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

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5
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1answer
161 views

Endomorphism of a local $k$-algebra inducing an automorphism modulo $m^2$ is an automorphism

The following is exercise 4.1 of Hartshorne's Deformation Theory, used in the proof given there of the sufficiency of the infinitesimal lifting criterion of smoothness: Let $(A,m)$ be a local ...
7
votes
2answers
114 views

If $k\subset R\subset k[x]$, then $R$ is Noetherian?

Is there a way to prove that any subring $R$ of the polynomial over a field $k$ such that $k\subset R$ is Notherian without appealing to integral extensions, Eakin-Nagata, etc.? The reason I ask is ...
2
votes
0answers
50 views

question on $\mathbb{Q} \otimes R[G]$ his maximal ideals, the action of a Galois group on it

Reasoning on a question a friend posed me, i've found a question in the following setup: Suppose you have a finite group G, now you can pass to the algebra $\mathbb{Q} \otimes R[G]$ where the second ...
3
votes
1answer
99 views

Prove $A_\mathfrak{p} \otimes_A B_\mathfrak{q} = B_\mathfrak{q}$, where $\mathfrak{q}$ prime in $B$

$\require{AMScd}$ Hi, I think I have the answer for this question, but I'm not sure if it's correct. So I would be very glad if someone could have a quick look through it. Let $A$, $B$ be ...
1
vote
0answers
43 views

refining a presentation of a quotient ring

Suppose that we have a commutative ring $R$ which i) is local ii) is the quotient of a regular ring and iii) it is a $k$-algebra, where $k$ is a field. I am trying to prove that in that case we can ...
2
votes
2answers
226 views

Evaluation morphisms of formal power series and nilpotent elements

Given a commutative ring $A$, and a finitely presented (associative) $A$-algebra $B$, show that a morphism of $A$-algebras $A[[x]] \longrightarrow B$ is given by evaluation at an nilpotent element $ ...
2
votes
1answer
92 views

Define, $p^{-1} = \{x \in K: xp \subset D\}$. Then show that there exists a non zero $c \in D$ such that $cp^{-1} \subset D$.

Let $D$ be an integral domain and $K$ be its field of fraction. Also, given that $D$ is Notherian, Integrally closed, and every non-zero prime ideal in $D$ is maximal ideal. Let $p$ be a ideal of ...
9
votes
2answers
882 views

Projective module over a PID is free? [duplicate]

A common result is that finitely generated modules over a PID $R$ are projective iff they are free. Is the same true that an arbitrary projective module over a PID is free? I can't find this fact ...
3
votes
0answers
71 views

Understanding the construction of the cotangent complex (S. Lichtenbaum's way)

I am trying to thoroughly understand the "old" construction of the cotangent complex. The first question I have is about the definition of an extension of degree two of a ring $B$ above a ring $A$ ...
2
votes
1answer
36 views

existence of a factorization morphism related to extension fields

Let $R$ be a Noetherian $k$-algebra, $k$ a field. Let $K$ be a field extension of $k$. Let $Q$ be a prime ideal of $R \otimes_k K$ such that $Q \cap R = p$. Question: What is an elegant way to ...
5
votes
2answers
122 views

Show that $R$ is a Noetherian Ring [closed]

$R$ is a ring and $I$ is a finitely generated nilpotent ideal. If $R/I$ is noetherian show that $R$ is noetherian.
3
votes
1answer
190 views

Dimension of graded module

Let $R$ be a Noetherian positively graded ring and $M$ a finite graded $R$-module. Prove that $\dim M = \sup\{\dim M_p: p\in\operatorname{Supp} M \text{ graded}\}$. This is the Exercise 1.5.25 ...
1
vote
1answer
59 views

Moving tensor products inside homs

Suppose that $(\mathcal C, \otimes, I)$ is a closed symmetric monoidal category with $\hom(A,B)$ the hom-sets and $[A,B]$ the internal hom (where $[A,-]$ is right adjoint to $-\otimes A$). Is there ...
9
votes
1answer
100 views

