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

learn more… | top users | synonyms (1)

4
votes
1answer
78 views

What are the natural surjections in the proof of Hopf's classification theorem?

I am currently reading Hatcher's book, trying to understand the proof of Hopf's classification Theorem on Hopf algebras that says the following: Every Hopf algebra $A$ that is commutative and ...
3
votes
0answers
46 views

Tensor product with endomorphism ring

Let $k$ be a commutative ring, $M$ a $k$-module and $k\longrightarrow A$ a $k$-algebra. Is it true that $$A\otimes_k\operatorname{End}_k(M)\cong\operatorname{End}_A(A\otimes_kM)?$$ If not, under what ...
2
votes
0answers
35 views

Geometric line $R$ is a “field”, $D= \left\{ x\in R\mid x^2 =0 \right\}$ is not an ideal of $R$

I am confused by properties of $R$, the geometric line in synthetic differential geometry. In the book Synthetic Differential Geometry by Kock, he assumes $R$ is only a commutative ring. However, in ...
0
votes
1answer
39 views

Estimating regularity of sheaves with rank of certain modules and zeroth cohomology

I'm studying Eisenbud's book "Geometry of syzygies", in particular the Gruson-Lazarsfeld-Peskine theorem for Castelnuovo-Mumford regularity. I'm concerned about an intermediate step in the proof. Let ...
2
votes
1answer
41 views

Contraction and extension of ideals respect inclusions, sums and intersections

Let $R$ be an integral domain. Let $Y$ be a multiplicatively closed subset of $R$ which contains $1$ but not $0$. Define $S=RY^{-1}=\lbrace ry^{-1} : r \in R, y \in Y \rbrace$ as well as ...
3
votes
1answer
47 views

On characterizing modules that don't annihilate any module under tensor product.

Let $R$ be a commutative ring and $M$ an $R$-module. Then under what condition can we deduce that for any nonzero $R$-module $N$, $M\otimes_RN\neq0$?
2
votes
0answers
47 views

Understanding Localized Rings mod an ideal.

Hi guys I am working with Fulton's book and I am trying to understand for myself the elements of two rings. $O_p(\mathbb{A}^n)/JO_p(\mathbb{A}^n)$ and $O_p(V)/\bar{J}O_p(V)$ Where $I = ...
2
votes
1answer
99 views

Minimal prime ideals of $\mathcal O_{X,x}$ correspond to irreducible components of $X$ containing $x$

Let $X$ be an algebraic variety over an algebraically closed field $K$. By definition, $X$ is a separated prevariety, and $x \in X$. I'm trying to show (i): The minimal primes of $\mathcal ...
0
votes
1answer
85 views

the natural map from $N_1$ to $\varinjlim N_i$ is injective?

Direct limit $\varinjlim N_i$ of a direct system $\{N_i\}_{i \in I}$ is defined to the union of all $N_i$ modulo certain equivalence relations. From the definition it seems that if ...
1
vote
0answers
48 views

Going-up but not lying over?

Let $R \subseteq S$ be an extension of commutative rings with identity. Assume $1_R = 1_S$. This extension satisfying going-up if for any inclusion of primes $P_1 \subseteq P_2$ of $R$, and any ...
2
votes
1answer
40 views

Computing an explicit tensor product

So I think this question is trivial but I can't seem to be able to do it so here we go : what is the tensor product $$k[x,y]/(y^2-x^3) \otimes_{k[y]} k[x,y]/(y^2-x^3)\ ?$$ My guess is that it is ...
1
vote
1answer
60 views

Intersection of n hyperplanes in projective space of dimension n is not empty

I want to prove the following: Let $H_1,\dots,H_n$ be $n$ hyperplanes in $\mathbb{P}^n =\mathbb{P}^n \mathbb{C}$. Then $\cap_{i=1}^n H_i$ is not empty. Please be noted that this is an exercise ...
0
votes
0answers
51 views

