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

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1answer
59 views

Obtain dimension of multivariate polynomial quotient ring?

Let $\mathbb{C}[z_1,z_2,...,z_n]$ be the ring of multivariate polynomials in complex variables $z_1,z_2,...,z_n$ with complex coefficients. This ring is spanned by the countably infinite basis of ...
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0answers
27 views

$\Bbb Z$-graded ring with no nonzero homogeneous prime ideals

Exercise $2.18$ in Eisenbud's algebra book asks to prove: Suppose $R=\bigoplus_{n=-\infty}^\infty R_n$ is a $\Bbb Z$-graded ring such that any homogeneous prime ideal is zero. Prove $R_0$ is a field. ...
1
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1answer
26 views

Extensions between integral domains give extensions of fields of the same degree.

Assume that $S \subset R$ is a ring extension where, both $S$, $R$ are integral domains. Furthermore, assume that $R$ is a free $S$-module of rank $n$. Is it true that the extension of fields $\mathrm{...
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0answers
40 views

Hilbert series of polynomial ring

Let $k$ be a field and $S=k[x]$ a polynomial ring with $\deg x = 1$. Then we know that $H_S(t) = \frac{1}{1-t}.$ Let $S'=k[y]$ with $\deg y = -1$ or $y=x^{-1}.$ Then I think that $H_{S'}(t)= \frac{-t}{...
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0answers
55 views

When is a finite $R$-algebra isomorphic to $R$?

Let $R$ be a $\bar{k}$-algebra (of finite type or complete) reduced (and maybe integral, if needed), let $A$ be an $R$-algebra, finite as an $R$-module, reduced and connected and such that there ...
2
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0answers
49 views

Making sense out of $F$-structures and the notion of $F$-variety

For almost two years I have been trying to make sense out of several claims about varieties over nonalgebraically closed fields made in the first chapter of the textbook Linear Algebraic Groups by T.A....
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0answers
66 views

Isomorphism between a quotient of a polynomial ring and a polynomial ring [closed]

I'm going to show that $\mathbb{K}[x,y,z]/(y^2-xz)$ is not isomorphic to any polynomial ring. I'll be grateful of someone brings a hint to show this result. Thank you.
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0answers
25 views

If $x\in \mathrm{Ann}(N)$ then $x$ annihilates $\mathrm{Ext}_i(N,M)$ for all $i$, why? [duplicate]

Matsumura in his Commutative Ring Theory, for the proof of theorem 16.6, uses a fact as follows: Let $A$ be a unital commutative ring, $N$ a (finitely generated?) $A$-module, and $M$ any $A$-...
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0answers
32 views

must this extension of a DVR be unramified?

Let $A$ be a normal domain, and $P$ a height 1 prime, then $A_P$ is a DVR. Let $K$ be the fraction field of $A_P$, and let $L$ be a finite Galois extension of $K$ of degree $e$, let $B$ be the ...
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2answers
499 views

How to understand “tensor” in commutative algebra?

Tensor is sure an important concept in commutative algebra, but the definition is kind of abstract, so is there any way to understand it which is easier? Thanks advance! The definition I see is the ...
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1answer
46 views

Atiyah–Macdonald exercise 14 chapter 1

So here is the part of exercise 14 of chapter 1 that has been bothering me: Let $A$ be a commutaive ring with identity. Let $\Sigma $ be the set of ideals with the property that every element in them ...
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2answers
135 views

Why is $\mathrm{Spec}(\mathbb{Z})$ a terminal object in the category of affine schemes?

I've seen this claim repeated in many places (always without source or proof), that $\mathrm{Spec}(\mathbb{Z})$ is a terminal object – however, the most I've been able to prove myself is that for any ...
4
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1answer
50 views

Geometric interpretation of a result from commutative algebra

I have come across the following result in Hartshorne, $I.6.5$ for those who have the book. The result says that if $K$ is a finitely generated extension of some base (algebraically closed) field $k$ ...
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0answers
28 views

Is the normalization of the cusp $H$-projective?

I have a slight confusion about a statement I think to be true and if the normalization of the cusp is "good enough", this would sadly provide a counterexample. So here is my hope, that the ...
1
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2answers
38 views

assuring factorization for R[x] when R is a UFD

I wanted to ask, suppose the ring $R$ is a UFD (Unique factorization domain) and I look at $R[x]$, the ring of polynomials over $R$. I wanted to know, how can I assure that when I have some polynomial ...
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0answers
56 views

Canonical sheaf of product

This is a follows up to Canonical divisor of product of varieties. I am in the case where $X$ is a smooth variety over a field $k$, and $K$ an algebraic closure of $k$. How can i express the ...
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0answers
22 views

Castelnuovo-Mumford regularity of the sum of two ideals

Let $R=k[x_1,x_2,x_3,x_4,y_1]$ be a polynomial ring, $I=\left<x_1x_2,x_2x_3,x_3x_4\right>$ and $J=\left<I,x_1y_1\right>$. We have $\mathrm{reg}(I)=2$ and $\mathrm{reg}(J)=3$, where $\...
3
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1answer
35 views

Sections of tensor bundle are tensor product of sections

Given $E,F$ vector bundles over a manifold $M$, I would like to know a proof of $\Gamma(E\otimes F) = \Gamma(E) \otimes_{C^\infty(M)} \Gamma(F)$. Where $\Gamma$ denotes the smooth sections over $M$. ...
4
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1answer
69 views

Why is the category of finitely generated modules over a non-noetherian ring not abelian?

