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

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5
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129 views

An example of prime ideal $P$ in an integral domain such that $\bigcap_{n=1}^{\infty}P^n$ is not prime

I am looking for an example of prime ideal $P$ in an integral domain such that the ideal $\bigcap_{n=1}^{\infty}P^n$ is not a prime ideal. This is a followup to this question where the ring was not ...
0
votes
1answer
37 views

How do you prove the ideal $I= (X^2, XY)$ has infinitely many distinct irredundant primary decompositions?

I have come up with the following two different decompositions of the ideal $I= (X^2, XY)$: $I = (X) \cap (X^2, Y)$ and $I = (X) \cap (X^2, XY, Y^2) = (X) \cap (X, Y)^2$. Can we generalize this ...
4
votes
1answer
50 views

Trying to understand Hilbert function from Joe Harris' Algebraic Geometry

Let $X \subset {\bf P}^d$ be rational normal curve. $X$ is defined as the image of the map $v_d : {\bf P}^1 \to {\bf P}^d$ $$[a:b] \to [a^d:a^{d-1}b:\dots:ab^{d-1}:b^d].$$ The map $v_d$ induces a map ...
1
vote
1answer
32 views

How to show that Hilbert function of points in projective space is constant for large values $\in N$?

Recall that for $X\subset \mathbb{P}^{n}$ an algebraic set with homogeneous coordinate ring $Γ(X) = k[x_1, ..., x_{n+1}]/I(X)$, the Hilbert function of $X$ is a function $h_{X }: \mathbb{N} → ...
0
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0answers
31 views

Possible error and confusing assumption with Liu's proof of a result on ideals in graded algebras

Let $B = \bigoplus_{d\geq 0} B_d$ be a graded $A$-algebra. Let $I$ be an ideal of $B$ and associate to it the homogeneous ideal $I^h = \bigoplus_d (I\cap B_d)$ (so $I$ is homogeneous iff $I = I^h$). ...
1
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1answer
61 views

A relation between the intersection of two nonzero principal ideals and the zero ideal

For an integral domain $D$, we have $\langle a\rangle\cap \langle b\rangle\neq 0$ for every nonzero elements $a,b \in D$, Now in a general case, let $R$ be a commutative ring with 1, such that $R$ ...
3
votes
2answers
154 views

An example of prime ideal $P$ such that $\bigcap_{n=1}^{\infty}P^n$ is not prime

I am looking for an example of prime ideal $P$ such that $\bigcap_{n=1}^{\infty}P^n$ is not prime. In a Prüfer domain such an intersection is always a prime ideal.
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0answers
44 views

Difference in definition of Krull dimension

The definition of Krull dimension of a module over a ring $R$ in the sense of deviation of the poset of submodules ordered by inclusion may not coincide with the definition for non-Noetherian rings ...
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0answers
51 views

Let $V=V(X^2-Y^3, Y^2-Z^3) \subset \Bbb A^3$, $P=(0,0,0), m=m_P(V).$ Find $\dim_k(m/m^2)$.

Let $V=V(X^2-Y^3, Y^2-Z^3) \subset \Bbb A^3$, $P=(0,0,0), m=m_P(V).$ Find $\dim_k(m/m^2)$. Here I have seen $F=Y^2(X^2-Y^3)+Y^3(Y^2-Z^3) \in V$ has multiplicity $m_P(F)=4$ and ...
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2answers
58 views

Checking if a point is singular by looking at the algebraic definition

I am trying to calculate if a point is singular or not. What I want to use is that a point is nonsingular if $\dim_k( m_p/m^2_p)=1$, where $m$ is the maximal ideal of an algebraic curve at a point ...
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vote
0answers
32 views

Minimal number of generators for a f.g module over $\mathbb{Z}[x]$

Let $R = \mathbb{Z}[x]$, $M$ be a finitely generated torsion free $R$-module with rank $n$, and let $\mu ( M)$ denote the minimal number of generators of $M$. For each prime ideal $P \subset ...
1
vote
1answer
43 views

any rational function on $\Bbb P^1$ is constant?

