Questions tagged [commutative-algebra]
Questions about commutative rings, their ideals, and their modules.
15,160
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An algebra given by generators and relations and an algebra generated as a subalgebra by some elements. Are they isomorphic?
Let $k$ be a field and consider the following two $k$-algebras:
$R_1 = k[a,b,c,d] / (ab - cd). $
$R_2$ is the unital $k$-subalgebra of $k(t)[x,y]$ generated (as an algebra) by
$x,y,tx,t^{-1}y$.
Are $...
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Prove that $A[b_1,...,b_n]$ is a finitely generated A-module if $b_i's$ are integral over A
This question was left as an exercise in my class on commutative algebra and I am not sure about it's solution. So, I am posting it here.
Question: if $b_1,..., b_n \in B$ are integral over A, then ...
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Explicit examples of formal power series which is not rational functions?
This MO question https://mathoverflow.net/questions/249541/formal-power-series-is-taylor-expansion-of-rational-function-iff-hankel-determin states that if $k$ is a field and $k[[T]]$ is the power ...
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Finite modules over Artinian Rings are Artinian [duplicate]
This question was left as an exercise in my class of Commutative algebra and I am struck on it.
Question: Prove that finite modules over artinian rings are artinian.
Thoughts: If ring is artinian then ...
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Prerequieries and refrences for Stanley-Reisner Ring
I need to study about topological properties of stanley reisner rings but I'm so lost in finding refrences.
I know a little algebraic topology and I know Abstract algebra (groups, rings, fields, ...
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Tensor product of maximal Cohen-Macaulay modules
Let $(R,\frak{m})$ be a commutative Cohen-Macaulay local ring and $M$ and $N$ two maximal Cohen-Macaulay modules of Krull dimension $d$. Is $M\otimes_RN$ Cohen-Macaulay?
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Hilbert's Basis Theorem by induction on maximal degree
Please help me check if the following proof of Hilbert's Basis Theorem is correct.
Proposition: If $A$ is a Noetherian ring, then so is $A[X]$.
Proof: It suffices to show that every countably ...
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Definition of rank of module homomorphism between free modules?
I'am reading the Eisenbud's Commutative Algebra and some question arises about some notation.
Theorem 20.9. Let $R$ be a ring. A complex
$$ \mathcal{F} : 0 \to F_n \xrightarrow{\varphi_n} F_{n-1} \to \...
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Localizations of Noetherian rings are Noetherian [duplicate]
This question was left as an exercise in my class of commutative algebra and I am stuck on part iv.
Question: Prove these assertions :
(i) Let I be an ideal of A. If A is noetherian , then A/I is ...
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Prove that coefficients of g(x) and h(x) are integral over A [duplicate]
This question is from my assignment in a course in commutative algebra and I am not able to prove it.
Let $A\subseteq B$ are ring extensions. Let f be a monic polynomial in A[x]. Suppose f(x) =g(x) h(...
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ideals that become principal in extension
Let $A,B$ be two Dedekind domains such $A\subset B$. On denote by $K$ and $L$ their quotient field respectively. One assumes que $[L:K]$ is finite.
Let $I$ be an ideal of $A$. Suppose that $IB=a B$ ...
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Does a torsion-free coherent sheaf embed into a locally free sheaf?
Let $ X $ be a Noetherian integral regular scheme and $ \mathcal{F} $ be a torsion-free coherent sheaf. (One definition of torsion-free is that the natural map $ \mathcal{F} \rightarrow \mathcal{F} \...
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How to show $\operatorname{Pic}(X)=0$? Exercise $14.2$.Q Vakil's notes
I'm reading Vakil's notes and I'm struggling with the exercise $14.2$.Q. I've been able to prove everything except $\operatorname{Pic}(X)=0$ with
$$
X=\operatorname{Spec}\frac{k[x,y,z]}{(xy-z^2)}.
$$
...
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Completion of a finite lenght module [closed]
I am reading a proof on Bruns and Herzog's book about Cohen-Macaulay rings and they assume the following propertie:
If $(R,\mathfrak{m},k)$ is a Noetherian local ring and $M$ is a finite lenght module,...
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Extension by a skyscraper sheaf
Let $C$ be a nodal curve. Then we have the normalisation map $f: \mathbb{P}^1 \to C$. I want to understand the sheaf $f_* \mathcal{O}_{\mathbb{P}^1}$. There is the map $\mathcal{O}_{C} \to f_* \...
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When does $m\otimes n\in M\otimes N =0 $ imply there exists $r\in R$ such that $rm=0$ or $rn=0$?
When does $m\otimes n\in M\otimes N =0 $ imply there exists $r\in R$ such that $rm=0$ or $rn=0$?
Any specific cases are very welcomed. There exists this nice answer from Eric saying that this is the ...
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Where can I find information about sets whose subsets produce unique values?
I'm interested in sets whose subsets produce unique values under a given commutative operation. I don't know what this is called. I tried searching for terms like unique commutative invariant, but ...
