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

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5
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95 views

Applying the Yoneda-Lemma to prove the existence of Tensor-products

In class the professor said when he came to prove the existence of the tensor-product for $A$-modules ($A$ any ring) that the existence and properties of the tensor-product would be one-liners having ...
4
votes
1answer
71 views

Nullstellensatz for non-algebraically closed fields

I'm trying to prove that the Nullstellensatz holds for non algebraically closed fields, when the variety is taken over the algebraic closure. Let $R=K[x_1,...,x_n]$ and $\overline{K}$ the algebraic ...
3
votes
1answer
67 views

Hom / tensor adjunction for $O_X$ modules?

Does the hom-tensor adjunction hold for $O_X$ modules also? With sheaf hom and sheaf tensor product, the statement would consist of a natural transformation $Hom_O (M \otimes_O N, K)\cong_{nat} ...
1
vote
1answer
38 views

Do the strongly vanishing elements of $R[[x]]$ form an ideal?

I've always been a bit annoyed by expressions like $$\sum_{n:\mathbb{N}} a_n$$ when the relevant limit doesn't converge, for the following reason: if you're not going to tell the reader what ring this ...
0
votes
0answers
30 views

(Atiyah) If $A \subseteq B$ and $B \setminus A$ is multiplicatively closed then $A$ is integrally closed in $B$ [duplicate]

I've tried proof by contradiction, with $y \in B\setminus A$ and considering an integral expression $y^n = a_{n-1} y^{n-1} +\dots + a_0$ of least degree (hence $a_0 \neq 0).$ Then $a_{n-1} y^{n-1} ...
3
votes
1answer
61 views

Exercise from Kaplansky - Commutative Rings (1.1.3)

Exercise 3 in section 1-1: Let $P$ be a finitely generated prime ideal with annihilator 0. Prove that the annihilator of the module $P/P^2$ is $P$. (Hint: If $p_1,\cdots,p_n$ generate $P$ and $x$ ...
2
votes
1answer
34 views

Does $\operatorname{Hom}_A (A / P, M) \not = 0$ imply that $P$ is an associated prime of $M$?

$A$ is Noetherian, $M$ is finitely generated. Does $\operatorname{Hom}_A (A / P, M) \not = 0$ imply that $P$ is an associated prime of $M$? I am trying to prove that associated primes ...
2
votes
1answer
38 views

Computing dimension of Zariski tangent space

Background: Let $X = V(I)$ be a closed subvariety of $Y = \mathbf{A}_k^n$ and let $x \in X(k)$ be a rational point, i.e. $k(x) = k$ (residue field). We can identify $E = k^n$ with $T_{Y,y}$. Let ...
1
vote
1answer
28 views

Example for $(\mathfrak{a} + \mathfrak{b}) (\mathfrak{a} \cap \mathfrak{b}) \subsetneq \mathfrak{a} \mathfrak{b}$

In Atiyah-MacDonalds book on Commutative Algebra we have on page 6 the following statement ($\mathfrak{a}, \mathfrak{b}$ denote ideals in a ring): "(...) in $\mathbf{Z}$ we have $(\mathfrak{a} + ...
1
vote
1answer
34 views

Fully faithful functor from Preschemes to $Funct((Rings),(Sets))$

I am trying to fill in the details of the proof of Proposition 2, II. Preschemes, §6, in Mumford's Red Book (page 114, 2nd edition). For any two preschemes $X_1,X_2$, ...
0
votes
1answer
34 views

Injective ring homomorphism induces injective morphism on sheaves

I'm really confused on this exercise. People have suggested hints and I've seen online solutions involving modules and tensor products, but I don't see how any of it is related to the problem. Let ...
2
votes
0answers
41 views

Extending sections of quasi-coherent sheaves on locally Noetherian normal schemes across codimension 2 sets?

Question (global): Is it possible to extend sections of quasi-coherent sheaves on locally Noetherian normal schemes across codimension 2 sets? Specifically, if $X$ is a locally Noetherian normal ...
1
vote
1answer
34 views

Extension of scalars of the dual vector space

I was working through the following http://math.stackexchange.com/a/426300/299525 and I could not justify one of the steps. It seems to be a consequence of a general result, as discussed in the ...
6
votes
1answer
31 views

Question surrounding centers of rings.