Direct proof of non-flatness

Consider $k$ a field and the rings $A=k[X^2,X^3]\subset B=k[X]$. How to prove that $B$ is not flat over $A$ by using only the definition of flatness that it maintains exact sequences after making ...
2
votes
2answers
47 views

Isomorphism of (tensored) algebras by restricting/extending scalars

Let $A, B$ be commutative rings with identities, the ring map $f: A \rightarrow B$ gives $B$ the $A$-algebra structure. Let $S=f(A)$, $C$ an $S$-algebra (hence is also an $A$-algebra). Then is the map ...
2
votes
1answer
46 views

On the units of a residue ring

Let $A$ be an intergal domain, $K$ its field of fractions, $p$ a prime ideal of $A$. If $A$ is a valuation ring of $K$ ( i.e. for any $y\in K, y≠0$, one of $y\in A$ or $y^{-1}\in A$ must holds, for ...
1
vote
2answers
89 views

Are coproduct exact functors?

Are coproducts left exact or right exact functors in general? Let k be a commutative ring (unital assosiative). Specifically in the category of k-algebras is the tensor exact. (This is not the case ...
1
vote
1answer
94 views

A proof of the Noether Normalization Lemma

Look at the following proof of the Noether Normalization Lemma taken from Qing Liu's book "Algebraic Geometry and Arithmetic Curves": I don't understand the highlighted part. To be more ...
3
votes
0answers
84 views

Buchberger's criterion to show Grobner basis for linear forms

Let $k$ be a field. A polynomial of the form $l=a_1x_1+\cdots+a_nx_n$ is called a linear form ($a_i\in k$), and its support is the set of all variables $x_i$ such that $a_i\neq 0$. Let $L\subseteq ...
2
votes
1answer
75 views

Dual of polynomial ring

Consider the free $k$-algebra $k[x_i]_{i \in I}$ indexed by $I$. Then is $Hom_{k-Mod}(k[x_i]_{i \in I},k) \cong k[x_i]_{i \in I}$?
4
votes
0answers
85 views

Criterion of nonsingular varieties

It's well-known fact, that if $X$ is non-singular algebraic variety over algebraically closed field $k$ and $Y \subset X$ is its irreducible closed subscheme defined by sheaf of ideals $J$, then $Y$ ...
4
votes
1answer
156 views

Elliptic Curves Without Geometry

Unfortunately geometry terrifies me, so I was hoping to understand the basic theory of elliptic curves algebraically (via their function fields). Let F be a transcendence degree 1 extension of ...
5
votes
1answer
241 views

Is the integral closure of an integrally closed Noetherian domain in a finite extension field Noetherian?

Just as the title says. Let $R$ be a Noetherian integral domain, let $K$ be its field of fractions, let $L$ be a finite extension of $K$, and let $S$ be the integral closure of $R$ in $L$. Must $S$ ...
2
votes
2answers
71 views

a “paradox” regarding regular and complete intersection rings

The following "paradox" arose as i was studying the proof of Theorem 2.3.3 in Bruns and Herzog, CMR. My question is self-contained but i could expand on details upon request. Let $(S,\mathfrak{n})$ ...
3
votes
1answer
59 views

Unramification stable under change base

I want to show that if $f:X\to Y$ is an unramified scheme morphism (ie $m_y\mathcal{O}_{X,x}=m_x\mathcal{O}_{X,x}$ and $k(x)\leftarrow k(y)$ finite and separable) then any base change $X\times_Y Z\to ...
4
votes
0answers
53 views

Extension of morphism of Coherent sheaves over the projective space

Let $\mathcal{F}_1, \mathcal{F}_2$ be coherent sheaves over $\mathbb{P}^n_{\mathbb{C}}$ for $n \ge 3$. Denote by $U_i$ the fundamental affine schemes defined by the non-vanishing of the coordinates ...
6
votes
0answers
99 views

