Tagged Questions
3
votes
1answer
38 views
Finite Projective Dimension implies non vanishing Ext
Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$?
Can't we write the free module as a direct ...
1
vote
1answer
40 views
Extension of homorphisms on a divisible R-module
Let $R$ be a principal ideal domain and let $M$ be a finitely generated $R$-module. Take $N$ a submodule of $M$ and let $P$ be a divisible $R$-module. Prove that any homomorphism $f: N \rightarrow P$ ...
1
vote
2answers
69 views
Integral closure $\tilde{A}$ is flat over $A$, then $A$ is integrally closed
Question. Let $A$ be an integral domain and $\tilde{A}$ be its integral closure in the field of fractions $K$. Assume that $\tilde{A}$ is a finitely generated $A$-module. I want to prove that if ...
1
vote
0answers
44 views
Noetherian localizations and extra-condition implies Noetherian
I'm trying to solve this question but I'm stucked:
If a ring $R$ satisfies the following two conditions:
i) For every maximal ideal $M$ of $R$, if $S = R\setminus M$ then $S^{-1}R$ is ...
2
votes
1answer
62 views
Show that $M=\bigcap_{\mathfrak{p}\in\operatorname{Spec}(R)}M_\mathfrak{p}=\bigcap_{\mathfrak{m}\in\text{Max}(R)}M_\mathfrak{m}$ for certain $M$.
$\newcommand{\Spec}{\operatorname{Spec}}$
$\newcommand{\mSpec}{\operatorname{Max}}$
This is a homework from my algebra course. I am in a situation where I think I have found a solution, though ...
4
votes
1answer
89 views
Vakil 14.2.E: $L\approx O_X(div(s))$ for s a rational section.
I am working through Vakil's Ch 14 (march2313 version) on invertible sheaves and am having trouble on 14.2.E.
The question (in notation to be defined) is this: how do I show that each point in the ...
7
votes
0answers
91 views
Class group of $k[x,y,z,w]/(xy-zw)$
I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...
2
votes
0answers
37 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
61 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 = ...
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 ...
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
$$\mathfrak a= \begin{pmatrix}
F_{1}=\sum_{i=1}^{n}a_{1i}X_{i} \\
...
1
vote
1answer
79 views
Exercise about prime ideals in a polynomial ring
Are considered prime ideals $q_{1}\subsetneqq q_{2}\subsetneqq q_{3} \subseteq A[X]$. Could you show that $q_{1}\cap A\neq q_{3}\cap A$ ?
1
vote
2answers
87 views
Determine the total ring of fractions
Determine the total ring of fractions of $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}_{12}$.
6
votes
2answers
72 views
Can you determine from the minors if the presented module is free?
Motivation (you can ignore this part): A problem in Hartshorne (II.5.8c) asks to show that if we have a coherent sheaf $\mathscr{F}$ on a reduced noetherian scheme $X$, and the function
...
1
vote
0answers
70 views
Injective hull commutes with Hom
Notation: $E_R(M)$ is the injective hull of $M$.
Let $R$ be a Noetherian ring, $I$ an ideal of $R$, and $M$ an $R$-module. Then $$\mathrm{Hom}_R(R/I, E_R(M)) \cong E_{R/I}(\mathrm{Hom}_R(R/I, ...
2
votes
2answers
102 views
Factorization of ideals in $\mathbb{Z}[\sqrt{5}]$
Consider the ring $R=\mathbb{Z}[\sqrt{5}]$. Let $I$ be the following ideal of $R$:
$$I:=(3,1+\sqrt{5})$$
My teacher said that the following equation holds:
$$I^2=(3)I,$$
but I actually can't ...
0
votes
1answer
69 views
calculating minimal prime ideals
Is there a "general approach" to determine the minimal prime ideals over an ideal $J$?
I checked some books and didn't find a general approach. Maybe the theory of Gröbner bases is related to these ...
3
votes
0answers
48 views
Proof about affine varieties
Ok so I have that $k$ is algebraically closed and $F$ is an element of $k^n$, and it is a reduced polynomial. We have that $V = V(F)$. In the book it says prove that $F$ generates $I(V)$ but in my ...
5
votes
2answers
170 views
Is this a prime Ideal?
I wish to see wether $J=(uw -v^2, u^3 - vw, w^3 -u^5)\subset\mathbb{C}[u,v,w]$ is a prime ideal. Can somebody give me a hint to do this?
Edit: More generally, I wonder wether $V(J)$, the algebraic ...
-1
votes
1answer
112 views
Discrete Valuation Rings problem 2
An order function on a field $K$ is a function $\phi:K\to \mathbb{Z} \cup {\{\infty}\}$ satisfying:
i) $\phi(a) = \infty$ if and only if $a=0$.
ii) $\phi(ab) = \phi(a) + \phi(b)$.
iii) ...