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Finding specific isomorphism $\mathrm{Hom}(M, R)\otimes_R F \rightarrow \mathrm{Hom}(M, F)$
Let $R$ be a commutative ring with $1_R$. Let $M$ be a $R$-module and $F$ be a free $R$-module.
How do I find a specific isomorphism to show that $$\mathrm{Hom}(M, R)\otimes_R F \cong \mathrm{Hom}(M, ...
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A question in corollary of Hilbert Nullstellansatz
This corollary was part of my lecture notes in commutative algebra and I am having questions in proof of it.
Statement: Let I be an ideal in $K[x_1,...,x_n]$ , K is algebraically closed . Then $ I (...
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Question in proof of a corollary of Hilbert Nullstellansatz
I have been reading commutative algebra from lecture notes and I have some questions in a proof of a corollary of Hilbert Nullstellansatz.
Let R be a finitely generated k-algebra. Then for an ideal $...
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Find the radical of an ideal in the ring of polynomials over the complex numbers
In $\mathbb{C}[X,Y,Z,W]$ I have the following ideal
$$I=(XZ-Y^2,XW-YZ,X^2-WY,YX-WZ,YW-Z^2,ZX-W^2).$$
I'm trying to find it's radical using Hilbert Nullstellensatz, i.e., trying to find the set of ...
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Questions in structure theorems for artinian rings [duplicate]
This theorem was covered in the lecture notes of commutative algebra from where I am studying and I am struck on it on some points.
Statement: Every artinian ring is a finite product of artinian ...
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A is noetherian and every prime ideal of A is maximal then...
This proof was part of my lecture notes in Commutative algebra and I am having trouble following it. So, I am asking the question here.
Statement: If A is noetherian and every prime ideal of A is ...
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Question in the proof that any Artinian ring is noetherian
This theorem is from my lecture notes of Commutative Algebra and I am struck on 2 points of the proof.
Statement: Any artinian ring is noetherian.
Proof: Let A be an artinian ring. Let $M_1 ,...,...
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Is $S^{-1}(q_i)$ is a primary ideal in $S^{-1} A$?
This question was left as an exercise in class of commutative algebra and I am struck on it.
Let A be a noetherian ring and $I=\cap_{i=1}^n q_i$ be a minimal primary decomposition. Let $p_i = \sqrt{...
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Maximal algebraic independent in commutative algebra over field
In field extension, maximal algebraic independent elements in a set of generators (generate by means of fraction of generators) will also be maximal alebraic independent amoung all subsets. It is then ...
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$ \mathfrak mR_{\mathfrak m} $-primary ideal is the localization of some $\mathfrak m$-primary ideal?
Let $\mathfrak m$ be a maximal ideal of a commutative Noetherian ring $R$. Let $J$ be an ideal of $R_{\mathfrak m}$ such that $\mathfrak m^n R_{\mathfrak m}\subseteq J \subseteq \mathfrak mR_{\...
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Find (a generating set for) $\mathbb{Q}[x]\cap I$ where $I=\langle x^2-y,y^2-x,x^5-x^2\rangle$ (generate gröbner basis).
Consider the polynomial ring $\mathbb{Q}[x,y]$ and the ideal $I=\langle x^2-y,y^2-x,x^5-x^2\rangle$ in $\mathbb{Q}[x,y]$. $G=(x^2-y,y^2-x)$ is a (reduced) gröbner basis for $I$ wrt. graded ...
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Prove that $Z(A)= \cup_{i=1}^n p_i$
I am reading commutative algebra from a class notes and I am not able to understand this proof.
Statement: Let $p_i$ are primary ideals associated to I (or A/I). Then show that $Z(A)= \cup_{i=1}^n ...
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A question in 1st uniqueness theorem of primary decomposition
I am self studying commutative algebra from a class notes based on atiyah and macdonald and I am struck on this proof.
Statement; Let $I=\cap_{i=1}^n q_i$ be a minimal primary decomposition of I. ...
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How to find radical of these ideals
This question was left as an exercise in my commutative algebra class and I am not able to complete it.
If $I=(x^2, xy) = (x) \cap (x^2,y)$ , then show that Ass(A/I)= {(x), (x,y)}.
The definition ...
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Geometric Interpretation of Jacobson rings -- Every locally closed subsets of $\operatorname{Spec} A$ consists of a single point is closed
I'm currently working on Atiyah&MacDonald's book on commutative algebra. I'm trying on Exercise 5.26, and remaining to show that
Question: $A$ is a Jacobson ring if and only if
$$
\text{every ...
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definition of closed immersions of schemes
Qing Liu's algebraic geometry and arithmetic curves defines closed immersion as follows(definition 2.22): We say that a morphism $(f, f^{\#}): (X, O_{X}) \to (Y, O_{Y})$ is an open immersion (resp. ...
