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

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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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1answer
25 views

Let $A$ a Noetherian ring and $ q \in\mathrm{Spec} (A)$. Then $q^{(n)} A_q=q^{n}A_q$.

Let $A$ be a Noetherian ring and $ q \in\mathrm{Spec} (A)$. Then $q^{(n)} A_q=q^{n}A_q$, where $q^{(n)}= \lbrace a \in A \mid \exists d \in A \setminus q\text{ such that }da \in q^n \rbrace$ and ...
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1answer
28 views

Annihilator of a flat ideal [on hold]

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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0answers
12 views

Need a reference (/proof) for a result related with regularity and degree.

Somewhere on internet i have read the following statement: Let $X$ be the disjoint union of two lines of a conic contained in a plane in $\mathbb P^2$. Then $\mathrm{reg}(I_X) \geq ...
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1answer
48 views

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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1answer
23 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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1answer
41 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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1answer
17 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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27 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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17 views

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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40 views
+50

Testing if a submodule is free

This is hopefully a very simple question. In "Groebner 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 $K$, ...
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19 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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1answer
39 views

Trying to understand Corollary $4.7 $ (page $60$) from Eisenbud's Geometry of Syzygies book:

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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1answer
26 views

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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0answers
47 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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54 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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2answers
69 views

How does extension of restriction of $M$ relate to $M$?

Let $A,B$ be rings, $f:B\to A$ be a ring homomorphism, and $M$ be an $A$-module. We can view $M$ as a $B$-module via restriction, and we may then extend the restriction of $M$ to an $A$-module by ...
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1answer
130 views

Ramification in the ring of all algebraic integers

If $F$ is a finite extension of $\mathbb{Q}$ then its of integers $R$ is a Dedekind domain, and has unique factorization of ideals into powers of prime ideals. For each prime number $\ell$, you can ...
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2answers
58 views

Notation in commutative algebra

I am doing some exercises on commutative algebra and came along the following expressions, which were not elaborated on. Is someone familiar with them? The first is for $p$ a prime number ...
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34 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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0answers
41 views

“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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3answers
75 views

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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1answer
26 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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1answer
44 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 $S$ and conversely? I think so, since there is good behavior in dimensions. Many thanks.
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43 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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0answers
23 views

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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1answer
152 views

Determine all discrete valuations on $\mathbb{C}(x)$.

To clarify, for a field $K$, a valuation $v$ on $K$ is a map $v:K^{\times}\to G$ for $G$ an ordered group (written additively) such that for any $a,b\in K^{\times}$: 1) $v(ab)=v(a)+v(b)$; 2) ...
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1answer
29 views

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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0answers
14 views

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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1answer
108 views

Smoothness of a subalgebra

Assume $A \subseteq B \subseteq C$ are commutative noetherian domains of zero characteristic, $C$ is a f.g. $B$-module, $B$ is a f.p. $A$-algebra ($B$ is not necessarily a f.g. $A$-module), $C$ is a ...
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1answer
48 views

An example of a c.i./Gorenstein/C.M. integral domain which is not integrally closed

If I am not wrong, it is known that: {Regular rings} $\subsetneq$ {Complete intersection rings} $\subsetneq$ {Gorenstein rings} $\subsetneq$ {Cohen-Macaulay rings}. It is known that a regular ring ...
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2answers
150 views

“Closure” and “neighborhoods” in Spec(A)

While trying to work through the sequence of problems in Atiyah-Macdonald's first chapter regarding the prime spectrum of a ring, I've run across a small point of confusion. Namely: In the point ...
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1answer
124 views

Plane curves isomorphic to the affine line

Let $C$ be a plane curve parametrized by $x=f(t),y=g(t)$ where $f(t),g(t)\in k[t]$. We can easily see that the coordinate ring of $C$ is isomorphic to $k[f(t),g(t)]\subset k[t]$. So $C$ is isomorphic ...
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0answers
61 views

The multiplicity of $X$ at $x$ does not change when $X$ is cut by a generic hypersurface: what are those generic conditions?

Given an algebraic variety $X$ with a point $x \in X$, the multiplicity of $X$ at $x$ is defined as the multiplicity of the maximal ideal of $x$ in the local ring $\mathcal{O}_{X,x}$. In ...
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3answers
905 views

What does the topology on $\operatorname{Spec}(R)$ tells us about $R$?

Let $R$ be a commutative ring with a unit. $\newcommand{\spec}{\operatorname{Spec}}\spec(R)$ denotes the set of all prime ideals in $R$, and it can be topologized using the Zariski topology. Last ...
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0answers
33 views

Krull dimension localization with coefficients

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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1answer
42 views

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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1answer
81 views

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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49 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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1answer
355 views

Valuation rings

What's the spectrum of a valuation ring? How to describe morphisms from it to a scheme? Is it enough to set the image of generic point and of a maximal ideal and correspondent map of local rings?
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1answer
171 views

Linear compact module

Can you tell me Why a finite module over a complete local ring is a linear compact module? I am study about linear compact module, can you tell me some concerned paper? Thanks you very much!
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1answer
19 views

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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2answers
157 views

Suppose that for every nonzero $R$-ideal $I$ and element $a \in I$ there exists a unique $R$-ideal $J$ such that $IJ=(a)$. Then $R$ is Noetherian.

Suppose that for every nonzero $R$-ideal $I$ and element $a \in I$ there exists a unique $R$-ideal $J$ such that $IJ=(a)$. Then $R$ is Noetherian. I'm having trouble proving this. To note, $R$ is ...
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1answer
31 views

Computing prime factorization of ideals?

I want to compute the prime factorizations of the ideals $\langle 4\sqrt{-14}\rangle$, $\langle 6\sqrt{-6} \rangle$ and $\langle 4\sqrt{-5} \rangle$ in the ring of algebraic integers of ...
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2answers
27 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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1answer
178 views

Projective dimension of tensor product $M\otimes M$

If the projective dimension of an $R$-module $M$ is finite, then can we say that projective dimension of tensor product $M\otimes M$ (as an $R\otimes R$-module) is finite?
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1answer
69 views

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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0answers
40 views

Kahler differentials over non-algebraically closed fields

Let $A = k[x_1,\ldots,x_n]/(f_1,\ldots,f_m)$ be a finitely-generated $k$-algebra, then at least when $k$ is algebraically closed, the module of Kahler differentials is $$ \Omega_{A/k} = A dx_1 \oplus ...
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1answer
27 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 ...
2
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1answer
55 views

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 ...