The center of a ring $R$ consists of elements that commute with every element of $R$:$$Z(R) := \{a \in R : ab = ba \text{ for all }b \in R\}.$$I know that $Z(R)$ is a commutative unital ring. I have ...
1
vote
1answer
22 views

Normal Noetherian rings of dimension at least 1

We want to pick $I_g$ to be "maximal" but what is the partial ordering to which it is maximal? For two $f,g \in A' - A$, I don't see how $I_f$ and $I_g$ would be related via subsets. How can we ...
1
vote
0answers
25 views

A characterization of normal schemes (clarification of a statement of proposition)

The following is taken from 4.1 in Liu's book. Definition: A scheme $X$ is normal at $x \in X$ if $O_{X,x}$ is normal. $X$ is normal if it is irreducible and normal at every point. ...
0
votes
1answer
60 views

If $A \subseteq B$, then $A_{\mathfrak p} \subseteq B_{\mathfrak q}$?

Let $A \subseteq B$ be commutative rings with identity, $\mathfrak q$ a prime ideal of $B$, and $\mathfrak p = A \cap \mathfrak q$. Is the 'identity' ring homomorphism $A_{\mathfrak p} \rightarrow ...
0
votes
1answer
27 views

Height of $p/a$ is less or equal than the height of $p$.

Let $a\subset p$ be prime ideals in a noetherian ring $A$. I want to prove that height of $p/a$ is less or equal than the height of $p$. The hint to prove that is that exists a bijection between ...
2
votes
1answer
42 views

$k$-subalgebra of finite extension of $k$ is a field

I am working on a proof which has Let $\mathfrak m,\mathfrak n$ be maximal ideals and $A$ Noetherian. Given that, $A[T_1,\dots,T_n]/\mathfrak n$ is a finite extension of $A/\mathfrak m$, if ...
3
votes
1answer
94 views

“Localizing” commutative pointed monoids

A pointed monoid is a commutative monoid $A$ with a distinguished element $0\in A$ such that $0\cdot A=0$. Morphisms should preserve $0$. If $A$ is a commutative ring or pointed monoid, and $f\in ...
2
votes
1answer
56 views

What is a periodic module

I've been reading the text "THE THEORY OF COMMUTATIVE FORMAL GROUPS OVER FIELDS OF FINITE CHARACTERISTIC" by Manin. On page 26, proposition 2.1, the author mentions the notion of a periodic module, ...
4
votes
2answers
79 views

For $\mathfrak{m}$ maximal and principal, there's no ideal between $\mathfrak{m}^2$ and $\mathfrak{m}$

Let $R$ be a commutative ring with unity. If a maximal ideal $\mathfrak{m}$ of $R$ is principal, prove that there is no ideal $I$ with $\mathfrak{m}^2\subsetneq I\subsetneq \mathfrak{m}$. I have ...
0
votes
1answer
40 views

The interior of a set in Spec (R)

Let $R$ be a commutative ring with identity, and $Spec (R) $ the set of all prime ideals of $R $ with Zariski topology. How can we verify that $int (V (I))= Spec (R)-V (ann (I))$, where $I $ is an ...
3
votes
0answers
42 views

Bijective correspondence of rational points in projective space

Let $k$ be a field and consider an arbitrary point $\alpha = (\alpha_0, \dots, \alpha_n) \in \mathbf{P}(k^{n+1})$. Then there is a bijection $\rho: \mathbf{P}(k^{n+1}) \to \mathbf{P}_k^n (k)$, the ...
0
votes
0answers
17 views

Finite number of maximal ideals above a maximal ideal in general integral extension [duplicate]

Let $A\subseteq B$ be an integral extension of integral domains and let $K$ and $L$ be the fields of fractions of $A$ and $B$ respectively. Assume that the field extension $L/K$ is Galois with finite ...
0
votes
1answer
51 views

Principal open sets in graded rings

I am interested in Prop II.2.5b in Hartshorne stating that if $D_+ (f) = \{p \in \textrm{Proj } B \mid f \notin p \}$ then there is a canonical homeomorphism $D_+ (f) \cong \textrm{Spec } B_{(f)}$ the ...
4
votes
1answer
79 views

When does the prime spectrum deformation retract into the maximal spectrum?