2
votes
2answers
93 views
Kernel of $p$-adic logarithm.
I'm completely clueless as to how to answer the following question:
Let $K$ be a field of characteristic zero which is complete with respect to a non-Archimedean aboslute value $|\cdot|$. Let ...
2
votes
0answers
88 views
Integral dependence and fraction fields [duplicate]
Consider $\mathbb{Q}[x]\subset\mathbb{Q}(x)\subset\mathbb{Q}(x)[y]=:K$, where
$$y^2=x,$$
and let $O_K$ be the integral closure of $\mathbb{Q}[x]$ in $\mathbb{Q}(x)[y]$. Show that ...
-1
votes
1answer
114 views
Algebraic Curves
Let $F$ be a non-constant polynomial in $k[X_1,...,X_n]$, $k$ algebraically closed.
Show that $\mathbb A^n \setminus \mathrm{V}(F)$ is infinite if $n\geq 1$, and $\mathrm{V}(F)$ in infinite if ...
4
votes
3answers
124 views
Integral closure of $\mathbb{Q}[X]$ in $\mathbb{Q}(X)[Y]$
Consider the ring $\mathbb{Q}[X]$ of polynomials in $X$ with coefficients in the field of rational numbers. Consider the quotient field $\mathbb{Q}(X)$ and let $K$ be the finite extension of ...
2
votes
1answer
76 views
Spec R is irreducible
A topological space is called reducible if $X=X_1\cup X_2$ for two closed subsets $X_1,X_2$ with $X_1\ne X\ne X_2$. Otherwise its called irreducible, want to show that $\text{Spec}(R)$ is irreducible ...
2
votes
2answers
88 views
Product of ideals corresponding to vanishing of points is equal to their intersection
Let $k$ be some field, and let $v,v',v''$ be three distinct points in $k^3$. Let $\mathfrak{m}_v = (X_1 - v_1,X_2 - v_2,X_3 - v_3)$ be the ideal in $k[X_1,X_2,X_3]$ corresponding to the polynomials ...
5
votes
2answers
130 views
Artinian rings and PID
Let $R$ be a commutative ring with unity. Suppose that $R$ is a principal ideal domain, and $0\ne t\in R$. I'm trying to show that $R/Rt$ is an artinian $R$-module, and so is an artinian ring if $t$ ...
-7
votes
1answer
133 views
Show that $T (S^{−1}M)= S^{−1}(T (M))$ [closed]
Let $R$ be an integral domain and $M$ an $R$-module. An element $m \in M$ is called a torsion element if $\operatorname{Ann_R(m)} \neq 0$, i.e. if $rm = 0$ for some nonzero $r \in R$. Denote the set ...
-4
votes
1answer
119 views
Show that $T(M)$ is a submodule of $M$.
Let $R$ be an integral domain and $M$ an $R$-module. An element $m \in M$ is called a torsion element if $\operatorname{Ann}_R(m) \neq 0$, i.e. if $rm =0$ for some nonzero $r\in R$. Denote the set of ...
-3
votes
1answer
126 views
$S, T$ be multiplicatively closed sets in the ring $R$, such that $S \subseteq T$ Show that the following are equivalent
Let $S, T$ be multiplicatively closed sets in the ring $R$, such that $S\subseteq T$.
Let $\varphi : S^{−1}R \to T^{−1}R$ be the homomorphism which maps each $r/s \in S^{−1}R$ to $r/s$ viewed as an ...
4
votes
1answer
96 views
Commutative algebra - integral extensions question
This is homework, but I am pretty stuck and I feel I am lacking intuition on this so I ask.
The question is as follows : let $k$ be a field and $R = k[x_1, \dots, x_n]$ (this notation means a ...
-6
votes
1answer
137 views
Show that $S^{−1}M =0$ if and only if there exists $s \in S$ such that $sM =0$.
Let $S$ be a multiplicatively closed subset of the ring $R$ and let $M$ be a finitely-generated R-module. Show that $S^{−1}M =0$ if and only if there exists $s \in S$ such that $sM =0$.
-3
votes
1answer
138 views
Show that $p$ and $q$ are not principal, but that $p^2$, $pq$ and $q^2$ are.
Let $K$ be the field $\mathbb Q(\sqrt{−15})$, let $R = \mathcal{O}_K$ be the ring of integers of $K$. Let $\alpha= \frac{-1+\sqrt{-15}}{2}$ and consider the prime ideals $p = (2,α)$ and $q = (17,α + ...
6
votes
4answers
130 views
Finding the ideals in a ring of fractions
I am dealing with the ring $$R=\left\{\frac{a}{b} \mid a,b\in\mathbb{Z}\mbox{, $b$ is not divisible by 3}\right\}$$ with addition and multiplication as defined in $\mathbb{Q}$ and I'm trying to find ...