I am learning about abelian categories for a talk I have to give next week. One of the first questions I had upon learning this definition is "does there exist an additive category that is not abelian?...
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2answers
44 views

Examples of $R$-modules $X$ such that $(X \setminus TX) \cup \{0\}$ isn't a submodule.

Work over an ambient commutative ring with unity. Given a module $X$, write $TX$ for its submodule of torsion elements. Suppose we want to find the "submodule" of torsion-free elements of $X$. So ...
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1answer
55 views

Comparing the prime spectra of $\mathbb{Q}[x],\mathbb{R}[x]$ and $\mathbb{C}[x]$

I understand that $\mathrm{Spec}\mathbb{Q}[x]=\{(0),(f(x)): f(x)\mbox{ is an irreducible polynomial}\}.$ An argument is the following one: $(0)$ is prime because $\mathbb{Q}[x]$ is an integral domain. ...
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0answers
30 views

S is saturated if and only if R\S, the complement of S in R, is the union of some (possibly empty) family of prime ideals of R [duplicate]

I got problem which is the same here but I can not solve follow the way 2 of them pointing out Show $R \setminus S$ is a union of prime ideals Please help me explain more. And one more thing I wanna ...
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0answers
21 views

What are minimal paths, generators of graph ideal in a cyclic graph $C_n$?

Minimal cuts are the generators of the cut ideal while the Alexander duality of path ideal generated by the minimal paths is the cut ideal -- more on Graph ideals here. Graph ideals are special case ...
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40 views

Quotient field of an affine domain is again affine algebra. [on hold]

I don't know if my doubt is true or not. My question is the following: If a domain is finitely generated $k$-algebra ($k$ is a field), can we say it's quotient field is finitely generated field ...
2
votes
1answer
60 views

Which affine schemes are projective?

Let $k$ be a field. Are there any useful necessary and sufficient conditions on $k$-algebras $A$ such that $\mathrm{Spec}(A)$ is a projective scheme over $k$? I know that there are very few of these, ...
2
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0answers
53 views

Understanding the prime ideals in the ring of dual numbers over a field

I want to understand $\mathrm{Spec}(k[\epsilon]/(\epsilon^2))$ where $k$ is an algebraically closed field and $R:=k[\epsilon]/(\epsilon^2)$ is the ring of dual numbers. Here is my attempt: Every ...
2
votes
1answer
65 views

General form of Nullstellensatz

At lots of places it is stated, that Hilbert's Nullstellensatz is well understood as theorem about more general Jacobson rings. Namely (When $R$ is a Jacobson ring and $S$ finitely generated $R$-...
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2answers
30 views

In a local ring, the maximal ideal $\mathfrak{m}$ is principal $\implies \dim_k(\mathfrak{m}/\mathfrak{m}^2)\leq 1$

This is Proposition 8.8, $ii)\implies iii)$ in Atiyah and Macdonald and it says there that this is clear. It isn't for me though. How would I prove this?
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1answer
58 views

What can I say about the maps $\text{Spec} (A / \mathfrak{a}) \to \text{Spec} (A)$ geometrically?

I was curious whether there is a general approach to say something about the geometric interpretation of the maps $\text{Spec}(A / \mathfrak{a}) \to \text{Spec}(A)$ for a commutative ring with unity $...
2
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1answer
22 views

Field is an Artinian module

I am going through theorem 2.14 in Eisenbud's Commutative Algebra. Given a ring $R$ that is Noetherian, all of whose prime ideals are maximal, we want to prove that $R$ is Artinian. Assume that $R$ ...
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1answer
53 views

Is a fiber product of flat morphisms flat?