For which rings is the maximal spectrum a deformation retract of the prime spectrum? For instance, a comment to the answer to this MO question mentions it's the case if we take the ring to be the ...
2
votes
1answer
20 views

Localization of ring of continuous functions at an element

Let $X$ be a topological space. Let $D_f= \left\{ x\in X:f(x)\neq 0 \right\}$. Let $\mathcal O_X$ be the sheaf of continuous functions on $X$. Is it true that $\mathcal O_X(D_f)= \left\{ \frac ...
0
votes
0answers
30 views

$\textrm{Dim } S(Y) = 1 + \textrm{Dim } Y$

Let $k$ be algebraically closed, and $S = k[Y_0, ... , Y_n]$, and let $\mathscr Y$ be a variety in projective $n$ space, corresponding to the homogeneous prime ideal $\mathscr P$ of $S$. Let $U_0 = ...
1
vote
0answers
54 views

If $JA_{P}\subset IA_{ P}$ for every associated prime, then $J\subset I$

This problem is Exercise 6.4 in Matsumura's Commutative Ring Theory. Let $I$ and $J$ be ideals of a Noetherian ring $A$. Prove that if $JA_P\subset IA_{P}$ for every $P\in ...
0
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0answers
27 views

Every irreducible component of $Z(\mathfrak a)$ has dimension $\geq n-r$

I had a lot of trouble with this question and now have written down a solution I think is correct. Is there anything I have maybe overlooked in this proof or didn't do rigorously? If it is correct, ...
1
vote
0answers
20 views

Divided Power Structure on Homology

Let $(R,\mathfrak{m},k)$ be a commutative local noetherian ring. Let $X$ be a Tate resolution of $k$ with filtration $X^0\subseteq X^1\subseteq \ldots $, i.e., $X^n=$the algebra obtained by adjoining ...
1
vote
1answer
36 views

System of Divided Powers on $\mathrm{Tor}^R(k,k)$

I was reading Gulliksen and Levin's (GL) text Homology of Local Rings, and I have a question about something in Chapter 2. Given a local commutative ring $(R,\mathfrak{m},k)$, they say that ...
0
votes
1answer
27 views

Why does an irreducible component of $Y \cap H$ have dimension $r-1$.

Let $R = k[X_1, ... , X_n]$, $Y$ an affine variety in $k^n$ of dimension $r$, and $H$ a hypersurface in $k^n$, with $Y \cap H \subsetneq Y$. I am trying to understand why every irreducible component ...
1
vote
0answers
35 views

Geometric meaning of prime elements?

This question is motivated by this MO question, which seeks geometric meaning for irreducibility of an element. The first sentence is: Consider a domain A and a non-zero element $f\in A$. That ...
8
votes
1answer
103 views

Where is basic algebraic topology in basic algebraic geometry?

I'm a student meeting commutative algebra and algebraic geometry for the first time. The idea of studying every (commutative) ring geometrically via its spectrum (as a locally ringed space) is ...
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0answers
48 views

Where is it used that $R^G$ is finitely generated?

In Lemma 5.0.4 of Toric Varieties by Cox, Little and Shenck I don't understand which part of the proof uses that $R^G$ is finitely generated. Can someone please help me? Lemma : Let $G$ act on ...
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vote
0answers
76 views

(Co)different ideal is divisorial?

Given a normal ring $A$. Denote $F$ the fractional field of $A$. Suppose $K/F$ is a separable extension, and denote $B$ the integral closure of $A$ in $K$. We know that ...
0
votes
1answer
31 views

Not algebraic extension inside a affine algebra which is not a domain

It is known that if a field $K$ is a $k$-algebra, and there is a finitely generated domain (affine domain) $B$ such that $K\subseteq B$, then $K$ must be algebraic over $k$ (a particular case happens ...
3
votes
1answer
86 views

$R$ be a commutative ring with unity satisfying a.c.c. on radical ideals ; is it true that $R[x]$ also satisfies a.c.c. on radical ideals ?

