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

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5
votes
0answers
133 views

Question about the nullstellensatz for projective schemes

Assume that $ G $ is a graded ring. Assume that $A$ is a relevant homogeneous ideal (that is, it does not contain the irrelevant ideal $ \oplus_{n > 0}G_n$). I am having trouble proving the ...
12
votes
2answers
144 views

Uniformly solvable families of polynomials

It is a famous theorem that there is no "quintic formula", i.e. there is no formula which expresses the roots of a quintic polynomial $x^5+a_4x^4+\cdots+a_0$ in terms of $a_4,\ldots,a_0$ and rational ...
4
votes
3answers
273 views

example of a flat but not faithfully flat ring extension

I am learning commutative algebra and there is a definition about faithfully flat modules or ring extensions. I can't think of an example of a flat but not faithfully flat ring extension or module. ...
2
votes
0answers
63 views

Support of a direct sum of local cohomology modules

Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$. Let $M$ be a finitely generated $R$ module. How can we show the following: $$\operatorname{Supp}(\bigoplus_{j\ge ...
2
votes
1answer
60 views

special case of Nagata's Lemma (Matsumura p.86)

Let $K$ be a field and $R$ a valuation ring of $K$ with maximal ideal $m_R$. Let $a \in R$ such that $1-a \in m_R$. Statement: For any $s$ that is not a multiple of the characteristic of $R/m_R$, ...
2
votes
1answer
74 views

All local cohomology modules being zero

Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$-module. Suppose $H_{I}^j(M)=0$ for all $j$, then how can one show that $M=IM$? The converse of ...
9
votes
2answers
428 views

Vanishing of a certain Tor

I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...
4
votes
0answers
50 views

Lattices as invertible module

Let $E$ be an etale algebra over $\mathbb{Q}$. In other words, $E$ is a finite sum of number fields. Let $L$ be a lattice in $E$, and $R$ the order associated to $L$. More explicitly, $$R=\{ e\in ...
6
votes
1answer
148 views

is the dual of a finitely generated module finitely generated?

I recently thought of this and have no idea whether over a general commutative ring the dual of a finitely generated module is finitely generated. This must be known.
2
votes
2answers
79 views

Some question on localization of polynomial ring

Let $S=A[x_1,\dots, x_r](r \geq 2)$ be a polynomial ring where $A$ is a commutative ring. Then is it true that $S=\bigcap_{i=1}^r S_{x_i}$? If $S$ is $A$-algebra and $x_i$ are not zero divisors, then ...
4
votes
1answer
66 views

Finite Projective Dimension implies non vanishing Ext

Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$? Can't we write the free module as a direct ...
3
votes
1answer
81 views

$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$

Let $R$ be a Noetherian ring and $I$ an ideal of $R$. If $n$ is the cohomological dimension of $I$, then why is the following isomorphism true: $$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$$ The ...
6
votes
2answers
186 views

Show field of fractions is finite extension of $\mathbb{Q}$

Let $A$ be a ring which is also a finitely generated $\mathbb{Z}$-module. If $A$ is an integral domain and $K$ is its field of fractions and $K$ has characteristic zero, then why is $K$ a finite ...
3
votes
1answer
139 views

Discrete Valuation Ring and Subring of the Fractions Field

Let $R$ be a Discrete Valuation Ring, and $K$ its fractions field. Now if $B\subseteq K$ is a subring with $R\subseteq B$ then we have $$B=R \text{ or } B=K.$$ Now this seems to be a very basic ...
2
votes
1answer
243 views

proof of the Krull-Akizuki theorem (Matsumura)

This set of questions refers to the proof of the Krull-Akizuki theorem given in Matsumura's Commutative Ring Theory, pages 84-85. For those who don't have the text, i will provide the details. The ...
1
vote
1answer
98 views

Integral closure of k-algebra

Let $k$ be a field and $A$ a finitely generated algebra over $k$ that doesn't have zero divisors. Why is the integral closure of $A$ a finitely generated module over $A$ ? (edited)
9
votes
2answers
176 views

If $R$ is a commutative ring with unity and $R$ has only one maximal ideal then show that the equation $x^2=x$ has only two solutions

If $R$ is a commutative ring with unity and $R$ has only one maximal ideal then show that the equation $x^2=x$ has only two solutions. I know that $0$ and $1$ are the solutions, but I can't proceed ...
2
votes
0answers
70 views

Descent Theorem Problem

If $A$ is a finitely generated $K$-algebra which is a domain, and $p_0 \subsetneqq \cdots \subsetneqq p_t$ a chain of prime ideals, how I can show that if $F$ is the field of fractions of $A$ and ...
1
vote
0answers
62 views

