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

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18 views

Result showing that a certain valuation ring in some function field has to be a DVR?

I know that if $R$ is a valuation ring such that $0 \to \mathbb{C} \to R$ is a left-split exact sequence, then there exists a discrete valuation ring $C$ with $R \subset C$ so that $0 \to \mathbb{C} ...
2
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1answer
44 views

Finding roots of an irreducible polynomial in a ring that is not a domain.

Let $A$ be a commutative ring, and $p\in A[X]$ a polynomial of degree $d>0$. If $A$ is an integral domain, we can find a ring $B$ such that $A\subseteq B$ and $p$ has a root in $B$. For example ...
2
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1answer
40 views

Isomorphic function fields of projective curves, bijection of points.

Suppose curves $C$, $D \in \mathbb{CP}^2$ are nonsingular. If their function fields are isomorphic, i.e. $K_C \cong K_D$, then do we necessarily have a bijection of points on $C$ and $D$? Can we do ...
3
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2answers
114 views

Show that $\mathbb{C}[x,y]/(x^2+y^2-1)$ is a UFD. [duplicate]

I am trying to prove that the ring $\mathbb{C}[x,y]/(x^2+y^2-1)$ is a UFD. I have an hint, that suggests to find an isomorphism between $\mathbb{C}[x,y]/(x^2+y^2-1)$ and $\mathbb{C}[e^{it},e^{-it}]$, ...
3
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3answers
115 views

Exercise 1.9 in Hartshorne - is my initial attempt a good start?

Hartshorne's Chapter 1, exercise 1.9 asks us to show that irred. components of $Z(\mathfrak a)$ have dimension $\geq n-r$ if $\mathfrak a$ is an ideal generated by $r$ elements. I think I've reduced ...
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1answer
25 views

Factorization of a ring morphism for artinian rings

Let $(R,m)$ be a complete local $k$-algebra. Let $A$ be a local artinian algebra with residue field $k$. Then, since $A$ is artinian, any map $f:R \rightarrow A$ factorizes as $R \rightarrow R/m^n ...
3
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1answer
47 views

Commutative Hereditary Rings

Is it true that the ring $\mathbb Z/n\mathbb Z$ ($n≠0$) is hereditary if and only if $n$ is square-free? The "if" part is OK to me because any field $\mathbb Z/p \mathbb Z$ ($p$ prime) is a PID, ...
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0answers
40 views

Definitions of the ring $R/(x_0^\infty,\ldots,x_{n-1}^\infty)$

Let $R$ be a ring, and let $I = (x_0,\ldots,x_{n-1})$ be a finitely-generated ideal inside of $R$, generated by a regular sequence. In algebraic topology one often encounters a ring, usually denoted ...
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0answers
71 views

Is an irreducible ideal in $R$ irreducible in $R[x]$?

Let $R$ be a commutative Noetherian ring and $I\subset R$ an ideal that is irreducible in the sense that if $I = J_1 \cap J_2$, then $I=J_1$ or $I=J_2$. Is (the ideal generated by) $I$ irreducible in ...
2
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1answer
28 views

$A\subseteq B$, $B$ integral over $A$, $\mathfrak{q}_{1}\subsetneq\mathfrak{q}_{2}$, then $A\cap\mathfrak{q}_{1}\subsetneq A\cap\mathfrak{q}_{2}$.

Let $A, B$ be commutative rings such that $A\subseteq B$ and $B$ is integral over $A$. I want to prove that if $\mathfrak{q}_{1},\mathfrak{q}_{2}$ are prime ideals of $B$ such that ...
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1answer
38 views

Another question about proposition 5.15 of Atiyah-MacDonald.

I have a trouble in understanding the proposition 5.15 in the book from Atiyah and MacDonald. I see that some time ago another user asked a similar question (Proposition 5.15 Atiyah Macdonald: ...
2
votes
1answer
31 views

If $\overline{k}$ is an algebraic closure of a field $k$, then $\overline{k}[x_{1}, \dots, x_{n}]$ is integral over $k[x_{1}, \dots, x_{n}]$.

I want to prove that if $\overline{k}$ is an algebraic closure of a field $k$, then $\overline{k}[x_{1}, \dots, x_{n}]$ is integral over $k[x_{1}, \dots, x_{n}]$. It is used in exercise 11.3 of the ...
2
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0answers
39 views

Characterization of tensor products of fields

For which commutative rings $R$ are there field homomorphisms $L \leftarrow K \to L'$ (not assumed to be algebraic or anything) such that $R \cong L \otimes_K L'$? Is there an intrinsic ...
2
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0answers
39 views

Subring of all elements represented by quotients of function field.

Suppose $K_C$ is the function field of a curve $C$ and $p \in C$. Let $\mathcal{O}_k \subset K_C$ be the subring of all elements represented by quotients $G/H$ where $G, H \in \mathbb{C}[x, y, z]$ are ...
2
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1answer
35 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
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2answers
116 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
48 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
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0answers
37 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
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1answer
44 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
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0answers
35 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 ...
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1answer
33 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 ...
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2answers
32 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
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1answer
60 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 ...
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0answers
35 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
96 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
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0answers
29 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
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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
88 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
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1answer
96 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
30 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
56 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, ...
10
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0answers
91 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
93 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
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1answer
45 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
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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
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0answers
86 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 ...
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2answers
107 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.] ...
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1answer
85 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
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1answer
36 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 ...
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0answers
25 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 ...
7
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4answers
261 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
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1answer
47 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?
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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 ...
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0answers
52 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
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0answers
40 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
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1answer
77 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
59 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
93 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
104 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
78 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. $Q_i\subseteq R$ ...