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

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2
votes
1answer
33 views

Natural isomorphism between curve and its projective completion?

If $C \subset \mathbb{C}^2$ is an irreducible affine curve and $\hat{C} \subset \mathbb{P}_2$ is its projective completion, is there necessarily a natural isomorphism of function fields $K_C \cong ...
3
votes
2answers
112 views

Does $IJ=IK\implies J=K$ always hold for integral domain and finitely generated nonzero ideal $I$?

Let $R$ be a commutative integral domain, $I,J,K$ three ideals of $R$ with $I\neq (0)$ being finitely generated. Then does $IJ=IK$ imply $J=K$? With Nakayama lemma, I can prove it if one of $J$ and ...
3
votes
1answer
47 views

How can affine coordinate rings be canonically identified as $k$-algebras?

Exercise 1.5 of Hartshorne asks us to show (in one direction) that any affine coordinate ring $k[x_1,\dots,x_n]/I(Y)$ is a finitely-generated $k$-algebra with no nilpotents. The second part is quite ...
0
votes
0answers
32 views

Smooth morphism and completion of DVR

Let $R$ be a Henselian discrete valuation ring with algebraically closed residue field and $\hat{R}$ its $m$-completion, where $m$ is the maximal ideal. Is it true that the induced morphism ...
1
vote
1answer
42 views

Exact sequence of modules and taking the quotient

Let $A$ be a commutative ring and $\text{Spec}\,A=\bigcup\limits_{i=1}^mD(f_i)$ be a covering by principal open sets. Show that the sequence of modules $$M\stackrel{\alpha}\to ...
0
votes
0answers
32 views

DVR and its fraction field

Let $k$ be a complete discrete valuation field with algebraically closed residue field. We know that its maximal unramified extension $k^{\mathrm{unr}}$ need not be complete. But can the ring of ...
1
vote
1answer
30 views

$\big(k[X,Y]/(H)\big)_{(X,Y)}\cong k[X,Y]_{(X,Y)}/(H)$

I'm trying to understand why $\big(k[X,Y]/(H)\big)_{(X,Y)}\cong k[X,Y]_{(X,Y)}/(H)$, where $k$ is a field and $H$ is an irreducible polynomial. I need this result in a theorem I'm proving, I ...
1
vote
2answers
31 views

Why this set is a $R/\mathfrak m$-module?

I'm reading this PDF: Discrete Valuation Rings and Function Fields of Curves. I'm trying to understand in this theorem why $\mathfrak m/\mathfrak m^2$ is a $R/\mathfrak m$-module (see number 4 below). ...
1
vote
1answer
59 views

If $\mathcal O_P(C)$ is a DVR, then $P$ is non-singular

Let $C$ be an irreducible curve over $\mathbb A^2$ and $P\in C$. I would like to prove if $$\mathcal O_P(C)=\{f\in k(C)\mid f=a/b, b(P)\neq 0\}$$ is a DVR, then $P$ is non-singular, i.e., the ...
0
votes
0answers
31 views

Viewing Koszul complex as an algebra

I keep coming across notes which says that the Koszul complex can be viewed as an algebra. Is it true that complexes can be viewed as an algebra. If the complex is not exact, can the homologies also ...
2
votes
2answers
90 views

What can be said about a regular quotient (by a principal prime ideal) of a polynomial ring?

Let $f \in \mathbb{C}[x_1,\ldots,x_n]$ be irreducible (so (f) is a prime ideal). Assume $S:=\mathbb{C}[x_1,\ldots,x_n]/(f)$ is regular, where, by definition, a noetherian ring is regular is all its ...
3
votes
0answers
26 views

$D$ is divisor of both $d(x/z)$ and $y/z$. [closed]

Let $C \subset \mathbb{CP}^2$ be the cubic curve defined by$$y^2z = x(x-z)(x-\lambda z)$$with $\lambda \in \mathbb{C} - \{0,1\}$. Let $p = [0, 0, 1]$, $q = [1, 0, 1]$, $r = [\lambda, 0, 1]$, and $s = ...
0
votes
1answer
56 views

Union of specific prime ideals is not an ideal

Let $R$ be a commutative ring with $1$ with three prime ideals $P_1,P_2,P_3$ such that $P_i\subseteq P_j$ if and only if $i=j$. I want to show that the union of these prime ideals, which I denote ...
3
votes
2answers
86 views

For what kind of $R$-modules $M$ can we find an element $m\in M$ satisfing that $i:M\to M\otimes_R M, x\mapsto x\otimes m$ is an epimorphism?