What is the flaw in the answer for proving any rational function on $\Bbb P^1$ is constant? Let $\phi: \Bbb P^1 \to \Bbb A^1$ be a rational function. Since $\Bbb A^1 \subset \Bbb P^1$ we can think ...
0
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1answer
51 views

How to prove every rational map from $\Bbb P^1 \to \Bbb P^n$ is regular. [closed]

How to prove every rational map from $\Bbb P^1 \to \Bbb P^n$ is regular. For $f=(\frac {f_1}{f'_1},...,\frac {f_n}{f'_n})$ where each $ f_i,f'_i$ are monomials of same degree. But now how to show ...
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0answers
29 views

Show that there are open sets $P\in U \subset X$ and $Q\in V \subset Y$ and an isomorphism of $U$ to $V$ which sends $P$ to $Q$.

Let $X$ be a variety. Define local ring of a point. Let $X$ and $Y$ be two varieties. Suppose there are points $P \in X$ and $Q \in Y$ such that the local rings $O_P(X)$ and $O_Q(Y)$ are isomorphic as ...
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0answers
36 views

if $\Delta$ is pure, then what happens to betti-numbers of $I_{\Delta}$ or $I_{\Delta^v}$

Assume that $\Delta$ is a simplicial complex and $\Delta ^v$ is its Alexander dual. Is there a known fact that: if in addition $\Delta$ be pure, then what happens to betti-numbers of $I_{\Delta}$ or ...
3
votes
0answers
42 views

Understanding and modifing a theorem from Fulton's “algebraic curves”

I am reading Theorem 2 in chapter 3.2 in Fulton. It says that $mult_p(F)=\dim_k (m_p(F)^n/m_p(F)^{n+1})$ for a sufficiently large $n$. Reading the proof I understood that we eventually play a game of ...
2
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0answers
40 views

Property of morphism can be checked on a general open affine contained within target

In the first few exercises for Chapter 2, Part 3 of Hartshorne's book on schemes there are several results that have the general shape as follows: Let $f:X\to Y$ be a morphism of schemes such that ...
5
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0answers
127 views

Applications of the Dedekind-Hasse criterion

It is a fact that an integral domain $R$ is a principal ideal domain if and only if there is a Dedekind-Hasse function $|R|\setminus\{0\}\xrightarrow{\ \ \delta\ \ }\mathbb{N}$ on $R$, i.e. a function ...
0
votes
2answers
31 views

Local property of dimension

Let $R$ be a commutative ring with unity. If the Krull dimensions of all the localizations $S^{-1}R$ are zero, where $S$ runs among multiplicative subsets of $R$, is it true that the Krull dimension ...
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0answers
25 views

Homogeneous Rings (As in Solomon Lefschetz Algebraic Geometry)

By a form we mean a homogeneous polynomials. In this book the author says: Consider now a collection $\mathfrak{R}_H$ of forms $g\in K[x_0,\cdots,x_n]$ such that if $g,g'\in \mathfrak{R}_H$ then ...
2
votes
2answers
61 views

Structure sheaf of $\operatorname{Spec}A$

I was trying to do the construction of the structure sheaf of $\operatorname{Spec}A$ by myself when I encountered a difficulty. To do this I plan to construct the sheaf on a basis of $SpecA$ and then ...
0
votes
1answer
18 views

Existence of homogeneous nonzero divisors?

Suppose that $M$ is a graded ring over $R$. Let $P$ be the maximal homogeneous proper ideal in $R$. If there is an $f \in P$ which is a nonzero divisor for $M$ ($fm = 0$ implies $m = 0$), but $f$ is ...
0
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1answer
53 views

A reference for patch topology

I saw a paper that defines the definition of patch topology on the Spectrume of a commutative ring with identity. Is there any reference for this concept that give some property for this topology? ...
0
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1answer
40 views

Ideal of the union of two skew lines in $\mathbb{P}^{3}$

Let $$ L_{1}=V(X_{0},X_{1})\subseteq\mathbb{P}^{3}, $$ $$ L_{2}=V(X_{2},X_{3})\subseteq\mathbb{P}^{3}. $$ I want to prove that $$ I(L_{1}\cup L_{2})=(X_{0}X_{2},X_{0}X_{3},X_{1}X_{2},X_{1}X_{3}). $$ ...
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vote
1answer
37 views

