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

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2
votes
2answers
62 views

What explicitly is the “adjunction” isomorphism $Hom_C(A,C)\to Hom_B(A,Hom_C(B,C))$?

Suppose $B$ and $C$ are commutative rings, $A$ a $B$-algebra, and $B$ is a $C$-module. What exactly is the "adjunction" isomorphism $$Hom_C(A,C)\to Hom_B(A,Hom_C(B,C))?$$ Given $A\to C$, it needs to ...
0
votes
1answer
71 views

Maximal ideal of polynomial ring over a subfield

Let $L/K$ be an algebraic extension of fields. Let $B = L[X,Y]$ and $A = K[X,Y]$. Suppose $a$, $b \in L$ and $m = (X-a,Y-b)$ is an ideal of $B$. Show that $m$ and $m \cap A$ are maximal ideals of ...
1
vote
1answer
30 views

Gauss lemma for arbitrary commutative ring [duplicate]

Part (iv) of exercise #2 for chapter 1 in Atiyah and Macdonald's book Introduction to Commutative Algebra asserts that if $f, g \in A[x]$ are primitive then $fg$ is primitive. We know that this is ...
3
votes
1answer
54 views

Does $f\otimes_A 1_{A/m}:M\otimes A/m\to N\otimes A/m$ injective for all maximal $m$ imply $f$ is an isomorphism?

Let $A$ be a commutative ring. Suppose $f\colon M\to N$ is a morphism of free $A$-modules of equal, finite rank. If $f\otimes_A 1_{A/m}$ is injective for all maximal ideals of $A$, does this imply ...
3
votes
1answer
24 views

Let $A$ be a finitely generated $R$-module. Show that if $A/MA$ can be generated by $n$ elements then so can $A$.

Let $R$ be a local commutative ring with the maximal ideal $M$. Let $A$ be a finitely generated $R$-module. Show that if $A/MA$ can be generated by $n$ elements then so can $A$. I tried to apply ...
1
vote
1answer
24 views

Finite ring extension of local rings

Let $R$ and $S$ be local rings with the maximal ideals $M$ and $N$, respectively. Assume that $R\subset S$ and that $S$ is a finitely generated $R$-module. If there exists a proper ideal $I$ of $R$ ...
0
votes
0answers
26 views

MCM Modules over Cyclic Quotient Singularities

Let $k$ be a field and $R$ the ring $k[[u^{n+1}, uv, v^{n+1}]]$. Then the indecomposable MCM $R$-modules are given by $M_j = R(u^av^b \vert b-a\equiv j \mod{n+1})$ for $j = 1,\ldots, n$. This is of ...
2
votes
0answers
46 views

How does commutative and/or differential algebra think about total derivatives?

If we apply the "operator" $\frac{d}{dx}$ to the polynomial $xy$, we get the expression $y+x\frac{dy}{dx}.$ (Source: high school.) Thinking of $xy$ as an element of the polynomial ring ...
0
votes
1answer
74 views

Dimension of quotient ring

What is the dimension of the following quotient ring, $\mathbb{Z}[x,y,z]/\langle xy+2, z+4 \rangle$, where $\mathbb{Z}$ is the ring of integers? I realized this is isomorphic to ...
1
vote
1answer
30 views

Prove that if $M$ is a simple $R=k[x_1,…,x_m]$ -module, then the dimension of $M$ over $k$ is finite.

Let $k$ be a field and let $R=k[x_1,...,x_m]$ be the polynomial ring in $m$ indeterminates. Prove that if $M$ is a simple $R$-module, then the dimension of $M$ over $k$ is finite. I think since ...
0
votes
0answers
41 views

characterization of integral closure?

I would like to know whether there exists any characterization of integrally closed domains which is related to some morphisms construction with $\mathbb{Z}$ and $\mathbb{Q}$. I was thinking about ...
1
vote
0answers
27 views

Deciding whether a non-f.g. non-divisible flat module is projective or not.