The automorphism group of a toric variety

Let $X$ be a projective toric variety (assume nonsingular, if it helps). Is there a nice description of its automorphism group $\operatorname{Aut}(X)$? I can see that for $\mathbb P^n$ it is ...
2
votes
1answer
51 views

Does resolution of singularities always factor through normalization

Let $X$ be an integral scheme and let $\tilde{X}$ denote its normalization. Is it always true that any resolution of singularities $X' \to X$ factors through the normalization map $\tilde{X} \to X$? ...
5
votes
1answer
387 views

Examples proving that the tensor product does not commute with direct products

Examples proving why the tensor product does not distribute over direct products? In fact the canonical map is not surjective; can you give me a simple example?
0
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1answer
89 views

Grading on the graded direct product

This question is related to this one. Probably it's obvious but could you tell me what is the grading on the graded direct product? I was thinking about $^*\Pi M^i=\oplus_j(\Pi_i M^i_j)$ where ...
0
votes
1answer
52 views

Graded direct products can differ from direct products

Assume $R$ is a graded ring and the $M_i$ are graded modules. Then Bruns and Herzog define the graded direct product $^*\Pi M_i$ as the submodule of $\Pi M_i$ generated by the sequences $(x_i)$ with ...
3
votes
3answers
300 views

Nontrivial ideal of a Noetherian domain contains a finite product of nonzero prime ideals

If $R$ is a Noetherian domain and $ 0 < U < R$ an nontrivial ideal of $R$. How to prove that there exists nonzero prime ideals $p_1,...,p_n \subset R$ such that the product $ p_1 p_2 ...p_n ...
1
vote
1answer
43 views

Epimorphism that is not monomorphism from $M\rightarrow M$

I have just finished an exercise where I prove that if $M$ is a module with acc then any epimorphism $f:M\rightarrow M$ but be an isomorphism. I then had a think about examples of non-noetherian ...
2
votes
0answers
206 views

LCM generators for the intersection of non-principal ideals in a Noetherian UFD

I am working with some non-principal ideals $I=\langle a,b\rangle$, $J=\langle c,d\rangle$ in a nicely behaved Noetherian UFD (the Laurent polynomial ring in finitely many commuting variables with ...
3
votes
2answers
107 views

$f$ is contained in no codimension $0,1$ primes $\implies f$ is invertible. Why do we need codim 1 primes as well?

11.3.G Let $A$ be a Noetherian ring, and $f$ be an element that is not contained in any codimension zero or one primes. Then $f$ is invertible. I found this statement in Vakil, and I am surprised ...
7
votes
1answer
111 views

When is the morphism from global section to stalk surjective

Let $X$ be a projective scheme over $\mathbb{C}$ and $\mathcal{F}$ a locally free sheaf on $X$ of rank $2$. Take a point $p$ in the support of $\mathcal{F}$. Suppose that there exists at least $3$ ...
3
votes
1answer
111 views

Am I understanding Hilbert's Basis Theorem correctly?

Hilbert's Basis Theorem states that any ideal in $F[x_1,x_2,\dots,x_n]$ has a finite basis. Let $I\subset F[x_1,x_2,\dots,x_n]$ be an ideal generated by $\langle f_1(X),f_2(X),\dots\rangle$. Going ...
0
votes
1answer
45 views

Proving $\dim A = \dim A\otimes_k K$ by reducing to an irreducible component.

Let $A$ be an equidimensional $k$-algebra with $\dim A = n$. Let $K/k$ be an algebraic extension. I just finished a proof that $A\otimes_k K$ is equidimensional with $\dim A\otimes_k K = n$ as well, ...
2
votes
3answers
197 views

Tensor products and monomorphisms of $A$-modules.

