are working with graded modules? Can we say that a sequence is exact iff the ...
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### Easy explanation on primary decomposition of ideals. [duplicate]

The primary decomposition of an ideal $(x^2, xy)$ is $$(x^2, xy) = (x) \cap (x, y)^2$$ which can be found on these notes. Could someone explain to me how this can be done? Edited: My question ...
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### Ideal of 8 general points in $\mathbb{P}^2$

I am working through chapter 3 of Eisenbud's Geometry of Syzygies. In the first example he makes the claim that the ideal of 8 general points in $\mathbb{P}^2$ is generated by two cubics and a quartic....
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### General procedure to prove something is a tensor product of modules

I'm trying to understand some proofs of statements of the form: Show that some module is the tensor product of two other modules. When I'm looking at these proofs I always see that they start ...
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### Is the localization of an injective cogenerator an injective cogenerator?

We know that in Noetherian rings any localization of an injective module is again an injective module. Is the localization of any injective cogenerator again injective cogenerator?
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### Nullstellensatz theorem- find vector not orthogonal to a given set of vectors in $\Bbb{Z}^n$

Let $v_1,v_2,...,v_r \in \Bbb{Z}^n-\{0\}$. Show there exists $w\in \Bbb{Z}^n$ such that $w_1=\langle w,e_1\rangle=0$ and $\langle w,v_i\rangle \neq 0$ for all $1 \le i \le r$. I've tried to ...
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### Question about S.Lang's proof of Kummer's Lemma

I have a question about the proof of Kummer's Lemma in Serge Lang's Cyclotomic fields (i.e. Theorem 6.1). Let $K = \mathbf{Q}(\xi_p)$ the $p$-th cyclotomic field extension of $\mathbf{Q}$. Let $u$ be ...
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### When does a f.g. algebra over a field $F$ make it “look like $F$ is algebraically closed?”

Let $F$ be a field, and let $A$ be a finitely generated algebra over $F$. If $\mathfrak m$ is a maximal ideal of $A$, then $A/\mathfrak m$ is an algebraic extension of $F$, although it is in general ...
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### Subgroup of idele class group is open

On page 380 of Neukirch's Algebraic Number Theory the author states that the subgroup $$\prod_{\mathfrak{p} \nmid \infty} U_\mathfrak{p} \times \prod_{\mathfrak{p} \mid \infty} K_\mathfrak{p}^\times$$ ...
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### Dimension of an Artin $K$-algebra and cardinal of its spectrum

Let $A$ be an Artin ring that is also a finitely generated $K$-algebra. In particular, the krull dimension of $A$ is $0$. By Noether's Normalisation Lemma we have that $A$ is a $K$-vector space of ...