Reduction of ideals in a commutative ring

Is it possible to have an infinitely generated reduction of a finitely generated ideal in a commutative ring with identity ? If yes, why ? If no, an example to this effect will be helpful. Thank ...
3
votes
1answer
59 views

Automorphism of $A[t]/(t^m)$

Let $A$ be a commutative ring and $t$ an indeterminate over $A$. If $f$ is an automorphism of the ring $A[t]/(t^m)$ satisfying $f(x)\equiv x\pmod{(t)}$ for each $x\in A[t]/(t^m)$ with $m$ a positive ...
6
votes
0answers
117 views

Artinian rings are perfect

Is there a simple way to prove that an Artinian ring is perfect? (in the commutative case)
3
votes
1answer
118 views

Property of Hom-functor

How to prove $$\operatorname{Hom}_{R}(A,\operatorname{Hom}_{\mathbb{Z}}(R,B))\cong \operatorname{Hom}_{\mathbb{Z}}(A,B)$$ where $R$ is a commutative ring, $A$ an $R$-module and $B$ an abelian group? ...
8
votes
2answers
87 views

Why over $\mathbb{Z}/n\mathbb{Z}$ projectivity, injectivity and flatness coincide for cyclic modules?

Assume $R=\mathbb{Z}/n\mathbb{Z}$ ($n\neq0$) and let $M$ be a cyclic $R$-module. Could you tell me how to prove that $M$ is projective if and only if it is injective if and only if it is flat? And ...
6
votes
1answer
156 views

How to show that differential operator can be defined in terms of certain commutator operators

Let $U$ be any open subset of $\mathbb{R}^n$ (or, more general, of some smooth manifold). Define $\mathcal{D}_{-1}(U):=\{0\}$. For any two linear operators $A$ and $B$, the commutator operator $[A,B]$ ...
3
votes
2answers
188 views

A noetherian ring whose ideals are idempotent is artinian

I have to prove the folowing: If $R$ is a Noetherian ring, and for every ideal $I$ of $R$ we have $I = I^{2}$, then $R$ is Artinian. My first thought was to try to prove that the nilradical of ...
8
votes
1answer
449 views

An example of a commutative ring in which every primary ideal is prime

It is clear that every prime ideal in a commutative ring is primary. The converse is false; for example, in the ring $\mathbb{Z}$ the ideal $(p^2)$ is an example of a primary ideal that is not prime ...
4
votes
1answer
74 views

studying the topology of a real algebraic set

Let $f_1,\ldots,f_n \in \mathbb{R}[x_1,\ldots,x_m]$ be polynomials with real coefficients and let $I$ be the ideal that they generate. Denote by $V_{\mathbb{R}}(I)$ the corresponding real variety, ...
2
votes
2answers
332 views

radical of sum of two ideals

$I$ and $J$ are ideals in $k[x_1,\cdots,x_n]$. Show that $\sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}$. I have no idea how to prove it. Can someone help?
5
votes
0answers
65 views

Flatness over Jacobson ring

I need either a reference or a counter-example to the following statement. Let $A$ be a noetherian Jacobson ring (i.e. a noetherian ring where every prime ideal $\mathfrak{p} \subset A$ is an ...
5
votes
1answer
104 views

What can be said about $p\in Spec(R)$ when $R_p$ is a field?

What can be said about $p\in Spec(R)$ when $R_p$ is a field? Especially when $R$ is local noetherian
34
votes
1answer
460 views

A ring isomorphic to its finite polynomial rings but not to its infinite one.

I was messing with the ring $k[x_1,\dots,x_n,\dots]$ of polynomials in numerable many variables in order to solve an exercise of Atiyah, and the following question came to me and made me curious: ...
6
votes
1answer
102 views

$0\to L\to R^{n}\to M \to 0$ is exact, prove $M$ is finitely presented if and only if $L$ is finitely generated.

Suppose $R$ is a ring, $0 \rightarrow L\rightarrow R^{n} \rightarrow M \rightarrow 0$ is a short exact sequence, prove $M$ is finitely presented if and only if $L$ is finitely generated.
8
votes
2answers
178 views

How does this step in the proof of the structure theorem for f.g. modules over a Dedekind domain work?

I am trying to show that every finitely generated projective module $P$ over a Dedekind domain $D$ is a direct sum of (fractional) ideals. May's notes on Dedekind domains claim the result can be ...
3
votes
0answers
79 views

why is an open faithfully-flat morphism fpqc?

