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

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Almost-invariant polynomials under dihedral group action

Think about the dihedral group $D_4$ acting on the polynomial algebra $\mathbb C[x_1, \cdots, x_4]$ via generating permutations $(x_1\ x_2)$, $(x_3\ x_4)$, and $(x_1\ x_3)(x_2\ x_4)$. I'd like to ...
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19 views

Reference on a result about integral closures.

Could you please give a reference or a sketch of a proof for the following proposition? Proposition: The integral closure of a complete local Noetherian domain $R$ is module-finite over $R$ You ...
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38 views

Independent set of variables modulo ideal and Krull dimension

Let $\mathfrak{a}\subseteq \Bbbk[x_1,\ldots,x_n]$ be an ideal, where $\Bbbk$ is a field. Let the maximal set of indeterminates independent modulo $\mathfrak{a}$ be of cardinality $k$. There is a ...
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41 views

Fiber of morphism induced by map on stalks

Given a morphism of schemes $f\colon X\to Y$ and a point $x\in X$, the map on the stalks induces a morphism $\operatorname{Spec}\mathcal{O}_{X,x} \to \operatorname{Spec}\mathcal{O}_{Y,f(x)} $. Is it ...
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78 views

Exercise $2$ from chapter $5$ of Eisenbud's Geometry of Syzygies book

I am trying to solve exercise $2$ from chapter $5$ of Eisenbud's The Geometry of Syzygies book.The problem is as follows: Let $X$ be the union of two disjoint lines in $\mathbb P^3$, or a conic ...
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2answers
48 views

Is the ideal of a variety the annihilator of a subspace of the symmetric algebra?

Let $V$ be a vector space over an algebraically closed field $K$. Let $\mathrm{Sym}(V^*)=\mathrm{Sym}(V)^*$ be the symmetric algebra on $V$, i.e. if we give a basis $e_1,...,e_n$ of $V$ and let ...
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93 views

Annihilator of a flat ideal

Let $R$ be a commutative ring and let $I$ be a finitely generated flat ideal of $R$. Let $J=\mathrm{Ann}(I)$. How can one prove that $I\cap J=0$? This can be found as a remark in the paper of ...
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32 views

Characterization of Groebner Bases in terms of uniqueness of remainders

Let $I$ be an ideal of a polynomial ring $R=k[x_1,\ldots,x_n]$ over a field $k$. A Groebner basis of $I$ is a finite generating set $\{g_1,\ldots,g_m\}$ such that every leading monomial (according to ...
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21 views

Fraction rings ideals members

Let $R$ be a ring with fraction ring $R_S$ and ideal $I$. I saw in arguments that when $a/s$ is in $I_S$ they dont say $a$ is in $I$. Instead they say $a/s=b/t$ with $b \in I$. Why? Many thanks.
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29 views

System of parameters in Noetherian local rings

I'm trying to understand the theorem for systems of parameters in Noetherian local rings, which says: Let $R$ be a Noetherian local ring with maximal ideal $m$. Then there exists an $m$-primary ideal ...
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35 views

Regularity and Short Exact Sequence

Suppose $ 0 \to M_1 \to M_2 \to M_3 \to 0$ is a short exact sequence of finitely generated graded $k[x_0,...,x_r]$-modules. Then show that $\mathrm{reg}(M_1) ...
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What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$?

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. If $L$ is a number field ...
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35 views

Trying to Compute Regularity and degree

Definition: For a finite subset $X \subset \mathbb P^r$,the Hilbert function $H_X(d)$ is constant for large $d$ and its value is the number of points in X,usually called the degree of $X$. Let ...
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1answer
53 views

Filling in Proof: Well-definedness of depth(I,M).

From Eisenbud's Commutative Algebra with A View Toward Algebraic Geometry (Theorem 17.4): Let $M$ be a finitely generated $R$-module, where $R$ is Noetherian. If $$r= \min \{i : H^i(M\otimes ...
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32 views

Some ideal property in a local ring

If we change the ideal $$(X_1,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$$ to $$(X_1^2,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$$ in this problem, what is the answer to the raised question? Again, the new local ...
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Lying Over Theorem + Is $\mathbf{C}[x] \hookrightarrow \mathbf{C}[x, y]/(xy-1)$ an integral extension?

