Questions tagged [commutative-algebra]
Questions about commutative rings, their ideals, and their modules.
10,564 questions
2
votes
2answers
43 views
A counterexample to Hartshorne I.5.4(a)?
I do not understand why the following does not give a counterexample to Problem I.5.4(a) in Hartshorne's Algebraic Geometry book. The question of how to solve this problem has already been posted but ...
0
votes
0answers
12 views
Laurent series and tensor
Let us begin with the complex vector space
\begin{equation}
V_{z}=\Big\{\omega\in \mathbb{C}[[z,z^{-1}]]dz\ \mid Res_{z=0} \omega (z)\Big\}
\end{equation}
We could define the tensor product of $V_{...
0
votes
0answers
15 views
Algebraically independent polynomials iff linearly independent differentials
This is an exercise question in Appendix A of Introduction of Algebraic Geometry, Justin R Smith. I am looking for an intuition for the solution.
if $k \rightarrow K$ is an extension of fields of ...
1
vote
0answers
17 views
Concerning Serre’s Intersection multiplicity
I am trying to understand a statement in a proof.
The setup is $(R,m)$ is a $3$-dimensional regular local ring with infinite residue field, $\mathfrak{p}$ is a height-$2$ prime ideal and $x$ is not in ...
0
votes
1answer
27 views
Associated primes of the square of a monomial ideal
Consider the ideal $J=(xy,yz,zx)$ in $R=\mathbb C[x,y,z]$. How to show that $(x,y,z) \in \mathrm{Ass}_R (R/J^2)$ ?
1
vote
2answers
34 views
Does an uncountable algebraically closed field of characteristic zero contain an uncountable subfield which can be embedded into $\mathbb{C}$?
This is probably very simple.
Let $k$ be an uncountable algebraically closed field of characteristic zero. Does there exist an uncountable algebraically closed subfield $k_0\subset k$ and an ...
0
votes
1answer
23 views
Coordinate ring of an algebraic set.
Let $K[x_1,x_2,....,x_n]$ be a polynomial ring over an algebraically closed field $K$. Let $V=V(I)$ be an algebraic set in $K^n$ and $I$ is a radical ideal. We know by a theorem that there exists a ...
0
votes
0answers
71 views
Is there a mistake in the proof of Theorem 4 in Kaplansky's Commutative Rings?
Here is a theorem (Theorem 4) in Kaplansky's Commutative Rings.
Let $R$ be an integral domain. Let $S$ be the set of all elements in $R$ expressible as a product of principal primes. Then $S$ is a ...
-2
votes
1answer
19 views
Class of isomorphism classes of noetherian rings [on hold]
Is the class of isomorphism classes of noetherian commutative rings a set?
What about the class of isomorphism classes of noetherian rings with finite Krull dimension?
2
votes
0answers
42 views
regular sequence $\iff$ complete intersection
Let $k$ be a field and $X \subseteq \mathbb{P}^n$ a closed subscheme of dimension $n-r$.
We say $X$ is complete intersection if $X = V(I)$, where $I$ is a homogeneous ideal which is generated by $r$ ...
1
vote
0answers
16 views
Action of Deck transformations after trivialization
Let $S$ be a connected scheme, and let $f\colon X\to S$ be a finite étale cover. Then there exists a finite étale cover $Y\to S$ such that $X\times_S Y\cong \coprod_{i\in I} Y$.
Replacing $Y$ by one ...
2
votes
1answer
54 views
Ideal with no zero divisors implies integral domain?
I'm trying to figure out a solution to the question:
If a commutative ring $R$ has a nontrivial proper ideal $I$ that contains no nontrivial zero divisor of $R$, is $R$ an integral domain?
I ...
2
votes
0answers
23 views
What is the Krull dimension of the Burnside ring of $\mathbb N$?
A contravariant functor $F$ from monoids to commutative rings was defined there.
Question. What is the Krull dimension of $F(\mathbb N)$?
(Here $\mathbb N$ denotes the additive monoid $(\mathbb N,+...
0
votes
2answers
40 views
Why $I(\mathbb{A}^n)=(0)$?