Suppose we have morphisms of schemes $f : X\rightarrow S$ and $g : Y\rightarrow S$, and a morphism $Z\rightarrow X\times_S Y$ such that the induced morphisms $Z\rightarrow X, Z\rightarrow Y$ are flat. ...
2
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0answers
33 views

Maximal unramified intermediate extension of DVRs

Let $A\rightarrow B$ be a finite tamely ramified extension of discrete valuation rings. Does there exist a DVR $C$ such that $A\subseteq C\subseteq B$ with $C$ unramified over $A$ and $B$ totally ...
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3answers
55 views

Integral closure of Gaussian Integers

I am considering $\mathbb{Z}[i]\subset\mathbb{Q}(i)$ Now I have a short note here that says that there are elements of $\mathbb{Q}(i)$ which are not integral over $\mathbb{Z}[i]$ 'because $\mathbb{Q}(...
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0answers
35 views

Dimension under integral local homomorphism

Let $f:(R,m) \to (S,n)$ be an integral local homomorphism. Let $p$ be a prime ideal of $R$ not equal to $m$. I want to know if one can claim $\dim S/f(p)S\neq0 $. This is true when $(f(p)S)^c=p$, ...
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0answers
39 views

A monomorphism from an $R$-module to $R$

Let $R$ be a commutative ring with unity possessing an element $r$ in the singular ideal $Z(R)=$ the set of elements whose annihilators are essential in the module $R_R$, and let $M$ be a faithful $R$...
1
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1answer
19 views

Normal closure of a number field and a possible quadratic field in it

While reading about prime decomposition in number fields, I came across following statement (stated as a fact): Let $K$ be a number field and $d= \text{disc}(\mathcal{O}_K)$, then the normal ...
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3answers
202 views

Examples of asymmetrically braided monoid

From nCatlab https://ncatlab.org/nlab/show/braiding : Any braided monoidal category has a natural isomorphism $$B_{x,y} \;\colon\; x \otimes y \to y \otimes x $$ called the braiding. ...
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0answers
35 views

Flatness as an algebra vs as a module

I was doing an exercise where one of the assumptions was that $B$ is a flat $A$-algebra... I realized that this might be ambiguous since its possible that $B$ is flat in the category of $A$-algebras ...
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0answers
41 views

primes of the strict henselization

I'm trying to get some intuition for the (strict) henselization of a local ring. Let $A$ be a local ring with maximal ideal $m$. I'm happy to assume it is Noetherian and normal. Let $p\subset m$ be a ...
1
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1answer
42 views

Splitting of primes in real cyclotomic field

The question is from Marcus' book, "Number Fields" (exercise 12, Chapter 4) Let $\omega= e^{\frac{2\pi i}{m}}$ and $p$ be a rational prime not dividing $m$. Then how does $p$ split in $\mathbb{...
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0answers
36 views

The $\mathrm{Proj}$-construction and inverse limits

I have a couple of questions about existence of certain inverse limits in the category of schemes (I am also happy about links to relevant literature... in the stacksproject I only found the affine ...
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1answer
31 views

What is a minimal prime ideal of a ring

From Wikipedia: A prime ideal P is said to be a minimal prime ideal over an ideal I if it is minimal among all prime ideals containing I. (Note that we do not exclude I even if it is a prime ...
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0answers
10 views

What are Hilbert Series on Graph Ideals for?

Partially related on Hilbert Series of Monomial ideals but I want to understand the purpose of Hilbert Series on Graph Ideals. Example on the cycle graph $C_4$ with $x_1,x_2,x_3$ and $x_4$ in corners:...
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0answers
33 views

What does Hilbert series of Monomial ideal describe?

I am trying to understand the point of hilbert series of monomial ideals. I am confused because Macaulay has commands for hilbertSeries, hilbertPolynomial and hilbertFunction. What does Hilbert ...
3
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1answer
28 views

Factorization of primes in normal closure of Quartic Field

Motivation for the question comes from Marcus' book on Number Fields (exercise 13, Chapter 4). Let $K= \mathbb{Q}[\sqrt[4]{m}, i]$ where $i=\sqrt{-1}$, $m\in \mathbb{Z}$ and $m$ is not a square. ...
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0answers
88 views

Finding the normalization of $K[X,Y]/(Y^5-X^7-XY^5)$

I understand that for "relatively simple" cases, we can compute the normalization of a coordinate ring, such as $K[X,Y]/(Y^2-X^3)$, quite easily (consider $Y/X$). However, how would one approach ...
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2answers
38 views

Generalizing concept of content of a polynomial to commutative rings [duplicate]

Let $A $ be a commutative ring with identity. Let $f,g\in A [x] $. Let $I_1,I_2, J $ be the ideals generated by the coefficients of $f,g,fg $ respectively. Must $J $ be equal to $I_1 I_2$ ? It is an ...
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0answers
49 views

Is a smooth ring a domain?

I know that smooth and regular is "quasi" the same and that a regular local ring is a domain. Here I start with the definition $A \to B$ is smooth if and only if for every square zero extension of $A$-...
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0answers
15 views

Demonstrations on the Stanley-Reisner Ideal of Simplical Complex of Graph

The simplicial complex of graph is defined here and I want to understand its Stanley-Reisner ideal where I cannot understand the point "such that there is no face of $\Gamma$ with vertices $x_{...
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1answer
26 views

Demonstrations on the Simplicial complex of Graph

where I cannot understand $F\in\Gamma\land G\subseteq F\Rightarrow G\in\Gamma$. I would like to see an example about the simplicial complex of a graph such as a cycle graph $C_3$. What are ...