Questions about commutative rings, their ideals, and their modules.
0
votes
0answers
95 views
Associated primes of a monomial ideal
Let $I\subset K[x_1,\dots,x_n]$ be a monomial ideal, $t\ge 2$ an integer, and $\mathfrak p \in \operatorname{Ass}(R/I^t)$. Then one knows that $\mathfrak p=(I^t : c)$ for some monomial ...
4
votes
1answer
75 views
simple application of Bezout's Theorem
Let $f(x),g(x) \in \mathbb{C}[x_1,\cdots,x_n]$ be two irreducible homogeneous polynomials of degree $n,m$ respectively. Does Bezout's Theorem say that the system of equations $f(x)=0, g(x)=0$ has ...
3
votes
1answer
36 views
Algebraic description of a stalk in the fppf topology
Let $X$ be a scheme and $x\in X$ a point. The stalk of $X$ at $x$ in the Zariski topology is the local ring $\mathcal{O}_{X,x}$. The stalk of $X$ at $x$ in the étale topology is the strict ...
0
votes
1answer
34 views
The number of generators of a submodule over a Principal Ideal Ring.
Can someone give me a hint in proving that if a module $M=\langle m_1,\dots,m_n\rangle$ is generated by $n$ elements over a principal ideal (commutative) ring, then every submodule can be generated by ...
2
votes
2answers
34 views
square-root of a square-free element (Matsumura)
Example 4, page 65 in Matsumura's Commutative Ring Theory reads as follows: "Let $A$ be a UFD in which $2$ is a unit. Let $f \in A$ be square-free (that is, not divisible by the square of any prime of ...
0
votes
0answers
54 views
Give an example of a ring
1) Give an example in which $\dfrac{A}{m^{n}}$ is not an $\dfrac{A}{m}$ algebra
2) Give an example in which ideal not only the $\dfrac{A}{m^{n}}$ chain $D=\dfrac{m}{m^{n}}\subseteq ...
3
votes
1answer
43 views
Correspondence between submodules and quotient modules
What is the (natural) bijection between the set of all sub modules upto isomorphism and set of all isomorphic quotient modules upto isomorphism of a finitely generated torsion module over a PID. Is ...
3
votes
0answers
48 views
Union of Associated Primes being finite.
Let $R$ be a commutative Noetherian ring with unit. Let $I=(x_1,x_2,...,x_t)$ be a nonzero ideal of $R$. Define $I_n=(x_1^n, x_2^n,...,x_t^n)$. Are there known results about $\cup_n Ass(R/I_n)$ being ...
-3
votes
1answer
37 views
Local ring and example
1) Let $A$ be a ring and $m\subseteq A$ maximal ideal
Test for each $n\in N$ the ring $\dfrac{A}{m^{n}}$ is a local ring with maximal ideal $\dfrac{m}{m^{n}}$.
Giving an example to prove that the ...
0
votes
0answers
66 views
Rational functions on the punctured affine plane. [closed]
How to show that the set of rational functions on $\mathbb A^2$ defined on $\mathbb A^2-(0,0)$ is $K[x,y]$. Can this be generalized?
0
votes
2answers
75 views
Valuation but not Noetherian Rings
For valuation rings I know examples which are Noetherian.
I know there are good standard non Noetherian Valuation Rings. Can anybody please give some examples of rings of this kind?
I am very ...
1
vote
1answer
56 views
Looking for a “prime-ish” family of subsets
Is there a nontrivial (what I mean is below) example of a compact Hausdorff space $X$ and a family $\mathscr{F}$ of subsets of $X$ with the following pair of properties?
$\mathscr{F}$ is ...
1
vote
1answer
65 views
Going down theorem
Here $A$ is a commutative ring with unity.
How to show that going down theorem holds for $A$ contained in $A[x]$, the polynomial ring.
Lying over is ok. I cannot do the other part.
3
votes
2answers
94 views
Localization over commutative Noetherian rings
Let $S$ be a multiplicatively closed subset of a commutative noetherian ring $A$. Let $M$ and $N$ be finitely generated $A$-modules. If $M_S$ is isomorphic to $N_S$, show that $M_t$ is isomorphic to ...
12
votes
5answers
283 views
In a principal ideal ring, is every nonzero prime ideal maximal? [duplicate]
Inspired by this question, I was wondering whether from just the hypothesis that $A[X]$ is a nontrivial (commutative) principal ideal ring (so without supposing it is a domain) one can deduce that $A$ ...
4
votes
1answer
39 views
Do Groebner bases give the smallest generating set for Ideals?
Given a Reduced Groebner Basis $(f_1,\ldots,f_n)$ for an ideal $I$, can there be another basis $(g_1,\ldots,g_m)$ for $I$ where $m<n$?
I've been reading through Cox, but can't seem to find an ...
2
votes
1answer
32 views
About injectivity of induced homomorphisms on quotient rings
Let $A, B$ be commutative rings with identity, let $f: A \rightarrow B$ be a ring homomorphism (with $f(1) = 1$), let $\mathfrak{a}$ be an ideal of $A$, $\mathfrak{b}$ an ideal of $B$ such that ...
