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

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2
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1answer
89 views

Definition of multiplicity

Q.1. Bruns_Herzog define multiplicity (in the case of graded rings and modules) as My question is that: why multiplicity for $d=0$ it is defined as $\ell(M)$? Is there a kind of ...
3
votes
1answer
55 views

Is a prime principal ideal which is not maximal among principal ideals always idempotent?

Let $R$ be a commutative ring with identity, $P$ a prime principal ideal of $R$. Suppose that there exists a proper principal ideal $I$ of $R$ which is strictly larger than $P$ (i.e. $R\supsetneq ...
2
votes
1answer
50 views

Surjective morphism of varieties with finite fibers but not “finite”

Let $X$ and $Y$ be affine varieties, and $f : X \to Y$ a dominant regular map. Following Shafarevich, I will call $f$ finite if the induced map on coordinate rings is integral. One consequence of ...
5
votes
1answer
113 views

Interpretation of sheaf flat over a base

I am trying to get an interpretation of what means for a sheaf to be flat with respect to a base. The definition is that, given $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ is flat over $Y$ ...
0
votes
0answers
42 views

Spectral sequences and Ext between extension of modules

Suppose $A$ is a commutative ring, $M_1,M_2,N_1,N_2$ are $A$-modules and we have two exact sequences of $A$-modules $$0\to M_1\to M\to M_2\to 0,$$ $$0\to N_1\to N\to N_2\to 0.$$ I want to write a ...
3
votes
0answers
32 views

Tensor product of flat modules - proof verification

Let $A$ be a commutative ring, and let $B,C$ be commutative $A$-algebras. Let $M$ be a flat $B$-module and $N$ a flat $C$-module. I want to show that $M\otimes_A N$ is a flat $B\otimes_A C$-module. ...
3
votes
2answers
50 views

What other classes of commutative rings can be defined by requiring that $\{0\}$ is the only proper ideal satisfying some condition?

A field is just a commutative ring $R$ such that $\{0_R\}$ is the only proper ideal. Interestingly, there's a similar characterization of integral domains. Given a subset $A$ of $R$, let $A^\perp$ ...
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0answers
49 views

Determine whether a regular surjective map is finite

Consider the regular map between affine closed sets $f \colon \mathbb{A}^1 \rightarrow \mathcal{Z}(y^2-x^3) \subseteq \mathbb{A}^2$ given by $f(t) = (t^2,t^3)$. $f$ is obviously a dominant map. I ...
3
votes
1answer
66 views

Does $(a)=(b)$ imply that $a$ and $b$ are associate in a principal ideal ring?

Let $R$ be a commutative principal ideal ring with identity. Suppose that $a,b\in R$ and $(a)=(b)$. I'd like to know if there always exists $u\in R^\times$ such that $a=bu$. I know several ...
0
votes
1answer
58 views

Atiyah-MacDonald, 3.18. Why is $B_q$ a local ring of $B_p$?

This question is on the hint that the book gives to finish the exercise. Namely, if $f: A \rightarrow B$ a flat homomorphism of rings, $q$ a prime ideal of $B$ and $p = q^c$, then $B_q$ is a local ...
4
votes
1answer
61 views

Projective modules over Dedekind Domains

Show that if $R$ is a Dedekind domain, then every projective $R$-module (not necessarily finitely generated) is a direct sum of ideals of $R$. I have spent a while on this problem and I wonder if it ...
0
votes
1answer
44 views

Properties of module length

Let $e_{A}(\phi, M): = l_A(\mathrm{coker}(\phi) ) - l_A(\ker(\phi))$. In my book it is stated that if $IM = 0 \implies e_{A}(\phi, M) = e_{A/I}(\phi, M)$ and this seems to be obvious for the author. ...
3
votes
1answer
78 views

What is a geometric interpretation of regular sequences in various instances?