Assume $S= R[T]/(f)= R[w]$ is a flat non-divisible $R$-module, where $R$ is a noetherian UFD, $T$ is an indeterminate over $R$, and $f\in R[T]$ is a non-monic polynomial of positive degree. Can we ...
2
votes
1answer
55 views

The direct limit of morphisms and the direct limit of tensor product functors

While reading these notes I had something of an existential crisis, after realizing that my understanding of direct limits might somehow be fundamentally insufficient. In particular, alarms started ...
1
vote
2answers
50 views

Every ideal has a FFR

Let $A$ be a regular local ring. Then every ideal has a finite free resolution. My thoughts: it's easy to prove that every ideal $I$ has a free resolution. In fact $I$ is finite and there is a ...
2
votes
1answer
27 views

Reduced one-dimensional Noetherian ring is Cohen-Macaulay

If $(R,m)$ is a local Noetherian reduced ring of Krull dimension $1$ then $R$ is Cohen-Macaulay, since in a reduced Noetherian ring the set of zero divisors is the (finite) union $U$ of minimal prime ...
3
votes
1answer
134 views

Algebraic extension and Krull dimension

Let $A \subseteq B$ be an extension where $A,B$ are Noetherian, commutative rings. If $B$ is algebraic over $A$, can we say that $\dim B\leq\dim A$? Just read the following paper "Constructive ...
1
vote
2answers
62 views

Jacobson radical of $\mathbb{F}_{2}(t)[x]/(x^4-t^2)$

Let $\mathbb{F}_{2}$ be the field of two elements. Let $R=\mathbb{F}_{2}(t)[x]/(x^4-t^2)$. Why is $R/J(R)$ equal to $\mathbb{F}_{2}(t)[x]/(t-x^2)$? here $J(R)$ denotes the Jacobson radical of $R$.
1
vote
0answers
35 views

Jacobson radical of an indecomposable commutative ring

Let $R$ be a commutative indecomposable ring with identity which has infinitely many maximal ideals. Can we deduce that the Jacobson radical of $R$ (the intersection of all maximal ideals) is the zero ...
5
votes
0answers
49 views

What is the obstruction to extending a linear map on tangent spaces of a variety to a regular map on neighborhood?

Suppose that $X$ and $Y$ are algebraic varieties of the same dimension $n$. If $p$ and $q$ are points in $X$ and $Y$ respectively, suppose that there is a linear map $i : T_p X \to T_q Y$. My vague ...
3
votes
1answer
39 views

The last nonzero local cohomology module is not finitely generated. [closed]

Let $R$ be a Noetherian ring and $I$ an ideal of $R$. If $M$ is a finitely generated $R$-module and $i\neq 0$ is the greatest integer such that $H^i_I(M)$ is nonzero, then $H^i_I(M)$ is not a ...
3
votes
1answer
62 views

What's the kernel of the codiagonal $k[x] \otimes_k k[x] \rightarrow k[x]$?

maybe this question is really stupid, but I could not solve it after thinking for a while. Let $I$ be the kernel of the codiagonal map $$k[x] \otimes_k k[x] \rightarrow k[x]$$ given by $f(x) \otimes ...
3
votes
1answer
43 views

Fraction field of $F[X,Y](f)$ isomorphic to $F(X)[Y]/(f)$

Assume $F$ is a field and $f$ is an irreducible polynomial in $F[X,Y]$ which involves the variable $Y$. Then, by Gauss's lemma, $f$ is irreducible also in $F(X)[Y]$ so that $F(X)[Y]/(f)$ is a field ...
1
vote
1answer
26 views

A Cohen-Macaulay localisation

Let $R=\mathbb C[X,Y]/(Y^3-X^3)$, let $x,y$ be the images of $X,Y$ in $R$, and let $R_1$ be the localization of $R$ at the maximal ideal $(x,y)$. I want to prove that $R_1$ is a Cohen-Macaulay ...
3
votes
0answers
49 views

Determining prime ideals lying above a given ideal

Let $R=\mathbb{Z}[x]/(f)$, where $$f(x)=x^4+42x^3-11x^2+22x-2002002002002002.$$ Let $I=3R$, the ideal generated by $3$ in $R$. Find all prime ideals of $R$ that contain $I$. I am hoping to ...
0
votes
1answer
36 views

