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

learn more… | top users | synonyms (1)

16
votes
1answer
415 views

Modules with $m \otimes n = n \otimes m$

Let $R$ be a commutative ring. Which $R$-modules $M$ have the property that the symmetry map $$M \otimes_R M \to M \otimes_R M, ~m \otimes n \mapsto n \otimes m$$ equals the identity? In other ...
8
votes
1answer
489 views

Affine scheme $X$ with $\dim(X)=0$ but infinitely many points

As the title says, I'm looking for an affine scheme of dimension zero, but with infinitely many points. At first I doubted that something like this could exist, and I still can't think of an example, ...
0
votes
1answer
260 views

Finitely generated algebra over a ring

Let $R$ be a finitely generated algebra over a commutative ring $R_0$, my question is: why $R$ is a homomorphic image of $R_0[x_1,x_2,...,x_r]$ for some $r$?Could one explicitly define the ...
2
votes
0answers
172 views

Are there formal power series ring in infinitely many indeterminates

There are polynomial rings in infinitely many indeterminates. Does it make sense to talk about power series rings in infinitely many indeterminates. If not, what do we get when we complete the ...
5
votes
2answers
110 views

Question about Maps of power series rings

Suppose we have an injective homomorphism of power series rings $$k[[x_1,...,x_m]]\to k[[y_1,...,y_n]]$$ can $x_i$ map to a unit in $k[[y_1,...,y_n]]$? Of course in general, for injective ...
2
votes
1answer
127 views

Another question on scheme morphisms

I have two questions on scheme morphisms. Is the property of a scheme morphism to be a closed immersion a local property (as it is for open immersions)? Let $X=Spec (R)$ be a noetherian scheme and ...
3
votes
1answer
181 views

A detail in the proof of Auslander-Buchsbaum Theorem

I'm trying to understand the proof of a theorem (Auslander-Buchsbaum) which says that given a local ring $(R,m,K)$, where $m$ is the maximal ideal and $K$ the residue field, and a finitely generated ...
5
votes
1answer
324 views

Is the quotient of a complete ring, complete?

If $(R,m)$ is a complete local ring (with respect to the $m-$adic topology) and $I$ a prime ideal in $R$, is $R/I$ complete (with respect to the $m/I-$adic topology)? It seems too strong, but I am ...
8
votes
1answer
183 views

primary ideal of regular local ring

Let $(R,\mathfrak{m})$ be a regular local ring of dimension $d$. Let $P$ be a prime ideal of height $d-1$. I want to know if $P^2$ is always a $P$ primary ideal ie if $P/P^2$ is torsion free as $R/P$ ...
6
votes
1answer
333 views

Prop. 2.3 Hartshorne: $\varphi:A\to B$ induces a morphism $\operatorname{Spec}(B)\to\operatorname{Spec}(A)$

I don't fully understand a step in the proof of the above-mentioned Proposition; more precisely, in part (b): If $\varphi:A\to B$ is a homomorphism of rings, $X=\operatorname{Spec}(A)$, ...
4
votes
2answers
332 views

How to find the nilpotent elements of $\mathbb{Z}/(\prod p_i^{n_i})$?

I've been following MIT's old opencourseware class on commutative algebra. For one problem, I want to find the nilpotent and idempotent elements of $\mathbb{Z}/(n)$, where $n=\prod p_i^{n_i}$. I know ...
7
votes
1answer
271 views

Does a regular function on an affine variety lie in the coordinate ring?(Lemma 2.1, Joe Harris)

I think the proof in for Lemma 2.1 in Joe Harris's book Algebraic Geometry, A First Course, does not work. (The statement is on Page 19, and the proof on Page 61.) The proof fails because that ...
2
votes
1answer
265 views

Why does $k[X,Y]/(XY)$ have two minimal primes?

I working on a problem for practice. For $k$ a field, I was able to show that any element of $A=k[X,Y]/(XY)$ has a unique representation in form $a+f(X)X+g(Y)Y$ for $a\in k$, $f(X)\in k[X]$ and ...
5
votes
1answer
86 views

Is this module finitely generated?

Suppose $M$ is a $A$-module, $A$ is a commutative ring with 1, such that for every countably generated submodule $N$ of $M$, there exists a finitely generated submodule $L$ which contains $N$. ...
2
votes
1answer
131 views

Hilbert function on ideal generated by linear forms.

This is a slight extension of a remark a read a few days ago. Let $K$ be a field, and let $A=K[X_0,\dots,X_N]$ be a polynomial ring, which is graded in the standard way (the elements of degree $n$ ...
3
votes
2answers
140 views

Why is the kernel of $k[x_1,\dots,x_n]\to k$ a maximal ideal?

In Reid's Undergraduate Commutative Algebra, $k$ a field and a point $P=(a_1,\dots,a_n)\in k^n$ determine a homomorphism on the the polynomial ring of functions $k[x_1,\dots,x_n]\to k$ by $g\mapsto ...
5
votes
1answer
157 views

$X \to Y$ flat $\Rightarrow$ the image of a closed point is also a closed point?