This question arose from my attempts to understand the inclusion Regular $\subset$ Complete Intersection $\subset$ Gorenstein $\subset$ Cohen Macaulay There are many related questions here and in ...
4
votes
1answer
65 views

Irreducible components of schemes

Consider the scheme $X:=\mathrm{Spec}(k[X,Y]/(X^2,XY))$. According to Qing Liu's "Algebraic geometry and arithmetic curves", the irreducible components are in $1-1$ correspondence with subschemes of ...
0
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0answers
38 views

Associated graded ring of a quotient

Given a ring A and an ideal $I \subseteq A$ we can form its associated graded ring with respect to $I$ $$ Gr_I(A)= A/I \oplus I/I^2 \oplus I^2/I^3 \oplus \ldots $$ I wondered if there is a way to ...
3
votes
1answer
74 views

Must a $R$-automorphism on $R[X]$ be of the form $X\mapsto aX+b,\ a\in R^*,b\in R$?

Let $R$ be a commutative ring. I wonder if every $R$-automorphism (that is, a ring automorphism that fix $R$) $\varphi$ of $R[X]$ satisfies $\varphi(X)=aX+b$, where $a$ is an unit in $R$ and $b$ an ...
2
votes
2answers
87 views

The unit group of $\mathbb{Q}[x, y]/(x^2+y^2+1)$

During some calculations, I encountered with the problem of calculating the unit group of the $\mathbb{Q}$-algebra $\mathbb{Q}[x, y]/(x^2+y^2+1)$. I believe it is the unit group of the field of ...
0
votes
1answer
49 views

Primary Ideal and Associated Primes [duplicate]

I'm trying to understand the proof of the following statement: If $R$ is Noetherian, then an ideal $Q$ is $P$-primary for a prime $P$ $\Leftrightarrow$ $Ass(R/Q)=\lbrace P \rbrace$. I can show the ...
3
votes
0answers
36 views

Equivalence of definitions for completion

For the settings on my question, take Atiyah's chapter on completions. Basically we have two definitions of completness (Atiyah's sense, the canonical map $\phi:M\rightarrow \widehat{M}$ is an ...
2
votes
1answer
30 views

An example of a (necessarily non-Noetherian) ring $R$ such that $\dim R[T]>\dim R+1$

What is an example of a non-Noetherian ring $R$ such that the Krull dimension of $R[T]$ is greater than dim$R+1$?
-1
votes
1answer
54 views

How to prove that the set of maximal elements of a set of prime ideals is finite

Let $A$ be a subset of ${\rm Spec}(R)$ with $R$ noetherian Are there any techniques to prove that ${\rm max}(A)$ (ie the set of maximal elements of $A$) is finite? I'm looking for equivalent ...
4
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0answers
44 views

Prime ideals contained in the union of almost all prime ideals

I am reading the proof of the long exact sequence involving $S$-class groups and $S$-units in Neukirch Algebraic Number Theory, Chapter I, Prop. 11.6, which states the following canonical sequence is ...
0
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0answers
102 views

Unique factorization in prime ideals in a local ring

Let $R$ be a local commutative domain with maximal ideal $M$. Assume that every ideal of $R$ is a product of prime ideals in a unique way. I want to show that the only non-zero prime ideal of $R$ ...
2
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0answers
31 views

Does the relation $\mid^*$ have any interesting applications for understanding the structure of commutative rings that aren't integral domains?

There is a binary relation $\mid^*$ defined on any commutative ring as follows: $a \mid^* b$ iff $ak=b$ for some $k \in R$ that is not a zero divisor. This is always transitive, and it is reflexive ...
5
votes
1answer
50 views

Proving a ring is Noetherian when all maximal ideals are principal generated by idempotents

Let $R$ be a commutative ring with unity such that all maximal ideals are of the form $(r)$ where $r\in R$ and $r^2=r$. I wish to show that $R$ is Noetherian. I know that if all prime (or ...
0
votes
1answer
40 views

Show that $f^*:Spec(B)\rightarrow Spec(A)$ is a closed mapping.

