# Tagged Questions

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

65 views

### Why is a graded $\mathbb{C}$-algebra free as a module over the subring generated by a regular sequence?

I am reading Bernd Sturmfels' book Algorithms in Invariant Theory. On p. 38 he makes the following assertion: "If $\theta_1,\dots,\theta_n$ are algebraically independent over $\mathbb{C}$, then the ...
37 views

### Closure of a rational point is irreducible

Let $X$ be a scheme of finite type over a field $k$ which is algebraically closed of characteristic zero. Let $K$ be another field and $\eta \in X(K)$ be a $\mathrm{Spec}(K)$ point of $X$. Is the ...
39 views

Let $X$ be a normal affine variety over $\mathbb{C}$. Q1. Let $x\in X$ be a singularity with a Cartier divisor $x\in D$. Then $\mathcal{O}_{X,x}$ is Cohen-Macaulay if and only if $\mathcal{O}_{D,x}$ ...
35 views

### Underlying abelian group of coproduct of commutative rings

If $A$ and $B$ are commutative rings (we assume) we can construct their coproduct $A \sqcup B$ with the inclusion maps $i_a$ and $i_b$. I'm trying to show that this coproduct has underlying abelian ...
23 views

### Is LM($f_ig_i$) necessarily equal to $LM\sum(f_ig_i$?)

Given $f_i,g_i\in k[x_1,\cdots,x_n],1\leq i\leq s$, I am asked if $$\text{LM}(\sum_{i=1}^sf_ig_i)=\text{LM}(f_i)\text{LM}(g_i),$$ is true for some $i$. I saw similar questions but not quite this ...
29 views

### On the definition of a singular point of a hypersurface over $\mathbb{F}_q$

Let $\mathbb{F}_q$ be the finite field of $q$ elements. I have a homogeneous polynomial $F \in \mathbb{F}_q[x,y,z]$. It then defines a projective hypersurface $V(F)$. Is a point on $V(F)$ a singular ...
68 views

### Tensor product of two fields

Let $F, k'$ be two fields containing a given field $k$. The book I'm reading (Borel, Linear Algebraic Groups) uses some facts about the structure of the tensor product $F \otimes_k k'$, for example ...
13 views

### Hilbert Syzygy theorem example [duplicate]

Suppose we have $K[x,y]$ with ideal $(x,y)$. How can we get a free resolution of it which terminates. I mean that Hilbert syzygy theorem tells us that there exists a resolution would be finite. But I ...
37 views

### Colimits of ideals

I'm reading a few commutative algebra books right now and noticed that many texts take filtered colimits (direct limits) of ideals of different rings, but never really seem to actually say where ...
59 views

### A minimal primary decomposition of a radical ideal is a prime decomposition.

I want to prove that, if $I$ is a radical ideal in a Noetherian ring, and if $I=Q_1\cap\cdots\cap Q_r$ is a minimal primary decomposition (i.e., each $Q_i$ has a distinct radical, and no $Q_i$ ...
24 views

### Ideal generated by polynomials and linear dependence

I've been thinking about this for almost a day and I have given up. I just get stuck in an invalid argument and dunno how else to do this. So the question is: Let $s>1$ and let $f_1,...,f_s$ ...
18 views

### Do these principal open sets really form a basis for the $F$-topology?

Let $F$ be a subfield of $k$ algebraically closed, and $k[\mathscr X]$ be the affine $k$-algebra associated with a Zariski closed set $\mathscr X \subseteq k^n$, and suppose $I(\mathscr X)$ can be ...
49 views

### Question on projective module

Let $R$ be a ring an $M$ be a projective $R$-module. Show that there exists a free $R$-module $F$ such that $$M\oplus F\cong F.$$ Any hints?
19 views

### Proving Spec(A) is Quasi-Compact/definition of sum of ideals [duplicate]

Given a ring $A$ we can make $Spec(A)$ with the Zariski topology. A question in atiyah and macdonald asks you to prove that $Spec(A)$ is quasi-compact. It is clear that if $Spec(A)$ is covered by open ...
77 views

### Krull dimension $\leq$ transcendental degree

I am trying to solve Exercise 11.32 from the online notes of Gathmann on Commutative Algebra: Let $K$ be a field and $B$ a not necessarily finitely generated $K$-algebra that is a domain. Show ...
28 views

### Prove that A is isomorphic to the subring of K[T] generated by $T^2$ and $T^3$ [duplicate]

Let $A = K[x,y]/\langle y^2-x^3\rangle$. Show that $A$ is isomorphic to the subring of $K[T]$ generated by $T^2$, $T^3$. I tried to show a map from $\phi : A \to K[T]$ which sends $x \to T^2$ and ...
41 views