Definitions of Weil Algebras

I am confused by several definitions of Weil Algebras and their connection to each other. Kock's book on synthetic differential geometry defines a Weil algebra over a ring $R$ as an $R$-algebra of ...
1
vote
1answer
52 views

Integral and prime ideal in Dedekind domain

Let $A$ be an Dedekind domain, $K$ its quotient field, $L$ a finite separable extension of $K$, and $B$ the integral closure of $A$ in $L$. If $p$ is a prime ideal of $A$, then $pB$ has a ...
1
vote
1answer
35 views

Zero Dimensional Commutative Ring

Let $R$ be a commutative ring with unity. I want a proof of the fact that $R$ is zero-dimensional (in the sense that all prime ideals are maximal) if and only if $R/J(R)$ is von Neumann regular ...
0
votes
0answers
65 views

Intersection of localizations

Let $R$ be a commutative ring with identity, not necessarily a domain, and let $S_i$ be a family of multiplicative closed subsets of $R$. What is the exact meaning of $\bigcap_i S_i^{-1}R$? For ...
0
votes
0answers
37 views

Showing that an element generates the kernel

$I $ is a monomial ideal generated by $\left < m_1, \dots, m_n\right >$ and suppose we also have an $R$-module homomorphism $\phi: \oplus_{j = 1}^n Re_j \to I$ defined by $$\phi(e_i) = m_i.$$ ...
0
votes
1answer
36 views

Is a divisible module over a local principal ideal domain a torsion module?

Is an injective module over a local principal ideal domain a torsion module? We know that injective modules and divisible modules over a PID are equivalent. What do we say about the torsion submodule ...
6
votes
1answer
148 views

Understanding an exercise from Fulton's Book on Algebraic Curves

I am reading Fulton's book Algebraic Curves. Currently I am working on a specific problem (2.43), and I have doubts about my work and would appreciate another opinion(s). Assume $p$ is the origin ...
3
votes
1answer
55 views

Normality of algebraic fibre spaces

Lazarsfeld said in his book (Positivity in Algebraic Geometry I, page 126, Example 2.1.15) that if $f:X\to Y$ is an algebraic fibre space then $X$ is normal implies that $Y$ is also normal. His ...
1
vote
0answers
38 views

Characterization of certain polynomial ideals in two variables

I am trying to classify certain homogeneous polynomial ideals in two variables with respect to possible minimal supports of polynomials in each homogeneous degree $d$ of $I$. The support of a ...
1
vote
1answer
63 views

Show that a sequence is a free resolution

Let $I \subset R = k[x_1,\dots,x_n]$ be an ideal and $f \in R$ such that $I = \left < f \right >$ ($k$ is a field, so R is commutative ring). How do I show that (1) $I$ has a free resolution ...
1
vote
1answer
44 views

Trivialization of flat proper morphisms with sections

Let $f:X \to Y$ be a flat proper morphism of noetherian schemes. Assume both $X$ and $Y$ are regular, irreducible and $\dim Y=1$. Suppose there exists a section to the morphism $f$ i.e., a morphism ...
1
vote
1answer
22 views

Are torsion-free modules over principal ideal domains/Dedekind domains projective

An exercise in "Commutative algebra with a view towards Algebraic geometry" by Eisenbud states that a torsion-free module over a Dedekind domain is a projective module (see page $484$, Exercise ...
0
votes
1answer
45 views

System of parameters in a Noetherian local ring

I'am studying "system of parameters" on Commutative Ring Theory by H. Matsumura. There is a theorem about height of an ideal and the number of generators of the ideal. In the proof of the part (ii), ...
0
votes
0answers
28 views

showing $\bar{X}$ is irreducible in $\mathbb{R}[X,Y]/(X^2+Y^2+1)$ [duplicate]

Consider the ring $A=\mathbb{R}[X,Y]/(X^2+Y^2-1)$. Let $\bar{X}$ be the image of $X$ in $A$. Show that $\bar{X}$ is irreducible in $A$. I tried this by assuming that ...
5
votes
1answer
67 views

$R\subset A\subset R[X]$, $A$ is Noetherian. Is $R$ Noetherian?