$R$ be a commutative ring with unity satisfying ascending chain condition on radical ideals ; is it true that $R[x]$ also satisfies ascending chain condition on radical ideals ?
0
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2answers
41 views

Is every integral domain contained in a discrete valuation ring?

Is is true that every integral domain which is not a field is contained in a proper subring of its fraction field which is a DVR?
1
vote
1answer
48 views

Why is the natural map $I^{-1}J\rightarrow\operatorname{Hom}_R(I,J)$ epi?

I am reading Eisenbud’s book Commutative Algebra, and have gotten stuck on the proof of Theorem 11.6c. Let $R$ be a ring, $K(R)$ its total quotient ring, and $I,J\subset K(R)$ invertible $R$-modules ...
2
votes
1answer
36 views

Can a graded $k$-algebra have torsion over $k[\theta]$ for $\theta$ a non-zerodivisor?

Let $k$ be a field and let $R$ be an $\mathbb{N}$-graded $k$-algebra such that $R_0=k$. Let $\theta$ be a not necessarily homogeneous element of $R$ that is transcendental over $k$ and is a ...
2
votes
1answer
37 views

Geometric intuition behind locality of morphisms of locally ringed spaces

Let $(X,\mathcal O_X)$ be a locally ringed space. The examples I have in mind are sheaves of continuous/$C^k$/smooth/etc. functions over a suitable topological space. The direction $f^\sharp:\mathcal ...
2
votes
2answers
65 views

$\mathrm{Hom}_R(N,M)$ is an essential extension of $\mathrm{Hom}_R(N,L)$

$M$ is an $R$-module and an essential extension of $L$. $N$ is a finitely generated submodule of $M$. Then $\mathrm{Hom}_R(N,M)$ is an essential extension of $\mathrm{Hom}_R(N,L)$. I tried to use ...
1
vote
2answers
44 views

$\operatorname{Ext}^{1}(M,R/m)=0$ implies $\operatorname{Tor}_{1}( M,R/m)=0$

Let $(R,m)$ be a commutative local Noetherian ring and $M$ a finitely generated $R$-module. I want to show that $\operatorname{Tor}_{n+1}(M,R/m)=0$ if and only if ...
1
vote
1answer
45 views

If $I=I(V)$ is an ideal of $F[x_1,\cdots,x_n]$ obtained from an algebraic set then can be factorized into prime ideals?

Let $I=I(V)$ where $$I(V)=\{f\in F[x_1,\cdots,x_n]\mid(\forall x\in V), f(x)=0\},$$ and where $V\subseteq F^n$ and $F$ is a field. Then there exist prime ideals $P_1,\cdots,P_m$ of $F[x_1\cdots,x_n]$ ...
1
vote
1answer
45 views

Weak Hilbert Nullstellensatz to show the bijection $Z(I)\overset{\simeq}{→} \left\{\text{maximal ideals in }A/I\right\}$.

Let $k$ be an algebraically closed field. Let $I \subset A := k[x_1, \ldots , x_n]$ be an ideal. We denote by $$Z(I) = \{a \in k^n \mid f(a) = 0, \text{ for all } f \in I\}$$ the zero set of ...
6
votes
1answer
56 views

Is $\operatorname{Spec}R_S$ homeomorphic to $ \left\{ \mathfrak p\in \operatorname{Spec}R:\mathfrak p\cap S=\emptyset \right\}$?

The correspondence theorem for localizations gives a bijection between $\operatorname{Spec}R_S$ and $ \left\{ \mathfrak p\in \operatorname{Spec}R:\mathfrak p\cap S=\emptyset \right\}$. According to ...
5
votes
0answers
82 views

If the module of Kahler differentials $\Omega_R$ is free than $\operatorname{rk} \Omega_R = \operatorname{dim} R$

Let $k$ be an algebraically closed field of characteristic zero, and $R$ is a (local as ring with maximal ideal $m$) algebra over $k$ of essentially finite type, such that $$ k \cong R/m. $$ It is ...