Let $R$ be a commutative ring with identity and $M$ a $R$-module. I'm interested in under what condition we can find an element $m\in M$ satisfing that $i:M\to M\otimes_R M, x\mapsto x\otimes m$ is an ...
1
vote
1answer
92 views

How to prove $\mathcal O_P(C)$ is a DVR for $P$ non-singular?

Let $C$ be an irreducible curve over $\mathbb A^2$ and $P\in C$ a non-singular point. I want to prove that $\mathcal O_P(C)=\{f\in k(C)\mid f=a/b, b(P)\neq 0\}$ is a DVR. I've already proved that ...
0
votes
1answer
26 views

Projective dimension of an ideal generated by a regular sequence

Let $R$ be a commutative ring with $1$ and $I$ be an ideal of $R$ generated by an $R$-sequence of length $n$. I want a simple (if any) proof that the projective dimension of $I$ is $n-1$. I ...
2
votes
2answers
54 views

Does $\text{End}_R(I)=R$ always hold when $R$ is an integrally closed domain?

Let $R$ be a commutative ring with identity, and $I\neq 0$ an ideal of $R$, I'm thinking how to calculate $\text{End}_R(I)$. I have proved that when $R$ is a integral domain, ...
9
votes
0answers
80 views

Study of rings of the form $R+I$

In my life I saw lots of ways of constructing rings: polynomial rings, quotient rings, localizations, endomorphism rings, rings of fractions, integral closure of a ring, center of a ring, etc... These ...
9
votes
2answers
82 views

Nonsingular curve $C$ of degree 4, exists rational function $f: C \to \mathbb{CP}^1$ of degree 2?

Suppose $C \subset \mathbb{CP}^2$ is a nonsingular curve of degree $4$. Does there exist a rational function $f: C \to \mathbb{CP}^1$ of degree $2$?
3
votes
1answer
41 views

Question about proof of Corollary 2.18 from Eisenbud

I am reading Eisenbud's Commutative Algebra. The following is the proof I am trying to understand. My question is the second sentence in the proof. I understand that a power of $P_P$ annihilates ...
3
votes
1answer
61 views

Exists rational function on curve in $\mathbb{CP}^2$ such that pole of order $2g + 2$?

Let $C \subset \mathbb{CP}^2$ be a nonsingular curve of degree $d$, and $p_1$, $p_2$, $q$ distinct points in $C$. For any $a_1$, $a_2 \in \mathbb{C}$, does there necessarily exist a rational function ...
0
votes
0answers
82 views

Relation between Tensor-hom adjunction and adjugate matrix

Let $R\to S$ be a ring homomorphism, let $M,N$ be $S$-modules and $Q$ an $R$-module. Then, we have $$\textrm{Hom}_R(M\otimes_S N,Q) \cong \textrm{Hom}_S(M,\textrm{Hom}_R(N,Q).$$ I want to know ...
0
votes
2answers
98 views

Noetherian ring under some conditions has at least two minimal prime ideals

Question is : Suppose $R$ is a noetherian ring. Prove that $R$ is either an integral domain, has nonzero nilpotent elements, or has at least two minimal prime ideals. [Use the previous exercise.] ...
1
vote
1answer
84 views

Tensor Product of Complexes and the definition of the differentials

Suppose we have the following complexes, $$0 \rightarrow R \xrightarrow{x_1} R \rightarrow 0$$ $$0 \rightarrow R \xrightarrow{x_2} R \rightarrow 0$$ $$0 \rightarrow R \xrightarrow{x_3} R \rightarrow ...
2
votes
1answer
32 views

Singular matrix with entries in a ring. [duplicate]

Given a matrix $M\in A^{n\times n}$, where $A$ is a commutative ring different from $\{0\}$, then we know that if there exists a vector $x\in A^n$ such that $Mx=0$, then $\det M$ must be a zero ...
1
vote
0answers
21 views

Computing the order of a divisor in the Jacobian of a hyperelliptic curve.

