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

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4
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226 views

Computing toric ideals via saturation

I have recently got interested in toric varieties and I have a question concerning their ideals. Let $A \in \mathbb{Z}^{m \times n}$ and $\ker A = \{ u \in \mathbb{Z}^n \; | \; Au = 0 \}$. For any $u ...
4
votes
2answers
92 views

Stalk of the quotient presheaf

Consider a sheaf of abelian groupS $\mathscr F$ over a topological space $X$. If $\mathscr F'$ is a subsheaf of $\mathscr F$ (over $X$), then we can construct the quotient presheaf $\mathscr ...
6
votes
1answer
276 views

Normalization of a quotient ring of polynomial rings (Reid, Exercise 4.6)

I solved all parts of Exercise 4.6 of the book Undergraduate Commutative Algebra of Miles Reid except the last one. Let $A=k[X]$ and $f\in A$ has a square factor but it is not a square polynomial ...
2
votes
1answer
165 views

Maximal ideals of rings which are finite dimensional vector spaces over $\mathbb C$.

If $K$ is a commutative ring which is a finite dimensional vector space over $\mathbb C$ what can we say about the maximal ideals of $K$? What can we say if instead of $\mathbb C$ we have some ...
1
vote
2answers
50 views

What are some concrete examples of ideals and modules where $I M = M$?

I'm trying to get more of intuition for the cloud of ideas surrounding the abstract Cayley-Hamilton theorem, Nakayama's lemma, etc., so I'd like to see some concrete examples. The problem is that the ...
1
vote
1answer
81 views

K-algebra isomorphic to a polynomial ring

I am trying to understand why this is true: Let K be a field, and let $K[a_1,\ldots,a_r]$ be a finitely generate $K$-algebra. If $a_1,\ldots, a_r$ are algebraically independent, then ...
0
votes
1answer
36 views

Can one find in this specific setting an extension of a given ring map?

All rings in this question are unitary and commutative and all maps are homomorphisms of commutative rings sending $1$ to $1$. Let $R$ and $S$ be regular local rings and let $$ ...
2
votes
1answer
100 views

Does the natural bijection between the set of prime ideals in A disjoint from S and Spec$(S^{-1}A)$ restrict to maximals?

I was studying rings of fractions, and I was wondering about the problem of restricting the canonical bijection (induced by retraction and extension of ideals) $\{p\in \text{Spec}(A) \mid p\cap ...
-5
votes
1answer
117 views

Finite product of algebras of finite type

Let $A$ be a commutative ring with unity. Let $B, C$ be $A$-algebras of finite type. Is $B \times C$ an $A$-algebra of finite type?
3
votes
2answers
112 views

If the ring map $f: A\rightarrow B$ is integral, fibres of $f^*$ are finite, then $f$ is finite?

If the ring map $f: A\rightarrow B$ is integral, i.e. $B$ is integral over the subring $f(A)$, and each fibre of the induced map $f^*: Spec(B)\rightarrow Spec(A)$ is a finite set, then should $f$ be ...
1
vote
1answer
52 views

Gelfand transform explicity

Let $T$ be a bounded normal operator. Let $A$ be the algebra generated by $T$ and $T^*$. What is the explicit Gelfand transform $G:A\to C(\sigma(T))$? My book says the image of $T$ is the ...
11
votes
1answer
141 views

$B\otimes_A A[x]=B[x]$

Let $A\rightarrow B$ be a homomorphism of commutative rings. Then $B\otimes_A A[x]\cong B[x]$ as $B$-algebras. How can one demonstrate this nicely, i.e. using universal properties alone and the Yoneda ...
2
votes
1answer
56 views

$K(A)\cong \mathbb Z$ for a PID $A$

In Atiyah and Macdonald, chapter 7, exercise 26, iii), it's required to show the Grothendieck group $K(A)\cong \mathbb Z$ for a PID $A$. By ii) of this problem, it's easy to show that $K(A)$ is ...
3
votes
1answer
65 views

Exact sequence induces exact sequences for free parts and torsion parts?

Let $A$ be a PID and consider the exact sequence of finitely generately modules over$A$: $$0\longrightarrow M' \overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}M''\longrightarrow 0 \tag{1}.$$ ...
3
votes
2answers
126 views

Surjection from a Noetherian ring induces open map on spectra?

Let $A$ be a Noetherian ring, $f: A\rightarrow B$ a surjective ring map, then should the induced map on spectra $f^*: Spec(B)\rightarrow Spec(A)$ be an open map? In Atiyah and Macdonald, Chapter 1, ...
1
vote
0answers
112 views

About the “going-down property”

In Atiyah and Macdonald, Chapter 5, Exercise 10, there defines the so called "going-down property" (GDP). Then in Chapter 7, Exercise 24, the hint says, the ring map $f: A\rightarrow B$ has GDP ...
3
votes
0answers
67 views

understanding the language of an argument in Bruns and Herzog Cor. 2.3.10

The following question is heavily related to the context of theorems 2.3.9 and 2.3.10 in the text Cohen-Macaulay rings by Bruns and Herzog. My intention is to make this question as self-contained as ...
2
votes
2answers
84 views

If $N$ is an $R/I$-submodule of $M$ can we view $N$ as an $R$-submodule of $M$?

Let $R$ be an integral domain, $M$ an $R$-module and $I\subseteq \mathrm{Ann}(M)$ an ideal of $R$. $N$ is an $R/I$-submodule of $M$ (as $R/I$-module). Can we view $N$ as an $R$-submodule of $M$ ...
4
votes
2answers
142 views

Module of $R$-valued functions on an infinite set is not countably generated

Let $R$ be an integral domain and $X$ be an infinite set. Let $R^X$ be the set of all functions $f: X \rightarrow R$, viewed as an $R$-module in the usual manner: for $\alpha \in R$, $\alpha f: x \in ...
5
votes
1answer
158 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
221 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
841 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
186 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
58 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
46 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
87 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 ...
0
votes
1answer
93 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
74 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
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1answer
153 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 ...
4
votes
1answer
228 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
69 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
57 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
51 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
97 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
50 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 $\tilde{X} \to X$? ...
4
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1answer
373 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
votes
1answer
88 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 ...