# Tagged Questions

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

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 ...
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. ...
26 views

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 ...
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....
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.
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 $A$-...
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 ...
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 ...
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 ...
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 ...
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 $k$ ...
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 ...
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 ...
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 ...
22 views

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 $R$ ...
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. ...
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 ...
55 views

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 ...
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 ...
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:...
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 ...
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. ...
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 ...
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 ...
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 $A$-...