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

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Localization of $\Bbb k[x]$

Let $R= \Bbb K [x]$ and $S=\{x^n: n \in \Bbb Z, n \geq 0 \} $. Let $D$ be the localization of $R$ in $S$, that is $D = S^{-1}R = \{ \frac{r}{s}: r \in R, s\in S \}$. By using the Universal Property ...
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47 views

If $A$ is an integrally closed domain, then is $GrSym A$ (the graded symmetric algebra on $A$) also integrally closed?

Question: If $A$ is an integrally closed domain, which is f.g. k algebra over an algebraically closed field, then is $A'$ (defined in the edit below) also integrally closed? (I am thinking about this ...
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30 views

Dimension of an affine scheme, 24.5.7 of Vakil's book

In the remark 24.5.7 of Vakil's book, it claims that the dimension of the scheme $\operatorname{Spec}k(x) \otimes k(y)$ is a $k(x)$-scheme with dimension one, where $k$ is a field and $x,y$ are two ...
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Why are Unique Factorization Domains (UFD's) geometrically significant?

We know that for $A$ a UFD, it's class group is trivial. More generally, for a factorial (stalks are UFD's) scheme $X$ (that is also noetherian and normal), we have an isomorphism between it's Picard ...
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$T^i$ functors in Hartshorne's Deformation Theory

In chapter 3 of Hartshorne's Deformation Theory, he defines functors $T^i$ for $i=0,1,2$ that take as input a ring homomorphism $A\rightarrow B$ and a $B$-module $M$ and outputs $T^i(B/A,M)$, a ...
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1answer
52 views

Flat schemes over artinian local ring with isomorphic special fibers

I'm sure this is standard, but I don't know where to find it. If $A$ is a local artinian $k$-algebra, $X_1,X_2$ are finite type schemes flat over $A$, and $f:X_1\rightarrow X_2$ is a morphism over ...
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1answer
27 views

Zero divisors in a finite dimensional Poincaré duality algebra

Let $A= \bigoplus_{i=0}^{n}A_i$ be a finite dimensional algebra over a field $\mathbb{k}$ such that $A_0 \cong \mathbb{k} \cong A_n$. Consider the bilinear form $$\varphi: A_i \times A_{n-i} \to ...
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Questions about flat limits and associate points, Vakil's section 24.4.12

Suppose $(A,m)$ is a discrete valuation ring and $[m]$ is the closed point of $Spec\,A$ and $\eta$ is its generic point, which is also the only nontrivial open set. If we have a morphism $\pi:X ...
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22 views

Finitely graded ring and zero divisors

Let $R= \bigoplus_{i=0}^{d} R_i$ be a finitely graded ring such that $R_0$ is a field and $R_d \cong R_0$. I'm trying to understand how zero divisors work in such a ring; when is it true that for $r ...
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Intersection multiplicity inequality problem

I have a question about an inequality that was stated in my class but never proven. We stated p is simple $I_p(F \cap G+H) \ge \min ( I_p (F \cap G) , I_p(f \cap H))$. Where we defined $I_p(F \cap ...
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Structure sheaf consists of noetherian rings

Let $X\subseteq \mathbb{A}^n$ be an affine variety. The ring $k[x_1,\ldots,x_n]$ is noetherian because of Hilbert's basis theorem. The coordinate ring $k[X]=k[x_1,\ldots,x_n]/I(X)$ is noetherian ...
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1answer
37 views

Krull Dimension of Direct Limits of Zero-Dimensional Rings

Is it true that direct limit of a directed system of zero-dimensional rings is zero-dimensional (in the sense of Krull)? Thanks for any help! If this is true, it is inferred that any prime ideal of ...
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46 views

Question about intersection multiplicity of a curve and it's tangent line

If we have a double point $a$ on some complex curve, call it $C$, defined by some polynomial $f$ and we have only one tangent line at $a$, call it $T_l$, then the intersection multiplicity $I(a,f \cap ...
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1answer
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Checking the intersection multiplicity for two curves

Hi guys I am using Fulton's book on algebraic curves as a main material to study. I am trying to check the intersection multiplicity of $p=(0,0)$ of $g(x,y)=y^2-x^3-x^2$ and $h(x,y)= y^2-x^3+x$ My ...
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39 views

Are there only finitely many prime ideals in this tensor product?

Let $\phi: A\to B$ be a ring homomorphism and suppose $B$ is finitely generated as an $A$-module. Let $\mathfrak{p}\subseteq A$ be a prime ideal and let $K$ be the field of fractions of ...
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55 views

Direct limit of completions of finitely generated submodules

Let $A$ be a noetherian, local, integral domain with maximal ideal $\mathfrak m$. Moreover let $M$ be an $A$-module; I'd like to know if there exists an explicit expression of the module: ...
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69 views

What is the name of the property $x^m=x$ when $x$ is in a ring?

