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

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2
votes
2answers
99 views

How can one see that $\operatorname{tr}(f\otimes g)=\operatorname{tr}f\operatorname{ tr }g$?

Suppose you have two free modules $M$ and $N$ of finite rank over a commutative ring $R$. Let's also take some $f\in\operatorname{End}_R(M)$ and $g\in\operatorname{End}_R(N)$, which gives a ...
12
votes
2answers
315 views

Why is it that $\det(\phi-x\text{id})=\sum_{i=0}^n (-1)^ic_ix^i$?

I'm trying to understand a certain formula for the determinant in a more general setting. Say you have a free module $M$ of rank $n$ over a (commutative) ring $R$. Let ...
3
votes
2answers
347 views

How to find Krull dimension of $k[[x,y]][x^{-1},y^{-1}]$ where $k$ is a field?

I don't know how to find Krull dimension of $k[[x,y]][x^{-1},y^{-1}],$ where $k$ is a field?
2
votes
1answer
131 views

on generators of $k$-algebra

Let $(A,m,k)$ be a local ring, and $A$ is a finitely generated $k$-algebra, and the maximal ideal $m$ is nilpotent. Let $x_1,\ldots,x_n \in m$ and their canonical images in $m/m^2$ generate this ...
15
votes
1answer
286 views

A space of ideals

Definition: Let $R$ be a commutative ring with 1. Endow the power set $2^R$ with the product topology. The ideal space $\mathcal{I}(R)$ is defined to be subset of $2^R$ consisting of ideals, ...
16
votes
2answers
810 views

Motivation behind the definition of flat module

Can someone explain what is the motivation behind the definition of a flat module? I saw the definition but I don't really know why it is important to work with these structures.
4
votes
1answer
252 views

Poincaré series and short exact sequences

For an additive function $\lambda$ and an exact sequence of modules $0 \rightarrow M_1 \rightarrow M_2 \rightarrow M_3 \rightarrow 0$, we have $\lambda(M_2) = \lambda(M_1) + \lambda(M_3)$ by ...
0
votes
1answer
57 views

Find the kernel of $A[Y]\to A_a$.

Given an $A$-algebra homomorphism $A[Y]\to A_a$ by sending $Y$ to $1/a$, where $a$ is an element of $A$. We want to find the kernel $I$. The kernel $I$ is $(aY-1)$ . It is easy to see that the map ...
3
votes
1answer
229 views

Transcendental degree and dimension

I do not fully understand the proof of Lemma 5.6 in the book A Course in Commutative Algebra of Gregor Kemper (you can find it here) The lemma states that : If $A$ is an algebra over a field $k$, ...
4
votes
1answer
233 views

Corollary 2.13 of Atiyah - Macdonald

I just started learning about tensor products and I have some trouble understanding this corollary in Atiyah - Macdonald. All modules are assumed to be $A$ - modules for $A$ a commutative ring. ...
1
vote
1answer
250 views

A question about a proof of the Hilbert basis theorem

I don't understand a step in this proof of the Hilbert Basis Theorem. Here is the proof Planeth Math. I don't understand why $ \mathrm{deg} (f_{N+1}-g)< \mathrm{deg}(f_{N+1}) $. This can only ...
5
votes
1answer
238 views

Genus of the desingularization of a plane curve

Background I have been considering the following question. Let $k$ be an algebraically closed field and consider a curve $C \subset \mathbb{P}^2$. Compute its genus, that is, the genus of its ...
1
vote
1answer
205 views

Conceptualization of exterior powers of projective modules

Let $A$ be a commutative noetherian ring, and $P$ a projective $A$ module with $rank(P)=n$. I know that $\wedge^nP \simeq L$ for some rank 1 projective $A$-module, $L$; but I'm not sure of how to ...
3
votes
2answers
392 views

Is there an analogue of the jordan normal form of an nilpotent linear transform of a polynomial ring?

Is there an analogue of the Jordan Normal Form for an infinite dimensional vector space? In general I think the answer is no. It's been awhile since I studied it, but I'm pretty sure something would ...
3
votes
1answer
299 views

Minimal generating sets for homogeneous polynomial ideal in two variables

This question is (somehow) related to System of generator of a homogenous ideal Let $A$ be the ring $A={\mathbb R}[X,Y]$, and let $m \geq 1$. Let $$ {\cal S}_m=\lbrace X^m, X^{m-1}Y,X^{m-2}Y^2, ...
4
votes
1answer
142 views

Algebraic extension and Integral extension

If $K$ is algebraic closure of $F$, then as a ring, $K$ is integral over $F$. Is that true or not?
2
votes
1answer
149 views

A question with an odd hypothesis.

Let $S$ be a discrete valuation ring and $R\subset S$ be a proper subring (also a DVR). Assuming that $M$ and $N$ are the respective maximal ideals of $R$ and $S$ and that $N\cap R = M$, then the ...
10
votes
2answers
597 views

Exactness of sequences of modules is a local property, isn't it?

It's well known, that passing to modules of fractions is exact, i.e. if $M'\xrightarrow{f} M\xrightarrow{g} M''$ is an exact sequence of $A$-modules ($A$ being a commutative ring with ...
4
votes
1answer
147 views

Change of base property for flat modules?

I've read the claim about base change for flat modules in several sources (Lang's Algebra, Hartshorne's Algebraic Geometry, A&M), but unfortunately it isn't proven anywhere. The claim is that ...
2
votes
1answer
210 views

Does codimension equal height in complete local domains?