1
vote
2answers
32 views
$k[x^2,x^3]/p$ ($p$:nonzero prime) is integral over $k$?
Let $p$ be a nonzero prime ideal of $A=k[x^2,x^3]$. I want to show $p$ is maximal.
My trial is that $A/p$ contains $k$ and since $k$ is a field, if I can show that $A/p$ is integral over $k$ then it ...
21
votes
0answers
339 views
A short proof for $\dim(R[T])=\dim(R)+1$?
If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and nontrivial ...
2
votes
0answers
39 views
counting zeros of complex functions
I'm trying to solve the following question : If $f(z,w)=z^2-w^m$ and $g(z,w)=z^2-w^n$, then $O_2/(f)\cong O_2/(g)$ iff $n=m$. $O_2$ is the ring of all holomorphic functions about zero. One way is ...
2
votes
1answer
48 views
Poles of formal power series (Hilbert-Poincaré series)
How are poles and orders of poles of formal power series defined?
The particular case, I am interested in, is the following definition from [Atiyah-Macdonald, Introduction to commutative algebra, ...
-2
votes
1answer
66 views
How I can demonstrate that f is surjective?
If $f:A\rightarrow B$ is a ring homomorphism that for every prime ideal $p\subseteq A$ the homomorphism $f_{p}:A_{p}\rightarrow B_{p}$ is surjective, then $f$ is surjective.
5
votes
2answers
105 views
Ideals in a Dedekind domain localized at a prime ideal
Let $R$ be a Dedeking domain, let $\mathfrak{i}$ be a non-zero ideal of $R$. By factorization theorem we can write
$$\mathfrak{i}=\mathfrak{p}_1^{a_1}\cdots\mathfrak{p}_n^{a_n}$$
for distinct non-zero ...
1
vote
0answers
52 views
Question about the proof of the going-up theorem of Cohen-Seidenberg.
Let $S$ be a subring of $R$ such that $R$ is integral over $S$. Let $P$ be a prime ideal of $S$ and $M=S-P$. Let $S_M$ be the quotient ring of $S$ and $R_M$ the quotient ring of $R$. Let $i: S \to ...
4
votes
1answer
80 views
Associated Primes of Tensor Product
Let $R$ be a Noetherian ring, and let $M$ and $N$ be finitely generated $R$ module. Do we know any formulas for $\operatorname{Ass}(M\otimes_R N)$ in terms of $\operatorname{Ass}(M)$, ...
3
votes
1answer
41 views
Question about algebraically independence.
Let $R=k[Y_1, \ldots, Y_m]/P$, where $k$ is a field and $P$ is a prime ideal of $R$. Suppose that $Y_1, \ldots, Y_m$ are algebraically independent over $k$. Let $y_1=Y_1+P, \ldots, y_m=Y_m+P$. Can we ...
1
vote
1answer
29 views
Finitely generated integral domain and finitely generated $k$-algebra.
Let $k$ be a field and $R$ a finitely generated domain over $k$. Then there are $y_1, \ldots, y_n$ such that $$R=k[y_1, \ldots, y_n]/P,$$ where $P$ is some prime ideal of $k[y_1, \ldots, y_n]$. My ...
4
votes
1answer
36 views
Isomorphism or non-isomorphism of two specific local rings
Let $K$ be a field and set $A=K[X,Y]/(XY)$ and $B=K[X,Y]/(Y^2-X^3-X^2)$.
Are the two local rings $A_{(X,Y)}$ and $B_{(X,Y)}$ isomorphic?
I think that they are non-isomorphic but I can't prove ...
2
votes
0answers
61 views
A commutative ring with alternating and commutativity properties with infinite distinct elements
Is there any nontrivial commutative ring without multiplicative identity that satisfies alternating property ($x \cdot x = 0$ for all $x$ where $\cdot$ is multiplication operator and $x \cdot y \neq ...
2
votes
1answer
62 views
Show that $K[x,xy,xy^2,\dots]$ is not Noetherian [duplicate]
Here is the problem I am stuck on: Fix a field $K$ and consider the subring $A \leq K[x,y]$ generated by $K \cup \{x,xy,\dots,\}$. Show that $A$ is not Noetherian.
I figure that taking ideals $I_n = ...
1
vote
2answers
63 views
Two principal ideals coincide if and only if their generators are associated
Suppose we have a ring $R$ and $(a),(b)$ are both ideals of $R$. Is it always true that $(a)=(b)$ if and only if there exists a unit $c$ such that $a=bc$ (i.e., $a$ and $b$ are associate)?
I ...
4
votes
2answers
66 views
Identifying the ideal generated by the variety $V(y^2-x^3)$
I am having trouble showing the following result:
Suppose that $k$ is an infinite field and consider the affine variety $V(y^2-x^3)$. If $I(V)$ denotes the ideal of all polynomials vanishing on ...
3
votes
2answers
46 views
Irreducibility is preserved under base extension
I want to prove that if $A$ is a finitely generated $k$-algebra ($k$ is a field) with prime nilradical then for any field extension $k\rightarrow K$, the $K$-algebra $A\otimes_kK$ has also prime ...
