Questions tagged [commutative-algebra]

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

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Sharp's Exercise 16.37

I am reading Sharp's book "Steps in Commutative Algebra". But I really had difficulty in the exercise below. So can you give me a hint about the exercise below? Exercise 16.37 Let $R$ denote the ...
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37 views

Factorisation of a homogeneous polynomial over a domain.

Fulton's book on Algebraic Curve asks a question that, over a domain R, a factor of a form (by which he means a homogeneous polynomial) in indeterminates $x_1,.., x_n$ is also a form. For one ...
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29 views

Separated morphisms are stable under base change

Suppose that the map $f$ in the following diagram is a separated morphism (i.e. $\Delta_{X/S}:X\rightarrow X\times_{S}X$ is a closed immersion). I want to prove that $p_{2}$ is also a separated ...
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55 views

Chain of prime ideals in $\mathbb{Q}[x_{1},x_{2},…]$

Let $R$ be a commutative ring with unity, and $\mathbb{N}_{0}$ is the set of non-negative integers. Definition: $$\dim R=\sup\{n \in \mathbb{N}_{0}\mid\text{ there exists a proper chain of prime ...
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2answers
57 views

Is $\mathbb{Q} [X,Y]/[x^{20},y^{20}]$ a local ring?

Is $\mathbb{Q}[X,Y]/[x^{20},y^{20}]$ is a local ring? My approach is to look for a maximal ideal, but got stuck how to find a single maximal ideal. Any feedback would be appreciated.
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38 views

A question about a localization of a graded ring

Let $R=\oplus_{i\in\mathbb{Z}} R_i$ be a (commutative) graded ring of type $\mathbb{Z}$. It can be shown that if $S$ is a multiplicative set consists of homogeneous elements, $R_S$ have a natural ...
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49 views

Numerical polynomial of modules with infinite length

This is from Lemma 10.58.9 in Stacks project. Let $R$ be a Noetherian local ring, $M$ be a finite $R$-module and $N$ a submodule. Suppose $\mathrm{Length}_R(M)=\infty$ and $\mathrm{Length}_R(M/N)&...
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28 views

Contractions and Extensions of Ideals and Faithful Flatness

Let $A \subseteq B \subseteq C$ be rings with $B$ a faithfully flat $A$-module and $C$ a faithfully flat $B$-module which is an integral extension of $B$. Given a maximal ideal $I \subseteq C$, is it ...
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26 views

Definition of meromorphic differentials

In Diamond, Shurman: A First Course in Modular Forms on p. 77, the meromorphic differentials on an open set $V\subset \mathbb C$ of degree $n$ are defined as $$\Omega^{\otimes n}(V)=\lbrace f(q)(dq)^n\...
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27 views

Descending sequence for module with finite length

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ be a finite $R$-module with finite length. Then the descending sequence $$M\supseteq\mathfrak{m}M\supseteq\mathfrak{m}^2M\supseteq\cdots\...
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53 views

Two candidates for definition of splitting field

Definition 1. [Bourbaki] A splitting field of $f\in \Bbbk[x]$ is an extension $\Bbbk\subset \mathbb K$ which splits $f$ (into possibly repeated linear factors) and satisfies $\mathbb K=\Bbbk(Z(f,\...
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44 views

A question involving faithful flatness, support of a module, and Spectrum of a ring

The following theorem is taken from Matsumura's Commutative Ring Theory [M] Theorem 7.3(i) and the paragraph before it. My questions only concern the proof of the Theorem below. A ring homomorphism $...
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31 views

Comparing dimension after intersecting with a hyperplane

Let ${\mathfrak{p}}$ be a homogeneous prime ideal in $S = k[x_0, \dotsc, x_n]$ so that $x_i \not\in {\mathfrak{p}}$ and consider $H$ be the hyperplane $H = Z(x_i) \subseteq {\mathbb{P}}^n$. From ...
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1answer
44 views