I am confused about something. When introducing the Lying Over Theorem -- namely, that if $f \colon R \subset S$ is an integral extension then $f^* \colon \mathrm{Spec}(S) \to \mathrm{Spec}(R)$ is ...
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56 views

Equivalence of smoothness and freeness of sheaf of differentials

Let $S$ be a regular locally Noetherian connected scheme, $f:X \to S$ a morphism of finite type with $X$ irreducible. Let $x \in X$ and $s = f(x)$ such that $$ \dim \mathcal{O}_{X,x} = \dim ...
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68 views

Unramified at a point $x \in X$ if and only if $\Omega _{X,x} = 0$

This is Corollary 6.2.3 in Liu's book. Let $f: X \to S$ be a morphism of finite type of locally Noetherian schemes. Then $f$ is unramified at a point $x \in X$ if and only if $\Omega_{X/S, x} = ...
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Trying to understand Corollary $4.7 $ (page $60$) from Eisenbud's Geometry of Syzygies

Corollary: If $X$ is a set of $n$ points in $\mathbb P^r$, then the regularity of $S_X$ is the smallest integer $d$ such that the space of forms vanishing on the points $X$ has codimension $n$ in ...
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37 views

Incidence correspondence as a scheme

The incidence correspondence is $\Sigma=\{(x,L) \mid x\in L\}\subset \mathbf{P}^n\times\mathbf{Gr}(k,n)$. What I ask myself is what this actually means, after all the underlying set of the fibre ...
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29 views

Proving continuity between prime spectrum of a ring and its localisation at a point

Consider a commutative ring R with unity and consider and element $f\in R$. I wish to show that there is a homeomorphism between the two sets $\{\mathcal{p}\in Spec(R),f\notin \mathcal{p}\}$ where $p$ ...
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“Closure” of a polynomial ring by fraction field

Let $k$ be an algebraically closed field of characteristic $p>0$, $A$ a regular noetherian $k$-algebra, $K$ the fraction field of $A$ and $\bar{K}$ an algebraic closure of $K$. Does there exist a ...
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Is this result on the bound of regularity of an ideal true?

I am solving a problem in which i need to use the following result but i am not sure whether the result is true on not: If the ideals $I_0,...,I_n$ are generated by linear polynomials in ...
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Can we recover a compact smooth manifold from its ring of smooth functions?

It is well-known that if $X$ is a reasonably nice topological space (compact Hausdorff, say) then we can recover $X$ from the ring $C(X)$ of continuous functions $X\to\mathbb R$; see this MSE question ...
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Testing if a submodule is free

This is hopefully a very simple question. In Gröbner Bases in Commutative Algebra by Ene and Herzog, I find the Problem 4.11, which says ($S$ here is a polynomial ring over a field $K$, $S=K[x_1\ldots ...
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1answer
51 views

Integral extension and s.o.p.

Let $R\subset S$ be an integral extension. Is a system of parameters of $R$ a system of parameters of $S$? I think so, since there is good behavior in dimensions. Many thanks.
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50 views

Generic fiber to the Frobenius morphism

Let $k$ be an algebraically closed (perfect) field of characteristic $p>0$ and $f:\mathrm{Spec}\, k[t] \to \mathrm{Spec}\, k[X]$ be the Frobenius morphism induced by $X \mapsto t^p$ and identity on ...
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A Question about the Intersection Multiplicity

In a recent lecture the lecturer defined the local ring of an irreducible affine variety $V$ at $P\in V$ as $$ \mathcal{O}_{V,P}=\{\phi\in K(V)\mid\phi\text{ is defined at }P\}. $$ Then he defined ...
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The Archimedean place of $\mathbb{Q}$

Is there a way to extract the Archimedean absolute value of $\mathbb{Q}$ from its field structure in a way analogous to its non-archimedean absolute values? Here is some context: Given a valuation ...
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K-theory of a classifying space (part two)

Continuing my previous question, given a compact, connected Lie group $G$, there is a sequence of maps $$R(G) \to \hat{R}(G) \overset\sim\to K^*(BG) \to \hat{H}^*(BG;\mathbb Q)$$ apparently first ...
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Krull dimension localization with coefficients [duplicate]

Let $A[x_1, \ldots, x_n]$ be a polynomial ring over an integral domain, $A$. Let $s = a\prod_{1\leq i \leq n}{x_i}^{\alpha_i}$, $a \in A$. What is the Krull dimension of $A[x_1, \ldots, x_n]_s$?
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Krull dimension on localization

Given $A$ an integral domain and $A[x_1, \ldots, x_n]$ a polynomial ring over $A$. Let $s = \prod_{1\leq i \leq n}{x_i}^{\alpha_i}$. What is the Krull dimension of $A[x_1, \ldots, x_n]_s$? Will it be ...
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Commutativity of a ring from idempotents.

In a ring $R$ with unity, every element can be written as product of finitely many idempotents. Can one show that the ring is commutative?
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Primary decomposition in a Dedekind Domain

I was a little bit puzzled with the following problem that I have recently come across: Let $R$ be a Dedekind domain and let $P$ be a prime ideal in $R$. Is it true that $P^k$ is an irreducible ...
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28 views

Why is $\varphi(X_i) = X_i + b_i$ an automorphism of $K[X_1,\dots,X_n]$?