Why is an open faithfully-flat morphism fpqc? In other words, why must an open faithfully flat morphism $X\rightarrow Y$ have the property that around every $x\in X$, there is an open nbhd $U$ of ...
3
votes
1answer
87 views

Monomials not in an ideal

Let $R=\mathbb{R}[x,y]$ denote the commutative ring of polynomials in two variables $x,y$ with real coefficients. Show that for each $k \in \mathbb{N}$ there exists a monomial of degree $k$ not ...
3
votes
0answers
266 views

Integral homomorphism induces a closed map on spectra

I'm trying to prove the following: Let $f:A\rightarrow B$ is a integral homomorphism (e.g. $B/f(A)$ is a integral extension). Consider $f^{*}: \operatorname{Spec}B \rightarrow ...
1
vote
1answer
60 views

Isomorphism of polynomial rings implying isomorphism of the coefficient rings [duplicate]

Let $R$ and $S$ be commutative rings. Let $x, y$ be indeterminates, and assume that one has an isomorphism $R[x] \rightarrow S[y]$ (not necessarily mapping $x$ to $y$ of course). Does this imply $R ...
7
votes
2answers
246 views

Coordinate ring in projective space. What are they?

When $X$ is an algebraic variety of affine $n$-space, then the coordinate ring of $X$ are polynomials restricted to $X$. But when $X$ is a variety of projective $n$ space, what are the elements ...
1
vote
1answer
76 views

Extension of homorphisms on a divisible R-module

Let $R$ be a principal ideal domain and let $M$ be a finitely generated $R$-module. Take $N$ a submodule of $M$ and let $P$ be a divisible $R$-module. Prove that any homomorphism $f: N \rightarrow P$ ...
2
votes
3answers
261 views

Spectrum of polynomial ring

In M. Reid's Undergraduate Commutative Algebra, the author states that if $k$ is an algebraically closed field then $\operatorname{Spec}{k[x]} = \{0\} \cup k$ (page 21). Is this correct? Instead, ...
5
votes
1answer
228 views

rational functions on projective n space

How to prove that the field of rational functions on whole of projective n space is constant functions. By rational function I mean quotients of homogeneous polynomials of same degree ...
2
votes
1answer
184 views

What does “Hauptidealsatz” mean in “Krull's Hauptidealsatz”?

What does "Hauptidealsatz" mean in "Krull's Hauptidealsatz"? Thank you very much.
1
vote
1answer
232 views

dimension of an ideal (definition)

Let $A$ be a commutative ring and $I$ an ideal. When we refer to the "dimension" of $I$, what exactly do we mean? Is it the Krull dimension of $A/I$? In particular, i am trying to understand the ...
1
vote
1answer
75 views

Annihilators of Modules

I'm stuck trying to prove that for two $R$-modules $M,N$ ($R$ commutative with a 1), then $$Ann(M+N)=Ann(M) \cap Ann (N)$$ I was trying to do double inclusion, and I can prove the RHS is contained in ...
1
vote
2answers
288 views

Integral closure $\tilde{A}$ is flat over $A$, then $A$ is integrally closed

Question. Let $A$ be an integral domain and $\tilde{A}$ be its integral closure in the field of fractions $K$. Assume that $\tilde{A}$ is a finitely generated $A$-module. I want to prove that if ...
0
votes
2answers
138 views

Homogeneous ideals are contained in homogeneous prime ideals

Let $I$ be a homogeneous ideal of a graded ring $S$, $I\ne S$. I want to show that there exists a homogeneous prime ideal which contains $I$. I proved the following: Let $T$ be the set of all ...
2
votes
1answer
70 views

Is localization of a prime ideal still a prime ideal?

Im still new to the topic so this question might seem trivial. But I hope if someone can help explaining to me if a prime ideal $P$ of a domain $A$ is still a prime ideal $P_s$ in the localization ...
4
votes
1answer
94 views

Primary decomposition of power of a prime.

Let $R$ be a commutative Noetherian ring with unit. Suppose $P$ is a prime ideal that is not maximal. How can we go about finding a normal (reduced) primary decomposition of the power of $P$, say a ...
1
vote
0answers
79 views

Ideal membership problem for monomial ideals

Hi guys. I'd really appreciate help on understanding the proof for this Lemma above. I'm not sure how we got: "we see that every term on the right side of the equation is divisible by some x^{a(i)}. ...
10
votes
2answers
358 views

What is a primary decomposition of the ideal $I = \langle xy, x - yz \rangle$?

Given the ring $k[x,y,z]$, where $k$ is a field, and an ideal $I=(xy,x-yz)$, find the primary decomposition of $I$. I tried to draw the graph of the variety of $I$ and get a decomposition of ...