Question about the proof of “direct image sheaf of coherent sheaf is coherent”

Let $f:Y\rightarrow X$ be a finite morphism of noetherian schemes. Let $\mathcal{F}$ be a coherent $\mathcal{O}_{Y}$-module. Then $f_{*}\mathcal{F}$ is a coherent $\mathcal{O}_{X}$-module. Let $\{U_{...
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103 views

Grothendieck group of local affine- surfaces with rational singularities

Let $(R, \mathfrak m)$ be an excellent, normal, local domain of dimension $2$ containing an algebraically closed field $k=R/\mathfrak m$. Let $ \pi: Y \to X=\operatorname {Spec}(R)$ be a resolution ...
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35 views

Prime ideals not prime to the conductor

Let $K$ be an algebraic number field, $\mathcal{O}_K$ its ring of integers, $\alpha\in\mathcal{O}_K$ such that $K=\mathbb{Q}(\alpha)$, $A:=\mathbb{Z}[\alpha]$, $\mathfrak{p}$ a non-zero prime ideal (...
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1answer
25 views

Relation between annihilators and exact sequence

This is repeat post from relation of annihilators on exact sequence and the hint seems unclear. $0 \rightarrow M^{\prime} \rightarrow M \rightarrow M^{\prime \prime} \rightarrow 0$ is an exact ...
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77 views

On non-factoriality of a class of simple hypersurface singularities

Let $k$ be an algebraically closed field of characteristic $0$. For which values of $n\ge 4$ the local ring $$R_n=k[[x,y,z,w]]/(x^2y+y^{n-1}+z^2+w^2)$$ is not a UFD ? I know that any such ring is an ...
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52 views

Finitely generated modules, Jacobson radical [closed]

I'm trying to solve a problem without the assumption of Nakayama's lemma, and this statement clearly implies the Nakayama's lemma. The statement goes as follow: let $R$ be a ring, and let $M$ be an $...
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86 views

Definition of isomorphism of graded rings

After searching through some literature I got a bit confused what one has to check to conclude that two graded rings are isomorphic (as graded rings). Suppose that $R$ and $S$ are graded rings, then ...
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49 views

Is there a name for algebras over a field $k$ whose residue class fields have finite dimension over $k$?

Let $k$ be a field and let $A$ be a $k$-algebra. Assume that for every maximal ideal $P \subseteq A$ the residue class field $A/P$ has finite dimension as a $k$-vector space. Is there a name for $...
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31 views

Prime ideal containing some of two ideals

Let $k$ be a field with $k[x_1,\ldots ,x_n]=:k[X]$ and $k[y_1,\ldots ,y_n]=:k[Y]$. Suppose $I$ is an ideal in $k[X,Y]$ such that $I=I_1+I_2$ where $ I_1$ and $I_2$ are ideals in $k[X]$ and $k[Y]$ ...
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59 views

$f:M \to Q$ a monomorphism with $Q$ injective and $g:M \to N$ an essential morphism, then there exists a mono $\bar{g}: N \to Q$ such $\bar{g} g= f$.

This one is based on (A) of the answer of this previous question For a projective cover $(\sigma, P)$ of a module $M$, $P$ is indecomposable implies $M$ is indecomposable Let $Q$ be an injective ...
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21 views

Is $\mathfrak{c}E$ a $C$-module for an abelian ring $C$ and an ideal $\mathfrak{c}$?

Let $C$ be an abelian ring, $\mathfrak{c}$ an ideal of $C$ and $E$ a $C$-module. Let $\mathfrak{c}E$ be the sub-$\mathbf{Z}$-module of $E$ generated by the family $(c x)_{(c,x)\in\mathfrak{c}\times E}$...
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72 views

Bounds on the degree and number of polynomials in the reduced Gröbner basis of an ideal and its radical over a field of positive characteristic.