Flat family: limit of intersection vs intersection of limits

Consider a $\textbf{flat}$ surjective map $f: X \rightarrow \mathbb{A}^1$. The general fibers $F_{\epsilon}$ are canonically isomorphic, and the special fiber $F_0$ above $0 \in \mathbb{A}^1$ is not ...
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0answers
86 views

invariants that can be measured by Local Cohomology

Let $(R,m)$ and $(S,n)$ be local rings and $S$ is an $R$-Algebra via homomorphism $f:R\to S.$ What invariants can be measured by Local Cohomology (and what application it has)? Example of what I ...
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1answer
29 views

Canonical reference on applied elimination theory?

I would like to study elimination theory as it is applied to sets of multivariate polynomials. I am interested in cases where some variables are completely eliminated, as well as cases where higher ...
1
vote
1answer
52 views

Polynomial ring in infinitely many variables over a noetherian ring is coherent

If $R$ is noetherian, show that the polynomial ring of infinite variables $R[x_1,x_2,...]$ is coherent, i.e. every finitely generated ideal is finitely presented. I don't really know how to get ...
0
votes
0answers
20 views

Extension of scalars by a submodule, which is also a ring

Suppose we have $M$ a free $R$-module, where $R$ is a commutative ring. Suppose we also have $S \subset M$ as a submodule and $S$ is also a ring. Can we say anything about $_S M = S \otimes_R M$? In ...
5
votes
2answers
92 views

Idempotent ideals in certain commutative rings

Let $R$ be a commutative ring with zero Jacobson radical such that each maximal ideal of $R$ is idempotent. Does it guarantee that each ideal is idempotent? I know only that if each maximal ideal ...
2
votes
1answer
35 views

Tensor left adjoint as a bi-functor?

Working in the category of modules over a fixed ring $A$ $\operatorname{Mod}_A$, we have the adjunction: $$F(X) = X\otimes_A M \dashv G(X) = \operatorname{Hom}_A(M,X)$$ for a fixed module $M$. Is ...
0
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1answer
45 views

Relation between the localization $R_f$ and the polynomial ring $R\left[\frac{1}{f}\right]$?

Let $R$ be a commutative ring with $1$. (If required, assume also that $R$ is an integral domain.) Consider the localization $R_f$ at $0\neq f \in R$ where the multiplicative set is $S=\{f^n\}_{n \geq ...
3
votes
2answers
84 views

Frobenius mophism, Exercise 7.3 R of Ravi Vakil's book on algebraic geometry

I am a geometry person and got stuck in Exercise 7.3.R which is about Frobenius morphism. Suppose $p$ is a prime and $r \in \mathbb{Z}^+$. Let $q=p^r$ and $k=\mathbb{F}_q$. Define $\phi ...
0
votes
1answer
13 views

Finite presentation of Weil algebras

Below is a picture of an excerpt of Lavendhomme's book on synthetic differential geometry. I'm confused about the "structure constants" $\gamma_{ij}^k$. Initially I thought the superscript $k$ is ...
2
votes
1answer
214 views

Is it true that if some power of an ideal is primary, then the ideal itself is also primary?

Is it true that if some power of an ideal $I$ is primary, then $I$ itself is also a primary ideal? I do not know whether the above statement is true or there is a counterexample. If one wants to ...
3
votes
1answer
46 views

Category of finitely presented $R$-algebras cartesian closed?

On page 26 of these notes, in the paragraph between formulas $(63)$ and $(64)$, the author says the category of finitely presented $R$-algebras is cartesian closed. I thought this category was ...
4
votes
1answer
78 views

What are the natural surjections in the proof of Hopf's classification theorem?