Lying Over for Algebraic Ring Extensions

Let $B$ be a finitely generated algebraic $A$-algebra (but not necessarily integral). Is it true that for any prime in $A$ we can find a prime in $B$ which contracts to $A$? What if we also allow the ...
3
votes
0answers
44 views

Vanishing of Tor

Let $R$ be a commutative ring with unit. Vanishing of $\operatorname{Tor}_0(M,N)$ (see here) for two finitely generated $R$ modules $M$ and $N$ implies $\operatorname{Ann}M+ \operatorname{Ann}N=R$. ...
1
vote
0answers
57 views

Constant Dimension for Localization of Projective Modules

It is a well known fact that the localization of a projective module over a commutative ring is free. However, I don't know anything about the dynamics of how the dimension of the resultant free ...
1
vote
1answer
48 views

$A \subseteq B \subseteq C$ with $A \subseteq C$ separable and $pd_{B \otimes_A B}(B) = \infty$

Assume $A \subseteq B \subseteq C$ are commutative rings such that $C$ is separable over $A$, namely $C$ is a projective $C \otimes_A C$-module. Separability of $C$ over $A$ does not imply ...
5
votes
2answers
57 views

How do ideal quotients behave with respect to localization?

Suppose $R$ is commutative ring with unity. For ideals $I$, $J \subseteq R$, the ideal quotient $(J:I)$ is $$(J:I) := \{x\in R \, : \, xI \subseteq J\}$$ Let $S\subset R$ be a multiplicative set. When ...
3
votes
1answer
50 views

On the proof of Theorem 20.8 from Matsumura

Theorem 20.8 in "Commutative ring theory" states that if $A$ is a regular UFD then so is $A[[X]]$. Here is the proof. He has to prove that the intersection of principal ideals $\mathfrak{b}=uB\cap ...
0
votes
0answers
32 views

The trace ideal of a non zero $R$-module

Let $R$ be a commutative ring with identity and $M$ be a cyclic $R$-module, we may define the ideal $tr(M)$ associated with $M$, the sum of the ideals $f(M)$, for all $R$-homomorphisms $f \in ...
7
votes
0answers
74 views

A Geometric Description of Injective Modules

I've found that when studying commutative algebra, thinking of things in terms of their algebro-geometric interpretation helps them stick as well as motivates otherwise odd and abstract concepts. ...
1
vote
1answer
37 views

Injectivity, Projectivity, and $P$-injectivity of Localization [closed]

Let $R$ be a commutative ring with unity. I have read that if $M$ is an injective $R$-module, then $S^{-1}M$ is not necessarily an injective $S^{-1}R$-module. I need an example... Does last ...
1
vote
1answer
43 views

Show that $\widehat{\mathbb Z}\cong \prod_{p\;\text{prime number}}\mathbb Z_p$

Let $\widehat{\mathbb Z}=\varprojlim_n \; \mathbb Z/n\mathbb Z$ be the inverse limit of the inverse system $(\mathbb Z/n\mathbb Z)_{n\in \mathbb N}$ and let $\mathbb Z_p=\;\varprojlim_n\; \mathbb ...
2
votes
1answer
42 views

Which primes belong to the support of $S^{-1}M$ with respect to $R$

Let $R$ be a noetherian ring, $S$ a multiplicatively closed subset of $R$ and $M$ a finitely generated $R$-module. The support of $M$, denoted ${\rm Supp}_R(M)$, is the set of prime ideals $p$ of $R$ ...
0
votes
0answers
22 views

when $ht_SQ =ht_Rf^{-1}(Q)$?