Disclaimer: Throughout, fix a field $K$ with $\text{char}(K) = p >0$, and we assume that all the computations related to Gröbner bases are done with a fixed elimination ordering. I am currently ...
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1answer
53 views

Show that $\oplus\widetilde{M}_{\alpha}\cong \widetilde{\oplus M_{\alpha}}$ with $M_{\alpha}$ $\mathcal{O}_{X}(X)$-modules.

Let $X=\operatorname{Spec}(A)$ be an affine scheme, and let $M_{\alpha}$-be $A$-modules. I want to show that $\oplus\widetilde{M}_{\alpha}\cong\widetilde{\oplus M_{\alpha}}$. Let $D(f)$ be a ...
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71 views

Cardinality of $\operatorname{Hom}_{\mathbb{C}}(A,\mathbb{C})$

Question Prove there is no finite generated algebra $A$ over $\mathbb{C}$ such that the cardinality of $\operatorname{Hom}_{\mathbb{C}}(A,\mathbb{C})$ is exactly $\aleph_0$. I need to prove it ...
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1answer
28 views

$A \subset B$ be a faithfully flat extension of domains and $B$ is integrally closed then $A$ is also integrally closed.

Let $A \subset B$ be a faithfully flat extension of integral domains. If $B$ is integrally closed then I have to show that $A$ is also integrally closed. Assuming $L,K$ be the field of fractions of ...
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43 views

Depth of $i$-th syzygy module, where $i$ is at most the depth of the ring

Let $(R, \mathfrak m)$ be a Noetherian local ring of depth $t\ge 1$. So for any finitely generated free $R$-module $F$, we have $\operatorname {depth}(F)=\operatorname {depth}(R)=t$. My question: ...
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1answer
34 views

A question about a step in a proof of the Krull Intersection Theorem

Lately, I have been using Steve Kleiman and Allen Altman lecture notes on commutative algebra, A Term of Commutative Algebra, that are available for free on internet, to study the subject. In those, ...
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46 views

Koszul algebras

Why we are interested to find whether an algebra is Koszul or not? What is the significance of being Koszul? Only this that we will know certain homological invariants of the algebra like projective ...
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45 views

Noetherian commutative ring with finite but not discrete spectrum

I know this is probably not that hard but I don't know how to properly approach this. So I am asked to give an example of a ring fulfilling the properties in the title of the question. Now I know ...
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2answers
33 views

Complement of multiplicative set is a (prime) ideal.

Let $R$ be a commutative ring with $1\neq0$. I'm trying to show that the complement $\mathfrak p$ of a multiplicative subset $S\subseteq R\setminus\{0\}$ is a (prime) ideal. In particular, I am having ...
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Ext and Tor duality

In the appendix of this paper of Felix, Halperin and Thomas, Proposition $A.6$ is the following: Let $R$ be a differential graded algebra, and let $M$ be an $R$-module. Then $$\text{Tor}^R(\...
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22 views

if $\operatorname{Dim}(A)=0$, then every prime ideal is maximal [duplicate]

Given a ring $A$, By the chain of prime ideals of $A$, we mean a sequence of prime ideals of $A$ such that $P_0 \subsetneq P_1 \dots \subsetneq P_{n-1}\subsetneq P_n$. Further this chain has a length $...
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35 views

Motivation for separated and proper schemes

Hartshorne mention at the beginning of section 4 in chapter 2 that the definition of separated is similliar to hausdorff. We all can see that. That is also what I found in google. Again - we all can ...
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62 views

The completion of localization

This is the example 5.6.3 in chapter I of Hartshorne's Algebraic Geometry. For the reducible variety $$ Y = \Big\{ (x,y) \in {\mathbb{A}}^2 ~:~ xy = 0 \Big\} $$ it follows that the local ring ${\...
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1answer
51 views

Depth of $R/I$ as an $R$-module versus as a ring

Let $(R, \mathfrak m,k)$ be a Noetherian local ring. Let $I\subseteq \mathfrak m$ be an ideal of $R$. Then $(R/I, \mathfrak m/I, k)$ is a Noetherian local ring but also $R/I$ is a finitely generated $...
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39 views