This question came from a proof in Algebraic Geometry by Hartshorne (Chapt3, Corollary 9.6) To be precise, Let $f:X \to Y$ be a flat morphism of schemes of finite type over a field $k$. Then is it ...
3
votes
1answer
53 views

If $\ell_{A_\mathfrak{p}}(N)<\infty$, then is it true that $\operatorname{Hom}_A(N,E(A/\mathfrak{q}))=0$?

Let $A$ be a Noetherian ring, $\mathfrak{p},\mathfrak{q}\subset A$ distinct prime ideals of the same height, $N$ an $A_\mathfrak{p}$-module of finite length. Then is it true that ...
4
votes
1answer
91 views

Remark on the dimension of quotient by prime homogeneous ideal.

There's a remark near the end of a section I'm reading about Hilbert polynomials that I don't fully understand. Let $K$ be a field, and let $A=K[X_0,\dots,X_N]$ be a polynomial ring, which is graded ...
2
votes
1answer
108 views

Relaxing a condition to prove that an associated graded ring is a domain implies the ring is a domain.

Just a while ago a question was posted that for a filtration $R=R^0\supset R^1\supset R^2\supset\cdots$ on a commutative integral domain $R$, the associated graded ring $$ ...
6
votes
1answer
133 views

Set of zeroes of a radical ideal?

Suppose $R=F[X_0,\dots,X_r]$, where $F$ is an algebraically closed field. Now $R$ is graded, with the homogeneous polynomials of degree $n$ being the elements of degree $n$. Now suppose $I$ is a ...
5
votes
3answers
114 views

Does $R$ a domain imply $\operatorname{gr}(R)$ is a domain?

Suppose you have a filtration $R=R^0\supset R^1\supset R^2\supset\cdots$ on a commutative ring $R$. This gives the associated graded ring $$ \text{gr}(R)=\bigoplus_{n=0}^\infty R^n/R^{n+1}. $$ From ...
6
votes
1answer
368 views

Concluding that a finitely generated module is free?

Suppose $R$ is a local Noetherian domain, and $M$ is a finitely generated $R$-module. Furthermore, let's suppose there exists $k>0$ such that $$ ...
3
votes
1answer
273 views

Integral closure of a local ring is the intersection of valuation rings lying above it

Let $L/K$ be a finite field extension. Let $\mathcal{O}$ be a valuation ring of $K$. Let $R$ be the integral closure of $\mathcal{O}$ in $L$. Why is $R$ the intersection of all valuation rings of $L$ ...
2
votes
2answers
112 views

Is every invertible rational function of order 0 on a codim 1 subvariety in the local ring of the subvariety?

I have been trying to read Fulton's Intersection Theory, and the following puzzles me. All schemes below are algebraic over some field $k$ in the sense that they come together with a morphism of ...
4
votes
3answers
606 views

Tensor product of 2 coordinate rings

For the term variety, I mean the irreducible algebraic set. My question is, if $V$ and $W$ are 2 varieties over a field $\Bbbk$, then does $\Bbb{k}[V]\otimes \Bbb{k}[W]$ has special structure? I try ...
2
votes
0answers
83 views

Intersections of finitely generated field extensions are finite?

I was reading the following post at MathOverflow: http://mathoverflow.net/questions/21086/when-are-intersections-of-finitely-generated-field-extensions-finitely-generated/21093 I can't comment there, ...
3
votes
3answers
207 views

It is possible to realize $\mathbb{Z}$ as $K[x_1,\ldots,x_n]/I$ for some field $K$?

Question: Are the integers $\mathbb{Z}$ an affine $K$-algebra, i.e. does there exist a field $K$, a $n\!\in\!\mathbb{N}$, and an ideal $I\!\unlhd\!K[x_1,\ldots,x_n]\!=\!K[\mathbb{x}]$, such that ...
3
votes
1answer
255 views

an example of regular ring with nilpotent elements

A regular local ring is a domain. But in general, a regular ring is not domain, so you can find regular rings with nilpotent elements. I am unable to construct an example of (A, I) as A is a regular ...
4
votes
1answer
120 views

Criterion for quotient ring to be decomposable.

I read in passing that for a commutative ring $R$ and an ideal $I$, then $R/I$ is decomposable if and only if there exist proper ideals $J$ and $K$ such that $J+K=R$, and $J\cap K=I$.
2
votes
1answer
275 views

Local criteria of flat modules

Let $A,B$ be rings and $M$ be a $B$-module. Let $f:A \to B$ be a ring morphism. For prime ideal $p \subset B$ ,let $q=f^{-1}(p)$, and the corresponding local morphism $A_q \to B_p$ makes $M_p$ an ...
5
votes
1answer
280 views

System of generators of a homogenous ideal

Let $I$ be a homogenous ideal in the ring $k[x_{1},\dots,x_{n}]$. My question is: If $\lbrace f_{1},\dots,f_{r}\rbrace$ is a minimal system of generators of $I$, then are the integers $r$ and ...
9
votes
2answers
332 views

Computing intersection multiplicity using Tor - explicit example

When trying to compute the (Serre-generalized) intersection number of two varieties at a closed point, I came to a need to compute the following $\operatorname{Tor}$: Let $k$ be an algebrically ...
4
votes
2answers
186 views

Proving one form of Hilbert's Nullstellensatz

I have been trying to prove the following problem in Atiyah Macdonald concerning one form of Hilbert's Nullstellensatz. The problem is as follows: If $X$ is an affine algebraic variety (the set ...
4
votes
1answer
273 views

A sufficient condition for a domain to be Dedekind?