Let $\mathbb{A}^n$ denote the set of n-tuples of elements from field $k$ and $I(X)$ the ideal of polynomials in $k[x_1,...,x_n]$ that vanish every point in $X$. The note I’m reading, in showing that $\...
4
votes
0answers
28 views
A function being “finite” over a point on non-normal schemes?
I recently came across a remark about Cartier divisors in a textbook on algebraic geometry. I'm not sure how to interpret the remark. I've attached the previous paragraph as well for context.
The ...
-1
votes
1answer
29 views
Computing Intersection of Ideals
I am having hard time computing the intersection of ideals.
If there is a graph G=(V(G),E(G))such that
(vertex set) v(G)={$x_{1}$,$x_{2}$,$x_{3}$,$x_{4}$,$x_{5}$},
(edge set) E(G)={$x_{1}x_{2},x_{2}...
1
vote
0answers
42 views
An injective homomorphism from $k[x,y]$ to $k[x,x^{-1},y]$
Let $k$ be a field of characteristic zero and let $R_{-1}:=k[x,x^{-1},y]$
be the $k$-algebra of polynomials in $x,y$ containing the inverse of $x$, denoted by $x^{-1}$.
Let $F_{-1} : k[x,y] \to R_{-1}...
1
vote
0answers
23 views
Blow up of plane curve is Normalization of local ring?
I have a question concerning normalizations of plane curves, which I know little about.
Consider the simple node $V(f = y^2 - x^3 - x^2)$. Then $t=y/x$ is integral over $k[x,y]$ so that $(k[x,y]/f) \...
4
votes
1answer
82 views
From monoids to commutative rings
We shall first define a functor
$$
F:\mathsf{Mon}^{\text{op}}\to\mathsf{CRing},
$$
where $\mathsf{Mon}^{\text{op}}$ is the category opposite to the category of monoids and $\mathsf{CRing}$ is the ...
0
votes
1answer
25 views
Maximum number of algebraically independent homogeneous polynomials of given degree [on hold]
Given degree $d\geq2$ how many algebraically independent homogeneous polynomials of degree $d$ can we have in $\mathbb Z[x_1,\dots,x_n]$? Is it possible to have $n^2$ or $n^3$ of them or are we ...
0
votes
1answer
32 views
Describing the kernel of this surjection
Let $k$ be a field. For a finite dimensional $k$-vector space $V$, write $k[V]$ for the symmetric algebra, which is noncanonically isomorphic to some polynomial ring in finitely many variables.
Take ...
4
votes
0answers
31 views
Minimal prime ideals of quotient modules
Let $R$ be a commutative Noetherian local ring, $M$ a finitely generated $R$-module. If $\dim_R M=\dim_R R/\mathfrak{q}$ for all $\mathfrak{q}\in \operatorname{Min}_R M$ and $x\in R$ is $M$-regular, ...
0
votes
1answer
25 views
Localization of $k[t]$ at a prime ideal is not a finitely generated $k[t]$-algebra?
Let $k$ be a field (infinite and algebraicaly closed), consider the prime ideal $(t)$ in $k[t]$.
Consider the localization of $k[t]$ at $(t)$: $k[t]_{(t)}$.
Now $k[t]_{(t)}$ is a $k[t]$-algebra. I ...
0
votes
1answer
18 views
Commutative ring with finitely many minimal primes [duplicate]
$A$ be a commutative ring with $1$ with finitely many minimal primes $\{p_1,\ldots,p_n\}.$ Then how can I show that $S^{-1}A \cong A_{p_1} \times\cdots \times A_{p_n},$ where $S=A \setminus \bigcup_{i=...
1
vote
1answer
14 views
$A \subset B$ be integral and flat extension then it is faithfully flat.
I want to prove the following:
$A \subset B$ be integral and flat extension of rings then it is faithfully flat.
Clearly enough to show that for every ideal $I$ of $A$, $I^{ec}=I$. Since the ...
1
vote
1answer
22 views
Presentation matrix for module in terms of other presentation matrices
Let $R$ be a commutative ring, and let $A$ and $B$ be right $R$-modules. Suppose that the following two sequences are exact:
$$
R^u\stackrel{Q}{\to} R^s\stackrel{\pi}{\to}A\to 0 \\ R^r\stackrel{f}{\to}...