Let $f:A \rightarrow B$ be an integral homomorphism of rings. Show that $f^*:Spec(B)\rightarrow Spec(A)$ is a closed mapping. My Try: So, $B$ is integral over $f(A)$. $f^*$ is given by $b\longmapsto ...
6
votes
1answer
74 views

How to imagine the difference between the following schemes?

Consider $A=\operatorname{Spec} k[x]_{(x)}[t]$ and $B=\operatorname{Spec} k[x,t]_{(x)}$ for a field $k$ (Vakil, note 11.3.8). For me, both are infinitesimal neighborhoods of an affine line - the ...
2
votes
1answer
71 views

Finding the kernel of $\alpha: K[X,Y,Z]^{3}\rightarrow \langle X,Y,Z\rangle$, $(f,g,h)\mapsto Xf+Yg+Zh$.

I am trying to do exercise $2.3$ of Reid's "Undergraduate Commutative Algebra": Let $A=K[X,Y,Z]$ where $K$ is a field, and $m=\langle X,Y,Z\rangle$. I have to show that the kernel of the ...
2
votes
1answer
46 views

Is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ?

Let $(X,d)$ be a metric space , then is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ? Do we need completeness of $X$ ?
1
vote
1answer
22 views

If $(X,d)$ is a finite metric space , then is every prime ideal of $C(X, \mathbb R)$ maximal?

If $(X,d)$ is a finite metric space , then is every prime ideal of $C(X, \mathbb R)$ maximal ? The thing is , since $X$ is finite , so it is compact , so ideal $M$ is maximal iff it is of the form ...
1
vote
1answer
35 views

Does the equation $\operatorname{Ass}M=\operatorname{Ass}E(M)$ hold for non-finitely generated modules $M$?

Does the equation $\operatorname{Ass}M=\operatorname{Ass}E(M)$ hold for non-finitely generated modules $M$? Here $E(M)$ is the injective envelope of $M$ and $\operatorname{Ass}$ denotes the set ...
5
votes
2answers
48 views

Dimension of Tensor Product for Flat Extensions

Suppose that $A,B,$ and $C$ are commutative unital rings, $A\to B$ is flat, and $A\to C$ is any map. I am trying to determine whether $$ \dim B\otimes_AC=\dim B+\dim C-\dim A $$ Any counterexamples ...
1
vote
1answer
47 views

Determine whether $(\mathbb{R}[x,y]/(y^2-x^2-x^3))_{(x,y)}$ is a discrete valuation ring.

Geometrically, the curve $y^2-x^2-x^3=0$ is singular at the origin in the real plane. Thus the ring should not be a dvr. I am thinking to show that it is not a dvr, it is equivalent to show that it ...
0
votes
1answer
34 views

Polynomial ring with integral coefficients is integral

Let $B$ be a ring and $A\subset B$ a subring. Assume that $B$ is integral over $A$. I have to prove that $B[X]$ is integral over $A[X]$. I tried writing down an integral relation for $f(X)\in B[X]$ ...
1
vote
2answers
81 views

If local rings are Noetherian, is scheme locally-Noetherian?

If all the local rings of a scheme are Noetherian, is the scheme locally-Noetherian?
0
votes
1answer
18 views

A question concerning to show that $V(I)$ is open if $I$ is radical ideal

Let $I:=(f_1 ,...,f_k)$ be a finitely generated ideal of $C(X,\mathbb R)$ such that $\mathrm{rad}(I)=I$, $f:=\sqrt{\sum_{m=1}^k |f_m|}=\sum _{m=1}^k g_mf_m$ where $g_i $'s are real valued ...
0
votes
0answers
54 views

Kernel identity of the canonical homomorphism

Let $A$ be a commutative complete Noetherian ring with unit for the $I$-adic topology, where $I$ is an ideal of $A$. Suppose that $M_0$ is finitely generated over $A$. Let $(M_n)_{n\geq 0}$ be ...
4
votes
0answers
112 views

Determine all discrete valuations on $\mathbb{C}(x)$.