I'm trying to justify to myself the assertion (used here) that given a field $K$ and elements $b_1,\dots,b_n\in K$, the map $\varphi(X_i) = X_i + b_i$ is a $K$-automorphism of $K[X_1,\dots,X_n]$. ...
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66 views

Corollary to Lemma of Nakayama

In Matsumura's Commutative Algebra there is the following Corollary to the Lemma of Nakayama: Let $A$ be a ring, $M$ an $A$-module, $N$ and $N'$ submodules of $M$, and $I$ an ideal of $A$. Suppose ...
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Possible Inaccuracy at classic paper by Bayer and Stillman

In reading the paper Bayer and Stillman, "A criterion for detecting $m$-regularity", i believe i have encountered what may be a little inaccuracy, which i describe next. Let $I$ be a homogeneous ...
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30 views

Height of prime ideal containing the variable of a polynomial ring

I have a ring $R$ and a prime ideal $P$ of $S=R[t]$ with $t \in P$. I'm trying to prove that if $\mathrm{ht}(P/tS)$ is finite then $\mathrm{ht}(P) > \mathrm{ht}(P/tS)$. Here ...
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Direct Product of Completions

This question is regarding Theorem 8.15, page 62 of Matsumura's Commutative Ring Theory. It says that if $A$ is a semi-local ring and $I=m_1\cdots m_r$ be the Jacobson radical of $A$. Then ...
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Deduce that there are short exact sequences

Show that for $n>0$ there is a short exact sequence of chain complexes $0\rightarrow C_i(X;\mathbb{Z})\stackrel{f}{\rightarrow} C_i(X;\mathbb{Z})\stackrel{g}{\rightarrow} ...
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Need a reference/proof for computing the regularity of ideal of points in $\mathbb P^d$?

In a lecture notes on 'Cohomology modules' i read the following remark: Given a set $X$ of points in $\mathbb P^d$,using the Local Cohomology modules one can easily compute the reg$(S_X)$ where ...
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Is there an adjective for rings whose every non-zero prime ideal is maximal?

(All my rings are commutative and unital.) Question. Is there an adjective for rings whose every non-zero prime ideal is maximal? Remarks: Every PID has this property; more generally, every ...
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a little “paradox” in local cohomology of zero-dimensional ideals

Let $S = k[x_1,x_2,x_3]$ be a polynomial ring of dimension $3$ over an infinite field, and let $I$ be a homogeneous ideal of height $3$. Since $S$ has no zero divisors, the Krull dimension of $I$ is ...
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About the discriminant ideal

Let $E/K$ be a separable field extension of degree $n$, let $A$ be a Dedekind Domain which quotient field is $K$, and let $B$ be the integral closure of $A$ in $E$. Then we have that the ideal ...
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36 views

Minimal free resolution of ideal generated by three homogeneous polynomials

I am trying to solve the following exercise; Let $R=k[x_0,x_1,x_2]$ and $f_i$ homogeneous polynomials of degree $d_i, 0\leq i \leq 2$. Suppose $f_0,f_1,f_2$ have no common roots in $\mathbb P^2$. ...
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Krull dimension of an algebra

Given the ring, $\mathbb{Z}_6[x,y]/\langle x \rangle$. What is the Krull dimension of the ring? Isn't the following a chain of prime ideals in the ring, $\langle \overline{2}\rangle \subsetneq ...
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39 views

Book recommendation on Primary decomposition of ideals [closed]

I'm trying to prepare a presentation on "Primary Decomposition of Ideals" which is the title of my project. But I'm new for the subject so I need help on the following points How to outline my ...
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Inverse limit of ideals equal to expected ideal of inverse limit?

Suppose we have a map $(A_n \to B_n)_{n \in \mathbb N}$ of inverse systems of unital rings and a system $\mathfrak a_n \lhd A_n$ of ideals, one sent into the next under the maps $A_n \to A_{n-1}$. ...
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72 views

Why can one say WLOG assume $R$ is a local ring in Atiyah and MacDonald's 3.15 Exercise?

In Exercise 3.15 in Atiyah and Macdonald's Introduction to Commutative Algebra, the ring $R$ can be assumed to be a local ring, because of proposition 3.9. That proposition states that if $\phi: M ...
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64 views

Exercise II-11 from Eisenbud-Harris, subscheme of dimension $0$, degree $3$, supported at origin isomorphic to what?

Suppose that $K$ is algebraically closed, and let $Z = \text{Spec}\,K[x_1, \ldots, x_n]/I \subset \mathbb{A}_K^n$ be any subscheme of dimension $0$ and degree $3$, supported at the origin. How do I ...