1
vote
0answers
18 views
Show that every non-zero proper ideal of $R$ can be written as a product of finitely many prime ideals of $R$, determined uniquely, upto order.
Theorem $:$
Let $R$ be a Noetherian domain such that it is integrally closed and every non-zero prime ideal of $R$ is maximal. Then every non-zero proper ideal of $R$ can be written ...
4
votes
0answers
42 views
Show that the ring $k[x,y,z]/(x^2-yz)$ is normal
Show that the ring $A=k[x,y,z]/(x^2-yz)$ is normal
My attempt:
Take an element $\frac{f}{g} \in K(A)=(k[x,y,z]/(x^2-yz))_{(x^2-yz)}$ assuming that it is integral over $A$. Since $z=\frac{x^2}y \in K(...
2
votes
0answers
27 views
In which rings does this multiplicative analogue of Bézout's theorem hold?
When I was thinking about this question: International Zhautykov Olympiad 2019 problem 6 I learned that, when $0 < a,b$ are integers that divide $n >0$ and $d$ is their $\gcd$, then if for a ...
1
vote
0answers
35 views
On finiteness of $\cup_{n\ge 1 } \operatorname{Ass}_R (R/I^n)$
Let $I$ be an ideal of a commutative noetherian ring $R$.
How to prove that $\bigcup_{n\ge 1 }\operatorname{Ass}_R (R/I^n)$ is finite ?
I am aware of Brodmann's result about Asymptotic stability ...
1
vote
0answers
26 views
On a subset of the associated primes of tensor product of modules
For a module $M$ over a commutative ring $R$, let
$\operatorname{Ass}_R (M):=\{\operatorname{ann}_R (m) \mid m\in M$ and $\operatorname{ann}_R(m) \in \operatorname{Spec}(R)\}$.
If $M,N$ are $R$-...
0
votes
0answers
41 views
On a sufficient condition for being a prime ideal
Consider the ideal $I$ in $\mathbb{C}[z_{1},\ldots,z_{m}]$ generated by $\{p_{1},\ldots,p_{t}\}$, where $t\leq m$ and $p_{1},\ldots,p_{t}$ intersects completely i.e. the map $(p_{1},\ldots,p_{t}):\...
0
votes
1answer
14 views
Prime ideal being maximal ideal in an affine algebra
Let $A$ be an affine algebra over a field and $P$ be a prime ideal of $A$. If $P$ is a finite intersection of maximal ideals, then $P$ is maximal.
Is this statement true? And if so, how to prove it?
...
0
votes
1answer
28 views
Does the following property of matrices hold for any commutative ring with identity?
I have recently gone through the proof of the following theorem given in my book $:$
Theorem $:$
Let $R$ be a commutative ring with identity with quotient field $K.$ Let $\alpha \in ...
1
vote
1answer
38 views
Finding associated primes of quotient modules
Consider the ideal $I=(a)\cap(a,b)^2=(a^2,ab)$ in $k[a,b]$ and set $R=k[a,b]/I$. The problem is to show that for all $n\ge 1$ and all $\lambda\ne 0$, the ideals $(b^n)$ and $(a+\lambda b^n)$ of $R$ ...
0
votes
0answers
35 views
$ S_m/m S_m $ contains just one prime ideal
I'm trying to read these lecture notes on my own for self studying. At the beginning of the second lecture, page 8, the author wants to deduce the geometric form of the Zariski's main theorem from the ...
1
vote
0answers
55 views
Krull Dimension equality.
Suppose that $R$ is a commutative Noetherian local ring, $M$ a finitely generated $R$-module and $\mathfrak{p}\in \mathrm{Supp}_R(M)$. Then I am little confused with the following equality between ...
0
votes
0answers
32 views
local and regular integral closure
Let $(A,\mathfrak{m})$ be a local domain with integral closure $B$ which is a regular local domain with maximal ideal $\mathfrak{M}$. Assume in addition that
$A/\mathfrak{m} = B/\mathfrak{M}$ and $B\...