Let $R\subset A\subset R[X]$ be commutative rings and suppose $A$ is Noetherian. Is $R$ Noetherian? I guess the answer is yes. Can we say from this relation that $A[X]=R[X]$? If yes, then by ...
2
votes
0answers
61 views

Confusion with arithmetically Cohen-Macaulay varieties

I'm a bit struck about this fact; I think it's really a silly question, but I'm not completely sure about it. Let $X\subseteq \mathbf{P}^m$ be a projective variety; choose the best hypotheses ...
1
vote
2answers
112 views

Strange definition of spectrum?

The following is taken from these notes. Definition 2.8. Let $\mathscr E$ be some cartesian closed category and let $A$ be an $R$-algebra. The spectrum $\operatorname{Spec}_A(R[x_1,\dots ,x_n]/I)$ of ...
2
votes
0answers
35 views

Is there a purely category-theoretic description of the total quotient ring?

The following is well-known: let $R$ denote a commutative ring and $S$ denote a submonoid. Write $S^{-1}R$ for the localization of $R$ at $S.$ Then the canonical morphism $R \rightarrow S^{-1}R$ is ...
0
votes
1answer
35 views

Does the statement “the $R$-algebra homomorphisms $A \rightarrow R$ are linearly independent whenever $R$ is a field” admit any generalizations?

The following is well-known. Proposition. Let $R$ denote a field and $A$ denote an $R$-algebra (not necessarily commutative). Then the set of $R$-algebra homomorphisms $A \rightarrow R$ form a ...
2
votes
1answer
29 views

Module over a commutative ring with a topology

Let $M$ be an $R$-module ($R$ commutative ring with unity). Let $M=M_0 \supseteq M_1\supseteq M_2\supseteq\cdots$ be a chain of submodules. The topology in $M$: The open sets in $M$ are arbitrary ...
0
votes
1answer
51 views

Extensions and Contractions of Ideals

Let $f: \mathbb{Z}[X] \longrightarrow \mathbb{Z}[\sqrt{2}]$ be a ring homomorphism that sends $X$ to $\sqrt{2}$. $\textbf{DEFINITION:}$ Let $f: A \longrightarrow B$ be a ring homomorphism. The ...
0
votes
0answers
52 views

Are the following monomial modules?

If $I$ is an monomial ideal of $R$ and $M, N$ are monomial modules of $\oplus_{i = 1}^{r} Re_i$, then the following are monomial modules. Why? 1) $M + N = \{m + n: m \in M, n \in N \}$ (because it's ...
2
votes
1answer
39 views

Regular sequence in degree 1

$R$ is a graded algebra generated by $R_1$(the degree 1 piece) over $R_0=k$ where $k$ is a infinite field and R has no negative degree. Given irrelevant ideal has depth d, then is it possible to find ...
3
votes
1answer
73 views

Why are projective coordinate rings not isomorphic when the corresponding projective varieties are?

I was trying to prove the following question from An Invitation to Algebraic Geometry by Karen Smith: Show that the homogeneous coordinate rings of projectively equivalent varieties are ...
4
votes
2answers
24 views

Noetherian semiprimary rings

Is any Noetherian semiprimary ring $R$ Artinian? By semiprimary I mean $R/J(R)$ semilocal and $J(R)$ nilpotent, where $J(R)$ is the Jacobson radical of $R$. I know that if $R$ is Artinian then $J(R)$ ...
1
vote
1answer
35 views

If $\mathbb{C}[x]/(x^n)$ is an Artinian $\mathbb{C}$-module, is it a Artinian ring?