I am currently reading Hatcher's book, trying to understand the proof of Hopf's classification Theorem on Hopf algebras that says the following: Every Hopf algebra $A$ that is commutative and ...
3
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0answers
46 views

Tensor product with endomorphism ring

Let $k$ be a commutative ring, $M$ a $k$-module and $k\longrightarrow A$ a $k$-algebra. Is it true that $$A\otimes_k\operatorname{End}_k(M)\cong\operatorname{End}_A(A\otimes_kM)?$$ If not, under what ...
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0answers
35 views

Geometric line $R$ is a “field”, $D= \left\{ x\in R\mid x^2 =0 \right\}$ is not an ideal of $R$

I am confused by properties of $R$, the geometric line in synthetic differential geometry. In the book Synthetic Differential Geometry by Kock, he assumes $R$ is only a commutative ring. However, in ...
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votes
1answer
39 views

Estimating regularity of sheaves with rank of certain modules and zeroth cohomology

I'm studying Eisenbud's book "Geometry of syzygies", in particular the Gruson-Lazarsfeld-Peskine theorem for Castelnuovo-Mumford regularity. I'm concerned about an intermediate step in the proof. Let ...
2
votes
1answer
41 views

Contraction and extension of ideals respect inclusions, sums and intersections

Let $R$ be an integral domain. Let $Y$ be a multiplicatively closed subset of $R$ which contains $1$ but not $0$. Define $S=RY^{-1}=\lbrace ry^{-1} : r \in R, y \in Y \rbrace$ as well as ...
3
votes
1answer
47 views

On characterizing modules that don't annihilate any module under tensor product.

Let $R$ be a commutative ring and $M$ an $R$-module. Then under what condition can we deduce that for any nonzero $R$-module $N$, $M\otimes_RN\neq0$?
2
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0answers
47 views

Understanding Localized Rings mod an ideal.

Hi guys I am working with Fulton's book and I am trying to understand for myself the elements of two rings. $O_p(\mathbb{A}^n)/JO_p(\mathbb{A}^n)$ and $O_p(V)/\bar{J}O_p(V)$ Where $I = ...
2
votes
1answer
99 views

Minimal prime ideals of $\mathcal O_{X,x}$ correspond to irreducible components of $X$ containing $x$

Let $X$ be an algebraic variety over an algebraically closed field $K$. By definition, $X$ is a separated prevariety, and $x \in X$. I'm trying to show (i): The minimal primes of $\mathcal ...
0
votes
1answer
85 views

the natural map from $N_1$ to $\varinjlim N_i$ is injective?

Direct limit $\varinjlim N_i$ of a direct system $\{N_i\}_{i \in I}$ is defined to the union of all $N_i$ modulo certain equivalence relations. From the definition it seems that if ...
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vote
0answers
48 views

Going-up but not lying over?

Let $R \subseteq S$ be an extension of commutative rings with identity. Assume $1_R = 1_S$. This extension satisfying going-up if for any inclusion of primes $P_1 \subseteq P_2$ of $R$, and any ...
2
votes
1answer
40 views

Computing an explicit tensor product

So I think this question is trivial but I can't seem to be able to do it so here we go : what is the tensor product $$k[x,y]/(y^2-x^3) \otimes_{k[y]} k[x,y]/(y^2-x^3)\ ?$$ My guess is that it is ...
1
vote
1answer
60 views

Intersection of n hyperplanes in projective space of dimension n is not empty

I want to prove the following: Let $H_1,\dots,H_n$ be $n$ hyperplanes in $\mathbb{P}^n =\mathbb{P}^n \mathbb{C}$. Then $\cap_{i=1}^n H_i$ is not empty. Please be noted that this is an exercise ...
0
votes
0answers
51 views

Definitions of Weil Algebras

I am confused by several definitions of Weil Algebras and their connection to each other. Kock's book on synthetic differential geometry defines a Weil algebra over a ring $R$ as an $R$-algebra of ...
1
vote
1answer
52 views

Integral and prime ideal in Dedekind domain

Let $A$ be an Dedekind domain, $K$ its quotient field, $L$ a finite separable extension of $K$, and $B$ the integral closure of $A$ in $L$. If $p$ is a prime ideal of $A$, then $pB$ has a ...