2
votes
3answers
41 views
Disjoint Union of Spectra
The following it from Atiyah-Macdonald's Introduction to Commutative Algebra, exercise 1.22.
Apparently this should be very easy, my apologies for asking.
I have been stuck for almost 1 day and I ...
1
vote
3answers
156 views
Is finitely generated a necessary condition for artinian modules?
I know that if $R$ is Artinian, then a f.g. $R$-module is Artinian. Is f.g. a necessary condition?
0
votes
0answers
39 views
How to find all possible polynomials set generate the same variety?
Some concepts I am not clear
a. Does statement $\langle f_1, f_2\rangle$ = $(\sum u_if_i, i=1,2)$ means $f_1 = u_1f_1 + u_2f_2$ and $f_2 = u_3f_1 + u_4f_2$ ?
b. Obviously $u_1 = 1$ and $u_2 = 0$ ...
0
votes
1answer
67 views
Noetherian modules
Question: Let $R$ be a Noetherian ring, and $M$ be an $R-$module, show that $M$ is Noetherian if and only if $M$ is finitely generated.
This is a question on my homework, I'm really confused about ...
1
vote
1answer
55 views
Generators for this ideal
I have a ring $R$ unitary and commutative with four elements and characteristic $2$. I have $$I=\{f \in R[X,Y]; f(t,t^2)=0\ \forall t \in R\}.$$ I have to find a finite number of generators for this ...
5
votes
2answers
134 views
$A$ PID, $M$ flat (i.e., torsion-free). Then $\operatorname{Ext}_A^1(M,N)$ is injective, for all $N$.
Let $A$ be a PID and $M$ a flat (i.e., torsion-free) $A$-module. Then, for every $A$-module $N$, $\text{Ext}_A^1(M, N)$ is injective in $A\text{-}\mathbf{Mod}$.
It is easy when $M$ is finitely ...
2
votes
2answers
148 views
Question on Noetherian Rings
I have the following two questions on rings of polynomials. They seemed similar enough that I thought I'd go ahead and group them here as opposed to making separate listings for them. The questions ...
4
votes
2answers
109 views
Showing that nonzero prime ideals are maximal in a polynomial ring.
$F$ is a field and $F[X^2, X^3]$ is a subring of $F[X]$, the polynomial ring. I need to show that nonzero prime ideals of $F[X^2, X^3]$ are maximal.
A classmate suggested taking a nonzero prime ideal ...
0
votes
0answers
47 views
Product of monomial ideals
Let $ I = (f_1, \ldots, f_n)$ and $ J = (g_1,\ldots,g_m)$ be two ideals generated by regular sequences of monomials in the polynomial ring $R = k[x_1, x_2, \ldots , x_u]$
Show that $$\Delta_{p(IJ)} = ...
2
votes
0answers
60 views
A injective Endomorphism over an Artinian Module is an Automorphism [duplicate]
Possible Duplicate:
If $M$ is an artinian module and $f$ : $M$ $\mapsto$ $M$ is an injective homomorphism , then f is surjective
Let $R$ be a commutative Ring with identity. We have an ...
0
votes
0answers
33 views
Simplicial complexes
Let $ I = (f_1, f_2); J = (g_1, g_2), L= (h_1, h_2) $ be monomial ideals in polynomial ring $ S = K[x_1,...,x_n] $ such that $ f_1, f_2, g_1, g_2, h_1, h_2 \in S, f_1 $ and $ f_2 $, $ g_1 $ and $ g_2 ...
0
votes
0answers
51 views
The $I$-torsion submodules of an injective module [duplicate]
Possible Duplicate:
Prove that the following module is injective
Prove that $I$-torsion submodules of injective modules are injective (the ring is Noetherian). Please help me.
3
votes
1answer
135 views
Prove that the following module is injective
Let $R$ be a commutative Noetherian ring with identity. Prove that if $I$ is an ideal of $R$ and $E$ an injective $R$-module, then $\bigcup_{n\geq 1}(0:_{E}I^{n})$ is an injective $R$-module. Please ...
2
votes
1answer
35 views
Question on Integral Closure
I'm trying to prove this fact: given $A$ an integral domain and an element $f\in A$ such that $A/fA$ has no nilpotents, then $A$ is integrally closed if and only if $A_f$ is integrally closed ...
5
votes
5answers
139 views
A question on local rings
I was trying to get a counterexample of this fact: given a ring $A$, $f\in A$ and $S=\{1,f,f^2,...\}$, is $S^{-1}A$ always a local ring? Could you help me please? Thank you.
2
votes
1answer
86 views
Ring containing a Dedekind ring
Suppose I have two domains, $A\subset B$, where $A$ is Dedekind and $\operatorname{Frac}(A)=\operatorname{Frac}(B)$. I also know that $B$ is both integrally closed and has height $1$. Is $B$ ...