I feel I should use the fact that ideals of $\mathbb{C}[x]/(x^n)$ are equivalent to submodules $\mathbb{C}[x],$ but I cannot see how to use this fact to prove that $\mathbb{C}[x]/(x^n)$ is an Artinian ...
2
votes
0answers
66 views

Branch locus of finite morphism closed?

Let $A \subseteq B$ be a finite ring extension of finitely generated $\mathbb{C}$-algebras of dimension one. Let $A$ be an integral domain. Consider the set $S$ of all maximal ideals ...
-1
votes
3answers
64 views

Structure theorem of Artinian rings [closed]

Can you help me with a complete proof of Structure theorem of Artinian rings? I find just partial proofs.
1
vote
0answers
52 views

Does affine open set equal to distinguished open subset in an affine scheme?

For $A$ a commutative ring, does it always hold that all affine open subschemes of $\text{Spec }A$ lie over a distinguished open subset of $\text{Spec} A$?
2
votes
0answers
56 views

Flat quotients of power series rings

I read the following statement in some algebraic topology notes and I want to know if it is true and, if so, why. Let $R$ be a ring and $f(x)$ a power series in $R[[x]]$. Suppose that $R[[x]]/(f)$ ...
1
vote
1answer
24 views

Pythagoras number of $\mathbb{F}_p$

For a commutative ring $A$ and $a \in A$, define the length of $a$ as $$ l(a) = \inf \lbrace n \in \mathbb{N} \mid \exists a_1, \ldots, a_n \in A : a = \sum_{i=1}^n a_i^2 \rbrace . $$ Let $\Sigma A^2$ ...
1
vote
1answer
40 views

I'm stuck with trying to construct a $K$-basis for the quotient of the polynomial ring $S/I$.

We were told in class that a $K$-basis for $S/I$ where $S=K[X_1, \dots , X_n]$ and $I$ a monomial ideal in $S$ is $W = \{X^a \in Mon(S) | X^a \notin I\}$. I'm having difficulties visualizing what the ...
0
votes
1answer
24 views

If $M \otimes_K L$ is a free $A \otimes_K L$ module, is $M$ a free $A$ module? $L$ is a field extension of $K$.

$M$ is finitely generated. $L$ is a field extension of $K$. Question: If $M \otimes_K L$ is a free $A \otimes_K L$ module, is $M$ a free $A$ module? I am asking this because I am trying to show ...
2
votes
0answers
28 views

Injective homomorphism $R^n\rightarrow R^m$ implies $n\leq m$? [duplicate]

Let $R$ be a nonzero commutative ring, and let $n,m$ be integers. Let $f:R^n\rightarrow R^m$ be an injective homomorphism of $R$-modules. I'm trying to show that $n\leq m$. My idea: Assume that ...
1
vote
1answer
18 views

Need help with the proof that set of monomials $N$ belonging to a monomial ideal $I$ is a $K$-basis for $I$.

The proof given in the text I'm reading is (here $S=K[x_1, \dots , x_n]$): The bit I'm having problems with understanding is how does $v=u_i w$ imply that supp$(f) \subset N$? It would make sense ...
0
votes
0answers
27 views

The ring of polynomials in $X, Y$ all of whose partial derivatives with respect to $X$ vanish for $Y=0$ is Noetherian? [duplicate]

The ring of polynomials in $X,Y$ all of whose partial derivatives with respect to $X$ vanish for $Y=0$ is Noetherian ?
2
votes
1answer
57 views

$R$ be a commutative ring with unity such that every prime ideal contains no non-zero zero divisor , then is $R$ an integral domain?

Let $R$ be a commutative ring with unity such that every prime ideal contains no non-zero zero divisor (i.e. if $P$ is a prime ideal and $x,y \in P$ with $xy=0$ then either $x=0$ , or $y=0$). Then ...
1
vote
2answers
35 views

commutative ring which have every maximal ideal generated by an idempotent [closed]

Can you help me with one example of commutative ring which have every maximal ideal generated by an idempotent?