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Pullback is not exact
Let $f: X \to Y$ be a map of schemes. Then we have the "quasicoherent" pullback which takes $F \in Qcoh(Y)$ and gives $f^* F = \mathcal{O}_{X} \otimes_{f^{-1} \mathcal{O}_{Y}} F$. This ...
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Valuation ring of completion of a field
I am actually quite confused. I have done an exercise that $\mathbb{Z}_p$ is a completion of $\mathbb{Z}$ w.r.t. the $p$-adic norm. Then again I got to know after reading somewhere that $\mathbb{Z}_p$ ...
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A question regarding primary decomposition of ideals
This question was left as an assignment in my class of commutative algebra but I was not able to completely solve it.
Prove that $I = (x^2, xy)= (x) \cap (x^2,y) = (x) \cap (x^2,xy,y^n)$ for any $n \...
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Maximal Cohen-Macaulay of finite flat dimension
Let $(R,\frak{m})$ be a commutative Cohen-Macaulay local ring and $M$ a maximal Cohen-Macaulay module of Krull dimension $d$. If $M$ has a finite flat dimension, is it true to say that its injective ...
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Is $\mathrm{Hom}_S(M,E) \otimes_R X \cong \mathrm{Hom}_S(\mathrm{Hom}_R(X,M),E)$ with $R$ a Noetherian ring?
Given a left Noetherian ring $R$, a ring $S$, a $R$-$S$-bimodule $M$, an injective cogenerator $E$ of right $S$-module, is there an natural isomorphism $$
\mathrm{Hom}_S(M,E) \otimes_R X \cong \mathrm{...
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For every commutative local ring R, the maximal ideal is nilpotent
A commutative ring $R$ is said to be a local ring if it has a unique maximal ideal. My question is: for every commutative local ring $R$, always the maximal ideal has nilpotency that is the maximal ...
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Is $\bigcap_{n=1}^\infty I^n$ contained in a minimal prime ideal?
Let $I$ be a proper ideal of a commutative Noetherian ring $R$. Let $J:=\bigcap_{n=1}^\infty I^n$. Then, is it true that $J \subseteq P$ for some minimal prime $P$ of $R$?
By prime avoidance, I am ...
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Equivalent Conditions for a Prime to not Contain Conductor
I am working through some notes on algebraic number theory, and am trying to show the following five conditions are equivalent. Here, $A\subset B$ is an extension of Dedekind domains corresponding to ...
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$\mathrm{Res}(f, g, h)$ as a product of three resultants?
I'm currently working with resultants, which I define as follows: let $k$ be a field, $V$ be a 2-dimensional vector space over $k$, and let $S^dV^*$ be the $d$-th symmetric power of dual space, i.e. ...
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Epimorphism of algebras restricted to ring of invariants
Suppose $G$ is an affine algebraic group that acts rationally on $K$-algebras $R$ and $R'$, ie every element is contained in a rational representation: a finite-dimensional $G$-stable vector subspace $...
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Unsure about proof for Vakil 5.4.M
I've been having some trouble proving the following in Vakil's FOAG recently:
Suppose A is a $k$-algebra, and $l/k$ is a finite extension of fields.
(Most likely your proof will not use finiteness; ...
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Transition functions for $\mathcal{O}(1)$ on $\mathbb{P}^1$
I am trying to understand the transition functions for $\mathcal{O}(1)$ on $\mathbb{P}^1$. I've been stuck on this while reading these notes (Proposition 1.8, page 3) on divisors and invertible ...
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Finding the kernel of a given map
Let $\mathbb K$ be an algebraic closed field and let $f,g\in\mathbb K[X,Y]$ be two polynomials so that $V_{\mathbb K}(f)$ and $V_{\mathbb K}(g)$ don't have common irreductible components and $(0,0)\in ...
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Hartshorne Theorem II.7.17 - Why is $\mathscr{I} \to \mathscr{M}^{-dn}f_*\mathscr{L}^d$ injective?
I'm reading the proof of the following theorem from Hartshorne (Theorem II.7.17), which says the following:
Let $Z$ be a variety and let $X$ be a quasi projective variety, both over $k$. Suppose $f:Z\...
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Is a normal domain whose prime ideals are totally ordeded a valuation ring?
Recall one of the definition of a valuation ring is a domain whose ideals are totally ordered. (Then it will be a normal domain.) But if we restrict to all prime ideals the reverse is not true. The ...
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Describe $\operatorname{Spec}(\mathbb{R}[x]/(x^n))$
I want to describe $\operatorname{Spec}(\mathbb{R}[x]/(x^n))$. For starters, I know that since $\mathbb{R}$ is a field, then $\mathbb{R}[x]$ is a PID, and therefore the only prime ideal in $\mathbb{R}[...
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In the proof of construction of canonical module
I'm reading David Eisenbud's Commutative Algebra, p.539, Theorem 21.15 :
I'm trying to understand the underlined statement. Why is it true?
My first attempt is,
Question 1. Let $I:=\operatorname{...
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