Given a hyperelliptic curve of genus $g$, of equation $H: y^{2}+h(x)y=f(x)$ and defined over the finite field $\mathbb{K}$, how does one compute the order of a (reduced) divisor defined over ...
6
votes
4answers
216 views

How to learn commutative algebra?

I want to learn commutative algebra from scratch. I was wondering, as you guys are experts in mathematics, what you think is the best way to learn commutative algebra? Is there any video course ...
6
votes
1answer
44 views

What does it mean for a prime ideal to split completely?

See here. What does it mean for a prime ideal to split completely?
0
votes
1answer
41 views

Subset of points in noetherian scheme of rank $\le n$ is open

Let $\mathcal{F}$ be a coherent sheaf over a Noetherian scheme $X$. Lets define its rank in a point $x \in X$ as the dimension of $\mathcal{F}_x \otimes k(x)$ (here $k(x)$ is the residue field in ...
1
vote
0answers
37 views

Computing the cotangent complex: what's the ring?

As far as I understand, deformation theory of schemes may be calculated via the cotangent complex. I have read that in general the cotangent complex may be difficult to compute. However, I have a ...
1
vote
0answers
38 views

Geometric structure on the set of valuation rings of a field

Let $K$ be a field. Let $\mathcal{O}_K$ be the intersection of all valuation rings with quotient field $K$. Can someone give an example of a field $K$ in which we don't have a bijection of sets: $$ ...
1
vote
1answer
67 views

How to prove that the ring of algebraic integers is a Bézout domain?

I was told that the ring of all algebraic integers (that is, the complex numbers which are roots of a monic polynomials with integral coefficients) is a Bézout domain. But I have no idea how to prove ...
1
vote
1answer
57 views

Looking for a direct proof that all maximal ideals of $\mathbb C[x_1,x_2,…,x_n]$ are generated by $n$ linear polynomials

Without using Hilbert's Nullstelensatz , can we directly prove that all maximal ideals of $\mathbb C[x_1,x_2,...,x_n]$ is of the form $\langle x-a_1,x-a_2,...,x-a_n \rangle$ ? It is easy to prove it ...
5
votes
2answers
82 views

Any curve of genus three is either hyperelliptic or trigonal?

A curve $C$ is said to be trigonal if it admits a rational function $f: C \to \mathbb{CP}^1$ of degree $3$. Is any nonsingular plane curve of degree four trigonal? Can the map be chosen so that it has ...
5
votes
2answers
92 views

Divisors of differentials.

Let $\mathbb{C}$ be the base field. Suppose $C \subset \mathbb{P}_2$ is a nonsingular projective curve of degree $d > 3$. Must it be the case that $D \in \text{Div}(C)$ is the divisor of a ...
2
votes
1answer
76 views

Number of ideals in a minimal irreducible decomposition

Assume $R$ is a local ring, $M\subseteq R$ is the maximal ideal, $I\subseteq R$ is an $M$-primary ideal and $I=\bigcap_{i=1}^n Q_i$ is a minimal irreducible decomposition of $I$ (i.e. $Q_i\subseteq R$ ...
2
votes
1answer
63 views

Endomorphism commutes with its adjugate

Let $R$ be a commutative ring, $M$ a free $R$-module of rank $n$ and $f \in \rm{End}(M)$. The adjugate $f^\sharp$ of $f$ is defined by the equalities $$ f^\sharp(x) \wedge y = x \wedge ...
3
votes
2answers
32 views