6
votes
1answer
41 views
Verifying Hilberts Nullstellensatz on a particular example
Let $k$ be an algebraically closed field of characteristic $2$ and consider the following equations:
$$xy + z^2 = 0$$
$$uv + w^2 = 0$$
$$uy + vx = 0$$
It's not hard to parameterize solutions to these ...
3
votes
1answer
84 views
Koszul complex of locally free sheaves
Let $X$ be a complex variety; one can also assume it is smooth if this helps. $\mathcal{E}$ is a locally free sheaf of rank $r$ on $X$, and $s \in H^0(X, \mathcal{E})$. Then one has a Koszul complex ...
3
votes
1answer
98 views
Question about isomorphism of modules.
I have been reading the book of DeMeyer and Ingraham "Separable Algebras of Commutative Rings," where in page 129 they prove the following.
Let
$\bullet$ Let $S$ be a commutative ring and $G$ be a ...
2
votes
1answer
69 views
Associated Prime Ideals in a Noetherian Ring
Let $I$ and $J$ be ideals of a Noetherian ring $A$. Then if $JA_P\subset IA_P$ for every $P\in \operatorname{Ass}_A(A/I)$, then $J\subset I$.
I'm reading Matsumura's Commutative Ring Theory book ...
6
votes
1answer
59 views
Is $\operatorname{Tor}_i(M,N)$ of finite length?
Let $A$ be a regular local ring, and let $M$ and $N$ be two finitely generated $A$-modules such that $M\otimes N$ is of finite length, and let $i$ be the largest integer such that ...
2
votes
1answer
47 views
A necessary and sufficient condition for a full lattice over an integral domain
I'm learning about lattices over integral domains and I would be grateful if someone could clarify the following for me.
Let $R$ be an integral domain with quotient field $K$ where $K\neq R$. Suppose ...
1
vote
1answer
41 views
Integral extension implies that the induced map on prime spectra is closed
Say we have an integral extension $f:R \hookrightarrow S$ of rings. I want to show that the induced map $f^*:Spec(S) \twoheadrightarrow Spec(R)$ is closed. In other words, let $V(I) = \{\mathfrak{P} ...
4
votes
1answer
67 views
Characterization of faithfully flat homomorphisms
Let $A \to B$ be a homomorphism of commutative rings. Why are the following conditions equivalent?
$A \to B$ is faithfully flat.
$A \to B$ is injective, flat and $B/A$ is a flat $A$-module.
This ...
13
votes
2answers
146 views
Is a linear combination of minors irreducible?
Let $X=(X_{ij})_{1\le i,j\le n}$ be a matrix of indeterminates over $\mathbb C$. For choices $I,J\subseteq\{1,\ldots,n\}$ with $|I|=|J|=k$ denote by $X_{I\times J}$ the matrix $(X_{ij})_{i\in I,j\in ...
14
votes
0answers
109 views
Hilbert's original proof of basis theorem
Does anyone know Hilbert's original proof of his basis theorem--the non-constructive version that caused all the controversy? I know this was circa 1890, and he would have proved it for ...
2
votes
1answer
61 views
Homogeneous forms of degree $d$ in quotient ring
We have a nice description for the space of all homogeneous elements of degree $d$ in $R = k[x_1,\ldots,x_{n+1}]$, namely it is isomorphic to $$(x_1,\ldots,x_{n+1})^d/(x_1,\ldots,x_{n+1})^{d+1}.$$
...
2
votes
1answer
60 views
Spectral Sequence involving “Triple Tor”
Can someone help me with the first 4 lines of Page 111 of Local Algebra by Serre?
I would like to know which spectral sequence is being used.
Initially I thought it is the Grothendieck ...
-4
votes
1answer
96 views
A commutative ring whose all localizations are fields
If $A$ is a ring such that $A_{p}$ is a field for every prime ideal $p\subseteq A$, is $A$ a field?
0
votes
0answers
103 views
Exercise about “dimension of rings”
Let $K$ be a field, and $\mathfrak a\subseteq K[X_{1},\dots,X_{n}]$ the ideal generated by the following polynomials of degree one
$$\mathfrak a= \begin{pmatrix}
F_{1}=\sum_{i=1}^{n}a_{1i}X_{i} \\
...
1
vote
1answer
83 views
Exercise about prime ideals in a polynomial ring
Are considered prime ideals $q_{1}\subsetneqq q_{2}\subsetneqq q_{3} \subseteq A[X]$. Could you show that $q_{1}\cap A\neq q_{3}\cap A$ ?
10
votes
1answer
146 views
Geometric meaning of completion and localization
Let $R$ be a commutative ring with unit, $I$ an ideal of $R$ and consider the following three constructions.
The localization $R_I$ of $R$ at $I$ (i.e. the localization of $R$ at the multiplicative ...
5
votes
2answers
92 views
Counterexample for going up theorem
I am searching for an example which shows that integral extensions are necessary for going up theorem.
Basically I want rings $A\subset B$ (not integral extension) such that lying over holds, but ...