I have been doing problems in Atiyah & MacDonald's Introduction to Commutative Algebra, and in problem 1.6 it asks to assume the existence of an idempotent element in an ideal whenever the ideal ...
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1answer
39 views

Nil but not Nilpotent [duplicate]

Is there a commutative ring $R$ with zero Krull dimension such that its Jacobson radical is nil but not nilpotent? Of course, in Noetherian case (which leads to Artinian case) for $R$ each nil ideal ...
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69 views

Krull dimension of finitely generated algebra over field

Let $k$ is a field and $A = k[x_1, x_2, ..., x_n]$ is finitely generated $k$ algebra, which is also an integral domain. Then I know that $A$ has finite Krull dimension (equal to the transcendence ...
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109 views

Finding the defining equations for a simple quotient variety

First of all let me note that I have no experience at all with modern algebraic geometry so if at all possible I would appreciate an answer not involving the concept of a scheme. I have however some ...
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41 views

Reduction of multivariate polynomial degree on the support of an ideal?

Consider a multivariate polynomial function $P$ in variables $z_1,...,z_n\in\mathbb{C}$. Furthermore, consider a polynomial ideal spanned by a set of multivariate polynomials $Y_i$ with ...
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1answer
66 views

Natural map $f^{\ast}f_\ast\mathcal{L}\to\mathcal{L}$

Let $f\colon X\to Y$ be a morphism of smooth varieties and $\mathcal{L}$ an invertible sheaf on $X$. How is the "natural map" $$f^{\ast}f_\ast\mathcal{L}\to\mathcal{L}$$ defined (which should be ...
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49 views

$f_\ast\mathcal{O}_X$ is locally free of rank $r=\deg f$ for a finite $r$-to-one morphism $f$.

Let $f\colon X\to Y$ be a finite $r$-to-one morphism between smooth projective varieties. How does one show that $f_\ast\mathcal{O}_X$ is a locally free sheaf of rank $r$ on $Y$ ? Any insights for ...
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47 views

Extension of scalars and completions

Suppose that $A$ is a Noetherian regular (added later) local domain. Moreover $\widehat A$ is $\mathfrak m$-adic completion $\widehat A$ w.r.t the maximal ideal and $K$ is the fraction field of $A$. ...
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if the fibers of all points are finite then will the map be finite?

If $f:X\to Y$ is a regular map between affine varieties then we say $f$ is finite if $k[X]$ is integral over $k[Y]$. If $f$ is finite then fibers of all points are finite. I think the converse of this ...
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52 views

A property of closed subsets in Spec (R)

Let $R $ be a commutative ring with 1, and let $Spec (R) $ be the set of all prime ideas of $R $. It is well-known that if $V (I)\subset V (J)$, then $rad (J)\subset rad (I)$, where $V (I)$ is the set ...
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3answers
68 views

What is a generator for this ideal?

According to the Hilbert Basis theorem, there is a finite number of generators for the ideal $I$ in $\mathbb{R}[x,y]$ generated by $\{(x^{n}+y^{n})\mid n\in \mathbb{N})\}$. What is the precise ...
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1answer
47 views

Does taking torsion commute with extension of scalars by a flat module?

Let $S$ and $R$ be Noetherian integral domains. Suppose $S$ is $R$-flat. Let $M$ be an $R$-module. Is it true that $\text{Torsion}_S(S\otimes_R M)=S\otimes_R (\text{Torsion}_R(M))$? I can see ...
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1answer
44 views

Noether Normalization, finiteness over a sub algebra

I'm currently doing an exercise on Noether Normalization in the context of a course on commutative algebra and I'm not sure whether the solution I have come up with is correct or does even make sense. ...
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1answer
52 views

Is noetherianity really about cardinals or ordinals?

If $\kappa$ is any cardinal, then one may define a "$\kappa$-Noetherian" ring as a ring such that for any module that has a generating set $S$ satisfying $|S|< \kappa$, then any submodule also has ...
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1answer
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Squarefree monomial ideals have a decomposition as the intersection of monomial prime ideals.

We've proven the following theorem in class: Every monomial ideal has a presentation $$I = \bigcap_{i=1}^m Q_i,$$ where each $Q_i = (x_{i_1}^{a_1}, \dots , x_{i_k}^{a_k})$. I've tried proving ...
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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 ...
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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 ...
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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 ...
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33 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} → ...
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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 $I = I^h$). ...
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1answer
75 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 $R$ ...
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155 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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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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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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63 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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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 ...
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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 ...
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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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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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38 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 ...
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46 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 ...
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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 ...
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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 ...
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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 ...