To clarify, for a field $K$, a valuation $v$ on $K$ is a map $v:K^{\times}\to G$ for $G$ an ordered group (written additively) such that for any $a,b\in K^{\times}$: 1) $v(ab)=v(a)+v(b)$; 2) ...
0
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0answers
32 views

What is interesting (useful) about Multiplicity?

Multiplicity is defined at 4.1.5; Bruns_Herzog. People say it is an important invariant. I don't know what idea is behind this definition and What is interesting/useful about it. what important ...
3
votes
1answer
43 views

Pushforward of a Cohen-Macaulay sheaf

Let $f: X \to Y$ be a finite surjective morphism of quasi-projective varieties. Let $X$ be Cohen-Macaulay and let $Y$ be smooth. Now let $\mathcal{F}$ be a coherent sheaf on $X$. Then $f_* ...
3
votes
0answers
58 views

Theorem 1 in chapter II.4 of Mumford's Red Book

while reading Mumford's wonderful Red Book, I arrived to a Theorem where I don't understand the proof. So Theorem 1 in chapter II.4 says Let $X_0$ be a prescheme over $k_0$, let $X= X_0 ...
0
votes
1answer
42 views

Question about the algebraic definition of tangent space

Let $V\subset \mathbb{A}^n(K)$ be an affine algebraic set and let $K[V]$ be its coordinate ring. For $a\in V$, let $T_aV$ be the tangent space of $V$ at $a$ and $P=I(a)$ be the maximal ideal in $K[V]$ ...
3
votes
1answer
64 views

Grobner basis and subsets

Let $A$ be a subset and $I$ an ideal of polynomial ring $R=k[x_1,x_2,...,x_n]$. Is there any algorithm for deciding when $A\subseteq I$?
2
votes
1answer
46 views

When is the Frobenius the identity?

If $f$ is an irreducible polynomial of degree $n$ over $\mathbb{F}_{p}$, then $\mathbb{F}_{p}[x]/(f)$ is the finite field $\mathbb{F}_{p^{n}}$ and the map $a \mapsto a^{p}$ is the Frobenius ...
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0answers
49 views

Gluing construction of the projective space scheme.

When constructing the projective space scheme $\mathbb{P}_R^n$ for a ring $R$, we may take the subrings $$ A_i = R\left[\tfrac{X_0}{X_i}, \ldots, \widehat{\tfrac{X_i}{X_i}}, \ldots, ...
2
votes
2answers
150 views

Proof of Proposition 2.4 in Atiyah-MacDonald [closed]

I'm struggling here with the proof. To be honnest i need a really concrete explanation because i have been on it for a long time and i can not find it nowhere else. Please can anyone help me with ...
1
vote
1answer
93 views

Analysis of the ideals of $C[0,1]$

For every ideal $I$ of $C[0,1]$ , define $Z(I):=\{x \in [0,1] :f(x)=0 , \forall f \in I\}$ and for every $A \subseteq [0,1]$ , let $I(A):=\{f \in C[0,1] : f(x)=0 , \forall x \in A\}$ . Then ...
7
votes
0answers
110 views

Check whether a polynomial ideal is prime in the power series ring

I would like to know whether the ideal $I = \langle y^{2}(y^{2}-x^{2}) + w^{7}, y^{2}(y^{4}-x^{4}) + z^{7}\rangle$ is prime in $\mathbb{C}[[x,y,z,w]]$, the ring of formal power series in the ...
5
votes
3answers
119 views

Properties characterized by a vanishing Ext or Tor module

While reading Weibel's "An introduction to homological algebra'', I've noticed that many properties of a module are characterized by the vanishing of some Tor or Ext. Fix a (commutative) ring $R$ and ...
1
vote
1answer
47 views

Relation between the inverse of finitely generated ideals and the inverse of their powers.

Let $D$ be an integral domain, $K$ its field of fractions, and $J_1,...,J_n$ are ideals of $D$ such that $(\sum_{i=1}^{n} J_i)^{-1}=D$. How can we prove that this implies $(\sum_{i=1}^{n} ...