0
votes
1answer
19 views
Right derived functors of the $I$-torsion functor and $\varinjlim \mathrm{Ext}^i_R(R/I^n,-)$ are naturally isomorphic?
Let $R$ be a commutative ring with unity and let $I$ be a proper ideal. (I'm not assuming $R$ is Noetherian.) For every $M \in R$-Mod, let $\Gamma_I(M):=\{m \in M : I^n m=0$ for some $n\ge 1\}$.
If ...
0
votes
0answers
26 views
Associated prime ideals of integral extension
Let $R$ be a commutative ring with unit element and $S$ be an integral extension of $R$. Then when will $Ass_R(R) = Ass_R(S)$? I am not able to prove $Ass_R(S) \subseteq Ass_R(R)$. Is there a counter ...
0
votes
0answers
18 views
Completion and $S^{-1} A$
Let $\mathfrak{p}$ be a prime ideal of a ring $A$. The completion $\hat{A}$ of $A$ with respect to its adic-topology is used to simplify $A$ beyond the localization $A_{\mathfrak{p}}$. For a ...
0
votes
1answer
19 views
Set of Closed points of a Quasi-projective variety is dense
Let $k$ be an algebraically closed field and $V$ be a Qausi-projective variety in $\mathbb P^n_k$ i.e. $V$ is an open subset of an open subset of a Zariski-closed subset of $\mathbb P^n_k$, or in ...
2
votes
1answer
39 views
+50
Surjective homomorphism from a faithfully flat module to a regular local ring.
Let $R$ be a regular local ring and let $M$ be a faithfully flat $R$-module. Does there necessarily exist a surjective $R$-module homomorphism from $M$ to $R$?
For context, I am computing $\sum_{f\in\...
1
vote
0answers
33 views
Concerning the $k$-algebra $k[x,x^{-1},y]$
Let $k$ be a field of characteristic zero and let $R_{-1}:=k[x,x^{-1},y]$
be the $k$-algebra of polynomials in $x,y$ containing the inverse of $x$, denoted by $x^{-1}$.
So in $R_{-1}$ we have: $xx^{-...
1
vote
0answers
21 views
If $B$ is an $A$-algebra of Noetherian rings then $\text{Ass}_{A}M=\{f^{-1}(p):p \in \text{Ass}_{B}M\}$ for a f.g. $B$-module $M.$
Let $A,B$ Noetherian rings and $f: A \to B$ be a ring homomorphism. Let $M$ be a finitely generated $B$ module. Then want to show that $\text{Ass}_{A}M=\{f^{-1}(p):p \in \text{Ass}_{B}M\}.$
So far I ...
1
vote
1answer
37 views
In an integrally closed, Noetherian, local, integral domain of dimension $1$, the maximal ideal $P$ is eventually principal
Let $R$ be an integrally closed, Noetherian, local, integral domain of dimension 1 with unique maximal ideal $P$.
Take an element $a \in P$ that is non zero. Show that for some $n$, $P^n$ is ...
-2
votes
0answers
20 views
If nilradical is equal to Jacobson radical, does every prime ideal be maximal? [duplicate]
I think it's probably not true, but have trouble finding an example. Can you please give a concrete example?
1
vote
1answer
34 views
Is there a quotient of a symmetric algebra which is in bijection with the simple tensors?
Does there exist a nonzero vector space $V$ over a field $F$ and a quotient $\overline{S(V)}$ of the symmetric algebra on $V$ such that the quotient map $\mathfrak{S} \to \overline{S(V)}$, where $\...
1
vote
0answers
30 views
Nilradical is the zero ideal in an integral domain.
Let $R$ be a commutative ring with unity. Let $I$ be an ideal of $R.$ Then I know that $$\operatorname {rad} ({I}) = \bigcap\limits_{\substack {p \supseteq I \\ \text {p prime}}} p.$$
Now let $I = (0)...
3
votes
1answer
55 views
Intersection of the kernels of localization maps
Let $M$ be a finitely generated module over a Noetherian ring $R$. I need to show that for a multiplicately closed subset $U\subset R$, $$\bigcap_{P\in \operatorname{Ass}(M)\\ P\cap U=\emptyset}\ker(M\...