Proof of a lemma which leads to Nakayama's lemma

I am trying to understand the proof of the following statement: Let $A$ be a commutative ring, let $M$ be a finitely generated $A$-module and $I$ an ideal of $A$ such that $IM=M$. Then there is an ...
2
votes
1answer
51 views

The proof of Krull's Principal Ideal Theorem

Theorem: Let $R$ be Noetherian and $P$ be a minimal prime ideal over $(a)$ for some nonunit $a$ of $R$. Then $\operatorname{ht}(P)\leq 1$. My lecture notes prove this as follows. WLOG $R$ is local ...
2
votes
1answer
37 views

Does “pseudo-independent implies independent” imply that $R$ is a field?

(All my rings are unital.) Suppose $R$ is a commutative ring and that $M$ is an $R$-module. Definition. Call a subset $X \subseteq M$ pseudo-independent iff for all proper subsets $Y$ of $X,$ the ...
2
votes
1answer
58 views

Localising a polynomial ring and non-maximal prime ideal

I'm trying to work out the following past paper question and I've got stuck. $R$ is an integral domain and $S = R[t]$, the polynomial ring in one variable over $R$. We have that $Q$ is a prime ideal ...
2
votes
1answer
60 views

Is a smooth ring extension of a UFD a UFD?

Let $A \subseteq B$ be noetherian integral domains, $A$ a UFD, and $B$ a smooth $A$-algebra (=the definition of a smooth algebra can be found in ...
2
votes
2answers
213 views

Can Zorn's Lemma be 'inverted' like this:?

Let $R$ be a (commutative) ring not equal to $0$. I want to show that the set of prime ideals of $R$ has a minimal element w.r.t. inclusion. This may be a wholeheartedly wrong attempt, but I thought ...
4
votes
4answers
138 views

Stably-free ideals are free?

In my class of algebraic topology, a friend of mine stated the following: If $R\ne 0$ is a commutative ring with unit and $I\subset R\oplus R$ is a submodule such that $(R\oplus R)/I\cong R$, ...
2
votes
0answers
18 views

Applications of module's length

I'm studying some theory about module's length and want to know motivation for this definition. I know that it's useful for intersection theory, but i know only one example from intersection theory: ...
0
votes
2answers
43 views

Is $(x^2,xy)$ a primary ideal in $k[x,y]$ for $k$ a field?

In Example of Page 52 in Atiyah's Introduction to Commutative Algebra $\mathfrak a = (x^2,xy)$ is not a primary ideal in $A = k[x,y]$ where $k$ is a field. I think, for any $z \in \mathfrak a$, ...
2
votes
3answers
115 views

Example of commutative ring that doesn't satisfy distribution of intersection over addition

I'm trying to find an example of commutative ring $R$ and ideals $\mathfrak a,\mathfrak b,\mathfrak c \in R$ such that $$\mathfrak a \cap (\mathfrak b + \mathfrak c) \neq \mathfrak a \cap ...
-1
votes
1answer
88 views

Describe the normalization of the cusp.

Show that the normalization of $A = k[x_1,x_2] / (x_2^2 - x_1^3)$ is isomorphic to $k[x]$ and describe (for $k$ algebraically closed) the induced map $Spec(k[x]) \to Spec(A)$ I know that $A$ is a non ...
1
vote
1answer
77 views

Show that this map has not the going-down property.

Let $A= k[x_1,x_2,y] / (x_2^2-x_1^2(x_1+1))$ and $Spec(A) \to Spec(k[x_1,x_2,y])$ the natural inclusion induced by the projection $k[x_1,x_2,y] \to A$. Consider the map $f : Spec(k[x,y]) \to Spec(A)$ ...
3
votes
0answers
23 views

What is $HC_0(\operatorname{Spec} k[x,y]/(xy))$?

Does anybody know how to compute $HC_0(\operatorname{Spec} k[x,y]/(xy))$? Here $HC_0(-)$ is the zeroth cyclic homology group. I'm curious since $\operatorname{Spec} k[x,y]/(xy)$ can be viewed as the ...