We know that in a Dedekind domain, every nonzero ideal admits a unique factorization into a product of prime ideals. I was wondering if this condition is sufficient for a domain to be Dedekind, ...
10
votes
1answer
621 views

Geometric meaning of primary decomposition

In the book "Commutative Algebra with a view toward Algebraic Geometry of David Eisenbud, he wrote about the Geometric interpretation of primary decomposition. I summary as follows : Let ...
7
votes
2answers
149 views

Why is $\text{Supp}(M)$ connected in the Zariski topology?

Suppose $M$ is a indecomposable module, so that it cannot be written as $M_1\oplus M_2$ for $M_1\neq M$ and $M_2\neq M$, which is finitely generated over a commutative ring $R$. Why is ...
3
votes
1answer
250 views

Characterization for artinian gorenstein ring

Let $(R,m)$ be an artinian local ring. Show that if $I \cap J \neq 0$ for all non-zero ideals $I$ and $J$, then $R$ is a Gorenstein ring. Another formulation could be: show that if $(0)$ is an ...
0
votes
1answer
128 views

Isomorphism of ($\mathbb{Z}/{(n)}$-graded) Rings

Let $A=\bigoplus_{d=0}^n A_g$ and $B=\bigoplus_{d=0}^n B_h$ be $\mathbb{Z}/{(n)}$-graded rings. In particular, we assume $A_n\ne 0$ and $B_n\ne 0$. Let $\phi:A\to B$ be an isomorphism of rings. My ...
4
votes
2answers
167 views

Characterization of primary ideals in a principal ideal domain?

On the commutative algebra wiki, a table of properties lists that "for a PID, the primary ideals coincide with the powers of prime ideals." I played around with it, couldn't produce a proof, and have ...
6
votes
1answer
198 views

In $K[X,Y]$, is the power of any prime also primary?

I've recently been reading about primary decomposition, and was browsing the questions here. From this, I know that it is not true that every primary ideal is the power of a prime ideal. I'm curious ...
3
votes
2answers
112 views

If $p$ and $q$ are prime ideals in a ring such that $p\subsetneq q$ is ht$(p)<$ht$(q)$

If we have a containment of prime ideals in a commutative ring with $1$ is the "larger" prime ideal necessarily of the higher height?
3
votes
1answer
179 views

$\mathrm{Tor}$ functor not left exact

Is there an example which shows that the functor $B\otimes_R(-)$ is not left-exact, given a ring $R$ and a right $R$-module $B$?
3
votes
1answer
94 views

Is the local ring $A_p$ the direct limit of rings corresponding to open subschemes?

Suppose $A_p$ is the stalk of a ring $A$ at a prime ideal $p$. Consider the (opposite) system of those open immersions $\operatorname{Spec}(A)\leftarrow \operatorname{Spec}(B)$ such that the scheme ...
1
vote
1answer
76 views

Why does $\operatorname{Supp}(M\oplus N)=\operatorname{Supp}(M)\cup\operatorname{Supp}(N)$?

For modules $M$ and $N$ over a commutative ring, why does $\operatorname{Supp}(M\oplus N)=\operatorname{Supp}(M)\cup\operatorname{Supp}(N)$? I tried justifying it with the following, but I'm not ...
6
votes
1answer
569 views

The fibers of a finite morphism of affine varieties are all finite

I am trying to find the proof of : The fibers of a finite morphism $\phi: X \rightarrow Y$ ($X,Y$ affine) are all finite. Here, a morphism is called finite if $K[X]$ is integral over the image ...
5
votes
1answer
93 views

Support of a Coherent Sheaf and Noetherianity

Exercise 5.6 b) of Chapter II of Hartshorne's Algebraic Geometry asks to prove that if $A$ is a Noetherian ring and $M$ a finitely generated $A$-module then $Supp(\tilde{M})=V(Ann(M))$. Where ...
6
votes
1answer
203 views

Tensor product of modules over quotients by annihilators

If M and N are modules over some commutative ring A and $\mathfrak{a} \subset \operatorname{Ann(M)} \cap \operatorname{Ann(N)}$ is an ideal, is it true that $M \otimes_A N \cong M ...
2
votes
0answers
88 views

Further explanation on proof that associated primes are precisely those belonging to primary modules in reduced decomposition of $0$.

Consider the following theorem: Let $A$ and $M$ be Noetherian. The associated primes of $M$ are precisely the prime which belong to the primary modules in a reduced primary decomposition of $0$ in ...
13
votes
3answers
556 views

Localization at a prime ideal is a reduced ring

Here is the question that I came up with, which I am having trouble proving or disproving: Let $A$ be a ring (commutative). Let $p \in Spec(A)$ such that $A_p$ is reduced. Then there exists an open ...