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

learn more… | top users | synonyms (1)

0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
39 views

About singularity

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 $\mathcal{O}_{D,x}$ ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 $Q_i$ ...
0
votes
0answers
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 $f_1,...,f_s$ ...
0
votes
0answers
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 ...
0
votes
0answers
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?
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
41 views

Quadratic integral dependence

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 ...
0
votes
1answer
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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.
0
votes
1answer
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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. ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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, ...
0
votes
0answers
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 ...
0
votes
1answer
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
40 views

The trace ideal of a non zero $R$-module

Let $R$ be a commutative ring with identity and $M$ be a cyclic $R$-module, we may define the ideal $tr(M)$ associated with $M$, the sum of the ideals $f(M)$, for all $R$-homomorphisms $f \in ...
0
votes
0answers
29 views

Question about Macaulay2

Let $R=\mathbb Q[s^4,{s^3}t,s^{5}t^3,t^4]$. With Macaulay2 how do I find the isomorphism $\text{gr}_{m}(R)=Q[x,y,z,w]/q$, where $q=(-z^2,yz,xz,-y^4+x^3w)$ and $\text{gr}_{m}(R)$ is the associated ...
0
votes
0answers
45 views

Continuous maps are morphisms of varieties?

From the definition of morphism (Hartshorne's Algebraic Geometry) it looks like all the continuous maps are morphisms. Let $\phi: X \rightarrow Y$ be a continuous map and $f:Y \rightarrow k$ be a ...
0
votes
1answer
126 views

Maximal Ideals of $\mathbb C[x, y]$

I recently learnt that the maximal ideals of $\mathbb C[x, y]$ are precisely the ones of the form $(x-a, y-b)$ for some $a, b\in \mathbb C$. I am unable to prove it. So I considered an easier ...
0
votes
0answers
28 views

Isomorphism of polynomial rings [duplicate]

I am trying to do exercise 3.6.F in Ravil Vakil's algebraic geometry notes : http://math.stanford.edu/~vakil/216blog/FOAGapr2915public.pdf We fix a field $k$. It comes down (or so I think) to proving ...
0
votes
0answers
69 views

Krull dimension of $A[x_1, \ldots, x_n]/\mathfrak{a}$

What is the Krull dimension of $A[x_1, \ldots, x_n]/\mathfrak{a}$ where $A$ is a Noetherian, commutative ring and $\mathfrak{a} = \langle f_1, \ldots, f_s \rangle$, where $\{f_1, \ldots, f_s\}$ ...
0
votes
0answers
46 views

What is $\overline{D(f)}$?

Let $A$ be a ring, $f\in A$. If $A$ is Noetherian, $\text{Spec}(A)$ has finitely many irreducible components, let us call them $\{Z_i\}_{i=1}^n$. So we write $$D(f)=\bigcup_{i=1}^n D(f)\cap Z_i. $$ ...
0
votes
0answers
35 views

How to simulate Permanental Point Process

I have simulated a determinantal point process in a square grid using Gaussian Kernel. The Gaussain matrix is decomposed into its eigenvectors and eigenvalues. In core implementation, the elementary ...
0
votes
0answers
38 views

which powers of maximal ideal contain/are included. the notation

Let $R$ be a (associative, commutative) local ring, denote by $\mathfrak{m}$ its maximal ideal. For any other ideal $J\subset R$ one can speak about: the biggest power $k\le\infty$ such that ...
0
votes
0answers
39 views

Some confused terminology in Matsumura's textbook about completion

I am reading Cohen structure Theorem in textbook "commutative algebra" by matsumura. Here, the author repeatly mention "Assume that $A$ is a complete and separated local ring". One can easily know ...
0
votes
0answers
64 views

Grobner bases of a determinantal ideal

I've been studying algebraic geometry recently and there is a problem I'm struggling with: Suppose $A$ is a $m\times n$ complex matrix of rank $\leq r$, this is equivalent to all its $(r+1)\times ...
0
votes
0answers
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 $p$ ...
0
votes
0answers
48 views

How to understand this sentence within Atiyah-Macdonald's textbook about commutative algebra

In page 102 of this textbook, authors mentioned that: Assume topological group $G$ has a fundamental system of neighborhoods consisting of subgroups as: $G= G_0 \supseteq G_1 \supseteq\cdots\supseteq ...
0
votes
0answers
30 views

Projective ideal of a Non-noetherian domain

If $R$ is an integral domain which is not Noetherian and let $I$ be an ideal which is not finitely generated. We have always, if I is invertible, then I is always projective. Is the converse true when ...
0
votes
1answer
133 views

Trying to use the Zariski topology in a problem without knowing scheme theory.

I don't know scheme theory, and I am doing a problem and the solution involves making conclusions based on the Zariski topology, and I want to make sure that I am "intuiting" things correctly when ...
0
votes
1answer
35 views

Associated prime of $M/Q$ where $Q$ is $\mathfrak{p}$-primary

I need check if my statement is true and proof check (for some reason I couldn't find this anywhere): Let $Q$ be a $\mathfrak{p}$-primary submodule of $A$-module $M$. Then $\mathfrak{p}$ is the ...
0
votes
0answers
60 views

Projectivity of a (prime) ideal in a noetherian integral domain

Assume $R$ is a noetherian integral domain (and assume $R \neq k[x_1,\ldots,x_n]$), $I$ is a non-zero ideal of $R$ ($I$ is finitely generated, since $R$ is noetherian), and $I$ is not necessarily ...