Weil divisors associated to Cartier divisors

Let $X=\{x_3^2=x_1^2+x_2^2\}\in \mathbb{P}^3$, let $L_1=Z(x_2,x_1+x_3)$, and $L_2=Z(x_2,x_1-x_3)$. I don't quite understand how to get that $\operatorname{div}(x_2)$ is associated to $[L_1]+[L_2]$ ...
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56 views

Cohomological criterion for non-triviality of negative part of graded module

Let $R$ be a graded ring and $M$ a graded module. Then for sufficently large $n$, we have $$H^0(\operatorname{Proj}(R), \widetilde{M}(n))\cong M_n.$$ Hence if I want to show that $M_{>0}$ is non-...
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1answer
26 views

Vector spaces over an integral domain and the canonical isomorphism between the tensor products

Let $A$ be an integral domain and write $S=A-\{0\}$. Then the total ring of fractions $S^{-1}A$ of $A$ is an abelian field. Note that $\varepsilon:A\rightarrow S^{-1}A,\,a\mapsto a/1$, is an injective ...
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Struggling while learning commutative algebra

I took a course in abstract algebra (till galois theory), topology (with some very basic algebraic topology), smooth manifolds, complex analysis and never did I struggle even epsilon close to how I am ...
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1answer
28 views

Fiber product of local artinian rings with a fixed residue field

Let $k$ be a finite field and suppose $A,B,C$ are Artinian local rings with residue field $k$. Suppose we have local homomorphisms $f \colon A \to C, g \colon B \to C$ which induce the identity on ...
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1answer
16 views

A module annihilated by a maximal ideal is semisimple

I want to show that if $M$ is a module over a commutative ring $R$ that is annihilated by a maximal ideal $I$ of $R$, then $M$ is a semisimple $R$-module. What I have in mind is the following: if $M$ ...
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25 views

“Understanding” the techniques involved in this proof that $F[x, y]/\langle y^2 - x^3 \rangle$ is a domain and is not integrally closed

Let $A = F[x,y] / \langle y^2 - x^3 \rangle$. In a past exam question (from last year), students were asked to show that $A$ is an integral domain, and also that it is not integrally closed in its ...
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49 views

Computing whether a set of polynomials cuts out a homogeneous variety

I have a set of multivariate polynomials over $\mathbb{Q}$, and I want to compute whether they cut out a homogeneous variety. My first idea is to compute the radical of the ideal $I$ that they ...
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1answer
55 views

extending ring homomorphism into fields

Let $A$ be a subring of $B$ such that $B$ is integral over $A$. Show that every ring homomorphism $f:A\rightarrow K$ with $K$ an algebraically closed field can be extended to a ring ...
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1answer
23 views

an irreducible affine curve is normal if and only if it is nonsingular

An is normal if and only if it is nonsingular. This statement comes from Kemper, A Course in Commutative Algebra. He says to use Proposition 8.10 and Theorem 14.1. Theorem 14.1. A Noetherian local ...
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24 views

Intersection of $I$-adic neiborhoods equal to intersection of kernel of localization map

Let $R$ be a noetherian ring and $I$ is an ideal. Let $M$ be a finite $R$-module. I want to show $$\bigcap_{n=0}^\infty I^nM=\bigcap_{I\subseteq \mathfrak{m}}\ker(M\to M_\mathfrak{m})$$ where $\...
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47 views

Bijection between $\text{Spec}(A\otimes_R B)$ and $\text{Spec}(A)\times_{\text{Spec}(R)}\text{Spec}(B)$ for $R$-algebras A and B.

I'm looking for a bijection between $\text{Spec}(A\otimes_R B)$ and $\text{Spec}(A)\times_{\text{Spec}(R)}\text{Spec}(B)$ for $R$-algebras $A$ and $B$. Where $\text{Spec}(A)\times_{\text{Spec}(R)}\...