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

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7
votes
1answer
479 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
512 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 ...
8
votes
1answer
494 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 ...
6
votes
1answer
336 views

In this special case the quotient polynomial ring is a UFD?

I would like to know if the following is true. Let $F$ be a field and let $p\in F[x]$ be a square-free polynomial. Then, the quotient ring $F[x,y]/\langle y^2-p\rangle$ is a UFD. I am not sure ...
3
votes
1answer
535 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
100 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 $N$. ...
3
votes
1answer
180 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
176 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 ...
6
votes
1answer
262 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
60 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
136 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
214 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
152 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
152 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 ...
7
votes
1answer
477 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 $$ ...
4
votes
1answer
460 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
131 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 ...
6
votes
3answers
1k 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
93 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
221 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
424 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 ...
3
votes
1answer
155 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 $J\cap K=I$.
3
votes
1answer
343 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
448 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 ...
11
votes
2answers
383 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
240 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
611 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, ...
12
votes
1answer
1k 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
186 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 ...
4
votes
1answer
568 views

Characterization for artinian Gorenstein rings

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
241 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 ...
5
votes
2answers
328 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, ...
6
votes
1answer
294 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
120 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
211 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
141 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
83 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 ...
7
votes
1answer
1k 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
117 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
273 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
101 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 ...
14
votes
3answers
869 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 ...
3
votes
1answer
233 views

Nontrivial example of $M$ such that $\text{Ass}(M)=\varnothing$?

Suppose $R$ is a commutative ring, and $M$ an $R$-module. Is there a nontrivial example of such $M$ where the set of associated primes $\text{Ass}(M)=\varnothing$? Taking $M=0$ feels kind of ...
3
votes
2answers
207 views

Example demonstrating that $R=\{a+bi\sqrt5: a,b \in \mathbb{Z}\}$ is not a Euclidean domain.

We know $R=\{a+bi\sqrt{5}: a,b \in \mathbb{Z}\}$ is not a UFD because, for example, you can factor $$6=(1+i\sqrt{5})(1-i\sqrt{5})=(2)(3)$$ and these are two distinct factorizations into ...
3
votes
1answer
230 views

Example of a module whose support is not closed? [duplicate]

Possible Duplicate: The support of a module is closed? Is there a simple example of a module $M$ of a Noetherian commutative ring $R$ such that ...
6
votes
2answers
229 views

Example of height $n$ ideal with $I/I^2$ (locally) $n$-generated, but $I$ is not.

For $R$, a commutative noetherian ring of dimension $d$, I'm looking for an example where $I \subset R$ is an ideal of height $n \lt d$ such that $I/I^2$ is generated by $n$ elements (locally ...
8
votes
1answer
459 views

Length of maximal chain of prime ideals equals transcendence degree of fraction field?

I've been reading some commutative algebra, but have been struggling with this idea for a while. Let $k$ be a field, and let $A=k[x_1,\dots,x_n]$ be a finitely generated integral domain, such that ...
12
votes
1answer
370 views

Geometrical interpretation of $I(X_1\cap X_2)\neq I(X_1)+I(X_2)$, $X_i$ algebraic sets in $\mathbb{A}^n$

Edit: I should point out that I'm working over an algebraically closed field $k$. Let $X_1,X_2\subset\mathbb{A}^n$ be affine algebraic sets. Show that $I(X_1\cap X_2)=\sqrt{I(X_1)+I(X_2)}$. Show ...
7
votes
3answers
159 views

On a proof about locally nilpotent homomorphisms.

Let $A$ be a commutative ring. I have a short question about the small result (Proposition 2.5 of Lang's book on Algebra, pg. 418) that if $M$ is an $A$ module, and $a\in A$, then $a_M$ defined by ...
6
votes
2answers
2k views

Why is the localization of a commutative Noetherian ring still Noetherian?

This is an unproven proposition I've come across in multiple places. Suppose $A$ is a commutative Noetherian ring, and $S$ a multiplicative subset of $A$. Then $S^{-1}A$ is Noetherian. Why is this? ...