There is a post about this problem already in this forum here http://goo.gl/PymvS7. But I was wondering if you can compute the solution without using roots. Beggining as the tip says, completing ...
32 views

### Orthogonal complement of finite free modules

Let $A$ be an integral domain (can assume it to be a regular $\mathbb{C}$-algebra), $M$ a finite-dimensional, free $A$-module and $N$ finite dimensional, free submodule of $M$. Does there exist a ...
37 views

### Regular sequence in rings of invariants

When I was reading a book in invariant theory, I've come across this assumption: "Let $f_1, f_2 \in \mathbb{F}[V]^G$ be a regular sequence in $\mathbb{F}[V]$". I know that we cannot assume anything ...
39 views

### For a graded ring, does localization and “taking degree zero part” commute?

If $S$ is a graded ring, and $f$ and g are nonzero homogeneous elements, then $((S_f)_0)$ has a map to $(S_{fg})_0$ induced by the localization maps, and the universal property of localization ...
68 views

### Looking for a surjective map which is not injective over an artinian module.

I would like to know if there exists a surjective map $f : \mathbb{Z}_{(p)} \to \mathbb{Z}_{(p)}$, which is not injective. Thanks a lot for your help.
77 views

### Dimension of quotient module

Let $A$ a Noetherian local ring and $M$ a finite $A$-module. How can I prove that if $x$ is $M$-regular then $$\dim M=\dim M/xM+1.$$ I had a proof with the hypotesis that ...
41 views

### About the existence of a UFD whose power series ring isn't a UFD

I'm trying to understand the proof given by Samuel in this paper of the existence of a UFD whose power series ring is not a UFD. I'm stuck at point (c) in the proof of theorem 4.1 at this statement ...
56 views

### Normal irreducible quadrics in $\mathbb{P}^3$

How can I show that an irreducible quadric $Q$ in $\mathbb{P}^3$ is normal? If $Q$ is non singular then the local ring associated to every point is a UFD and so a normal ring, but how can I do ...
64 views

### Prime ideals in $A[x_1, \ldots,x_n]$

Let $A$ be a commutative, Noetherian ring and let us define a monomial ordering, $\prec$ on $A[x_1, \ldots,x_n]$. My doubt is regarding the maximal chain of prime ideals in $A[x_1, \ldots,x_n]$. When ...
23 views

### Non-existence of commutative rings with many nilpotent elements with some restrictions where matrix powers are efficient

Cross-posted from MO. At the moment can't find better reference than "Cycle Enumeration using Nilpotent Adjacency Matrices with Algorithm Runtime Comparisons" though certainly there are others. ...
20 views

### Definition of filtration over monoid

I want to know if the following definition is correct. A $\textbf{filtration}$ over a monoid $M$ (operation denoted multiplicativity), is a total order $<$ on $M$ that fulfills $1_M < x$ and if ...
53 views

### generalized affine scheme

I'm thinking about following theorem. For a finitaly algebraic theory $\mathbb{T}$, $\text{FP}\mathbb{T}$ denotes the full subcategory of $\mathbb{T}\text{-Alg(Set)}$ consisting of finitely presented ...
64 views

### Axiomatization of the equational theory of ideals in a commutative ring

Is there a known axiomatization of the equational theory of ideal operations in a commutative ring? I have in mind the following: Consider a language with operations for ideal intersection, product, ...
24 views

### Extending McCoy's theorem to multiple indeterminates [duplicate]

So, working in a commutative ring with unity $R$, I've proven that $f\in R[x]$ is a zero divisor iff there exists $s\in R$ such that $sf=0$. I'm now being asked the followup question to extend ...
82 views

### Maximal ideal of polynomial ring over a subfield

Let $L/K$ be an algebraic extension of fields. Let $B = L[X,Y]$ and $A = K[X,Y]$. Suppose $a$, $b \in L$ and $m = (X-a,Y-b)$ is an ideal of $B$. Show that $m$ and $m \cap A$ are maximal ideals of ...
46 views

### characterization of integral closure?

I would like to know whether there exists any characterization of integrally closed domains which is related to some morphisms construction with $\mathbb{Z}$ and $\mathbb{Q}$. I was thinking about ...
47 views

### Jacobson radical of an indecomposable commutative ring

Let $R$ be a commutative indecomposable ring with identity which has infinitely many maximal ideals. Can we deduce that the Jacobson radical of $R$ (the intersection of all maximal ideals) is the zero ...
40 views

66 views

### Having only the zero as a nilpotent element is a local property

I want to show that having only the zero as a nilpotent element is a local property for a Ring $R$. Assume $R$ only has the zero element as a nilpotent element and there exists a prime ideal $p$ ...
48 views