Let $R$ and $S$ be commutative rings with identity, and $f:R\to S$ be a homomorphism. By what assumptions we can have $ht_SQ =ht_Rf^{-1}(Q)$, for every prime ideal $Q$ of $S$? What about $ht_SQ ...
2
votes
1answer
34 views

Quotients in a non-discrete valuation ring

Let $R$ be a valuation ring, $\mathfrak{m}$ the maximal ideal of $R$. Let $k$ be the residue field, and $K$ the field of fractions of $R$. Assume that the valuation on $K$ is such that ...
5
votes
1answer
132 views

Homogeneus primes in a graded ring

Let $B=\oplus_{n\in\mathbb Z} B_n$ be a graded ring (commutative with 1). We know that $B_0$ is a subring of $B$, so we have the inclusion $B_0\hookrightarrow B$. My question is: Is every prime ...
0
votes
1answer
84 views

Dimension of a quotient ring

What is the Krull dimension of $B=A[x,y,z]/\langle xy + 1, z + 1\rangle$, given $A$ is a Noetherian commutative ring?
5
votes
1answer
99 views

Decide if a given set of monomials is a basis of a polynomial ring quotient

Let $R = \mathbf{k}[x_1,\ldots,x_n]$ be a polynomial ring over some field $\mathbf{k}$ (which can be $\mathbb{C}$ if that makes a difference) and $I$ some ideal of $R$ such that $R/I$ is ...
1
vote
0answers
45 views

Finite colength ideals in a power series ring

$\newcommand{\Hilb}{\operatorname{Hilb}}$Let $I\subset R:=\mathbb{C}[[x,y]]$ be an ideal. Then, $I$ is said to be of colength $n$ if $\dim_\mathbb{C}(R/I) = n$. For example the ideal $(x,y)$ has ...
0
votes
0answers
24 views

Question about Macaulay2

Let $R=\mathbb Q[s^4,{s^3}t,s^{5}t^3,t^4]$. With Macaulay2 how do I find the isomorphism $\text{gr}_{m}(R)=Q[x,y,z,w]/q$, where $q=(-z^2,yz,xz,-y^4+x^3w)$ and $\text{gr}_{m}(R)$ is the associated ...
0
votes
0answers
39 views

Continuous maps are morphisms of varieties?

From the definition of morphism (Hartshorne's Algebraic Geometry) it looks like all the continuous maps are morphisms. Let $\phi: X \rightarrow Y$ be a continuous map and $f:Y \rightarrow k$ be a ...
4
votes
0answers
89 views

When is $f^! \mathcal{O}_Y$ a line bundle?

Let $f: X \to Y$ be a finite, surjective morphism of reduced, separated schemes of finite typer over some field of characteristic zero. The sheaf $\mathcal{H}om_Y(f_* \mathcal{O}_X, \mathcal{O}_Y)$ is ...
1
vote
0answers
24 views

Over what rings is the Hefferonian determinant unique?

Fix an $n\in\mathbb{N}$ and a field $\mathbb{K}$. A lot of texts in linear algebra like to define the determinant function on $\operatorname{M}_n\left(\mathbb{K}\right)$ as the unique function ...
4
votes
0answers
103 views

Milnor patching for general modules

The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings $$ \begin{array}{} R & \xrightarrow{f_2} & R_2 \\ \downarrow{f_1} & ...
9
votes
2answers
148 views

Why would we a priori expect $V(I)$ to satisfy axioms to define the closed sets for a topology on $\text{Proj}(S)$?

The topological space $\text{Proj}(S)$ has the underlying set$$\text{Proj}(S) = \{\mathfrak{p} \text{ a homogeneous prime such that }S_+ \not\subseteq \mathfrak{p}\},$$and the closed sets are the loci ...
4
votes
1answer
42 views

$R$ is a unique factorization domain $\iff$ every prime minimal over a principal ideal is also principal

I'm trying to show that a ring $R$ is a unique factorization domain $\iff$ every prime minimal over a principal ideal is also principal. I think the idea is to use the principal ideal theorem of ...
0
votes
0answers
26 views

Isomorphism of polynomial rings [duplicate]

I am trying to do exercise 3.6.F in Ravil Vakil's algebraic geometry notes : http://math.stanford.edu/~vakil/216blog/FOAGapr2915public.pdf We fix a field $k$. It comes down (or so I think) to proving ...
2
votes
0answers
84 views

When $mB \neq B$? $m$ is a maximal ideal of $A$, $A \subseteq B$.

Let $A \subseteq B$ be commutative rings, $m$ a maximal ideal of $A$. When $mB \neq B$? This is true when $A \subseteq B$ is faithfully flat. (If I am not wrong, this is also true when $A \subseteq ...