This problem addresses the same question that has been asked in $A^m\hookrightarrow A^n$ implies $m\leq n$ for a ring $A\neq 0$, which is exercise 2.11 of Atiyah and Macdonald's Introduction to ...
0
votes
1answer
96 views

Noether normalization lemma

Let $A=k[x,y,z]/(x^2-y,z-x^3) $ with $k$ a field.I am trying to mate the normalization lemma of noether, which means i must find a set of algebraic independent $x_1,..,x_s$ such as $A$ is finitely ...
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vote
2answers
668 views

Homogeneous polynomial in $k[X,Y,Z]$ can factor into linear polynomials?

My question is quite simple. Let $k$ be a closed algebraic field and $f\in k[X,Y]$. We know that $f$ can factor into linear polynomials. I would like to know if there is some generalization of ...
0
votes
1answer
49 views

An affine morphism of schemes over $k$ is closed iff the pullback over Spec $\bar k \to$ Spec $k$ is

On page 250, Vakil states that an affine morphism of schemes $\pi:X\rightarrow Y$ is a closed embedding iff so is $\pi\times \bar k:X\times_k\bar k\rightarrow Y\times_k\bar k$. The only if direction ...
3
votes
2answers
202 views

What is the proof of the single factor theorem over an arbitrary commutative ring?

Theorem (Single factor theorem) Let $R$ be a commutative ring, and let $P\in R[X]$, where $R[X]$ is the polynomial ring over the indeterminate $X$. Suppose $P(\alpha)=0$. Then $(X-\alpha)$ divides ...
0
votes
1answer
117 views

Systems of linear equations over integers modulo n

Let $\mathbb Z_n$ be the ring of integers modulo $n$. Let $A\in M_k(\mathbb Z_n)$ be a square matrix of size $k$. Let $X=[x_1, \ldots, x_k]^T$, where $x_i\in\mathbb Z_n$. There is some method to ...
3
votes
2answers
210 views

Represent localization as a direct limit

Let $A$ be a commutative ring with identity, $S\subset A$ a multiplicatively closed subset and $1\in S$. Does the equation $$S^{-1}A=\varinjlim_{s\in S}A_s$$ make sense? Here $A_s$ is the ...
1
vote
1answer
103 views

Trying to prove pre image of product of ideals is, the product of the pre images of the two ideals…

This is from Elements of Abstract Algebra by Allan Clark. 166 $\beta$ $\Phi ^{-1}(a'b') = (\phi^{-1}(a'))(\phi^{-1}(b'))$ I can prove an element of the right side is an element of the left. But I ...
0
votes
1answer
41 views

$I(X\times Y)=(f_{1},\dots,f_{r},g_{1},\dots,g_{s})$

If $I(X)=(f_{1},\dots,f_{r})\subset k[x_{1},\dots x_{n}]$ and $I(Y)=(g_{1},\dots,g_{s})\subset k[y_{1},\dots ,y_{m}]$ then should $I(X\times Y)=(f_{1},\dots,f_{r},g_{1},\dots,g_{s})\subset ...
6
votes
2answers
234 views

The maximal ideal in a local ring is finitely generated

Assume $m<R$ is the maximal ideal of a commutative local ring with identity, such that $m=m^2$. Is $m$ finitely generated? Is the condition $m=m^2$ redundant? I am trying to apply Nakayama's lemma ...
1
vote
1answer
133 views

Annihilator of a projective module is always a projective idempotent ideal?

I found as a remark in a book that if $M$ is a projective module over a ring $R$, $\mathfrak a$ is the ideal generated by all $f(m)$ where $f\in \text{Hom}(M,R)$ and $m\in M$, and $\text{Ann}(M)$ is ...
0
votes
2answers
70 views

Embedding dimension one problem

Let $(R,m)$ be a local noetherian ring and $\dim R=0$. If $\dim_km/m^2=1$ a) show that $m$ is a principal ideal. b) prove that every nonzero ideal is parametric and principal. (An ideal $q$ called ...