For an ideal $I$ in a commutative ring $R$, define $\operatorname{codim}I=\dim R-\dim R/I$. Does codimension equals height for all ideals in the formal power series ring? Does this hold for complete ...
5
votes
2answers
188 views

Prime ideals in certain local ring extensions

$(R,m)\subseteq (S,n)$ is a local extension of rings and $S$ is a finitely generated $R$-module. If $P$ is a prime ideal of $R$ such that $P\subset m^2$ and $P'$ is a prime ideal in $S$ such that ...
9
votes
2answers
490 views

Computing the “lying over”, “going up”, “going down” ideals.

For any commutative unital ring $R$ and an ideal $\mathfrak{a}$ of $R$, we shall denote $$\begin{align*} \mathrm{Spec}(R)&:=\{\text{prime ideals of }R\},\\ ...
7
votes
1answer
267 views

If $P$ is a prime ideal in a commutative Noetherian local ring $R$, is $P\hat{R}$ a prime ideal in $\hat{R}$?

Do prime ideals expand to prime ideals in the completion? I believe this is the case since I think $R/P\equiv \hat{R}/P\hat{R}$, although Atiyah-Macdonald explicitly mentions the preservation of ...
7
votes
1answer
402 views

Localizations of quotients of polynomial rings (2) and Zariski tangent space

I am sorry, in the whole text below $k$ is just meant to be $\mathbb{C}$. This question is closely related to my previous one here. I am considering the two rings $k[X]=k[x,y,z]/\langle ...
3
votes
1answer
84 views

What operations are preserved by the canonical map to quotient rings

Let $R$ be a commutative ring and $I,J,K$ be ideals such that $I\subseteq J$ and $I\subseteq K$. Let $\pi: R \to R/I$ be the canonical map. I am able to prove $\pi$ preserves sum and products. ...
2
votes
1answer
217 views

Regular ring of infinite dimension?

Is it possible for a (non-local) regular ring to have an infinite dimension? As far as I know, the well known characterization of regular rings in terms of finite global dimension is only for local ...
6
votes
2answers
259 views

In a commutative ring $R$ if $I,J,K$ are ideals, is $(I+J)\cap (I+K)=I+J\cap K$

In a commutative ring $R$ if $I,J,K$ are ideals, is $(I+J)\cap (I+K)=I+J\cap K$. I think this is true but cannot find it referenced anywhere. Working modulo $I$, both sides are just $J\cap K$ and ...
2
votes
0answers
145 views

Depth of the quotient of two squarefree monomial ideals

Let $S=k[x_1,\dots,x_n]$ and $J\subset I$ two squarefree monomial ideals in $S$ such that $I$ is generated in degree $d$, and $J$ is generated in degree $d+1$. Let $$I=(f_1,\ldots,f_r),\quad ...
0
votes
3answers
488 views

Question regarding to Dedekind domain and PID

I just wondering if the following statement is true. If $R$ is a Dedekind domain and $P$ is a prime ideal of $R$, then $R_P$ is a PID. $R_P$ means $R$ localize at $P$. Thanks.
5
votes
1answer
443 views

What is the importance of the Krull's principal ideal theorem

What is the importance of the Krull's principal ideal theorem in later study of commutative algebra and algebraic geometry? Can any one tell me the geometric picture of this theorem? Thank you very ...
6
votes
1answer
197 views

$F$ free $\implies$ $\mathrm{Hom}_R(M,R)\otimes F\to\mathrm{Hom}_R(M,F)$ is a monomorphism?

Suppose $M$ and $F$ are modules over a commutative ring $R$. In general, it is not true that the natural (functorial) homomorphism $\mathrm{Hom}_R(M,R)\otimes F\to\mathrm{Hom}_R(M,F)$ is a ...
3
votes
2answers
989 views

Preservation of Integral Closure under Localization.

Wikipedia says that if an integral domain $A$ is integrally closed, then $S^{-1}A$ is integrally closed if $S$ is a multiplicatively closed subset of $A$. They state it as a reason for another ...
3
votes
0answers
86 views

Is $M\to M^{\vee\vee}$ injective when $M$ is free?

It's a common theorem that when $M$ is a finite-free $R$-module of rank $n$, there is a natural isomorphism $M\cong M^{\vee\vee}$, where $M^\vee$ denotes the dual. So $M^{\vee\vee}$ is also free of ...
3
votes
1answer
257 views

Ways to prove that an extension of Noetherian modules is Noetherian

Let $M$ be a module and $N$ a submodule of $M$. If $N$ is Noetherian and $M/N$ is Noetherian, so is $M$. This is usually proven like this: Let $(A_n)$ be an ascending series of submodules of $M$. ...
4
votes
1answer
306 views

Gorenstein ring of dimension zero

Let $(R, \mathfrak m)$ be a local ring and in the same time a finite dimensional algebra over the complex numbers. How one can prove that if $\operatorname{Ann}_R(m)$ has dimension one then $R$ is an ...
2
votes
1answer
227 views

Ring of fractions mapping property

Let $R$ be a commutative ring with unit and let $S \subseteq R$ be a multiplicative closed subset. Consider the ring of fractions $S^{-1}R$ with the homomorphism $f\colon R \to S^{-1}R$ which sends ...
18
votes
1answer
479 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 ...
10
votes
1answer
901 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, ...
1
vote
1answer
352 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
258 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 ...
6
votes
2answers
121 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
140 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 ...
7
votes
1answer
356 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 $(R,m)$, where $m$ is the maximal ideal, and a finitely generated non-zero $R$-module $M$ such ...
8
votes
1answer
589 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
241 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 $R/P$ ...
7
votes
1answer
487 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
525 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
499 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
338 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
549 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 ...