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

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3
votes
1answer
102 views

Projecting an affine hypersurface away from a point in its projective closure is never a finite map?

Let $X\subset \mathbb{A}_k^r$ be an irreducible hypersurface defined by a polynomial $g$, where $k$ is an algebraically closed field. Embed $\mathbb{A}^r\hookrightarrow\mathbb{P}^r$ in the usual way. ...
2
votes
0answers
62 views

Generic fibre of a fibre product

Let $X,Y_1, Y_2$ be noetherian schemes over $\mathbb{C}$ and $Y_1,Y_2$ be integral schemes. Let $f: X \to Y_1 \times_{\mathbb{C}}Y_2$ be a morphism and $X_0$ be its generic fibre (i.e. fibre over the ...
1
vote
1answer
27 views

Prime Ideals gotten from homomorphisms

I am asked to prove that every prime ideal P of a ring R can be obtained as the kernel of a homomorphism to a field. I know that the kernel of a homomorphism is an ideal. I need to start from an ...
1
vote
0answers
99 views

Help in the proof of Horrocks theorem

I'm trying understand this proof: Obviously, $b_s\in \mathfrak a$, to see this just take $g_1=0$, but I couldn't prove why the others $b_i$ are in $\mathfrak a$. If we do as the author said we had ...
-1
votes
1answer
538 views

Surjective Implies Injective for R-Homomorphism on Finitely Generated Module [duplicate]

Let $M$ be a finitely generated module over a ring $R$, and let $f$ be an $R$-homomorphism from $M$ to itself. Does $f$ injective imply $f$ surjective? Does $f$ surjective imply $f$ injective? I have ...
2
votes
0answers
48 views

$I(Z(J)\setminus Z(K))=(I(Z(J):I(Z(K))$, where $J$, $K$ are ideals in $k[x_1,\ldots, x_n]$

I was wondering if the first statement here is true. I am asking because the second statement of the above link is not true as can be seen from this question. If it is not true, then under what ...
2
votes
1answer
165 views

What does it mean geometrically for a variety to be locally a complete intersection?

We say that an affine variety $X \subset \mathbb{A}^n$ of dimension $n-k$ is a complete intersection if the ideal of $X$, call it $I(X)$ is generated by $k$ polynomials, $f_1,\dots,f_k$. We say ...
1
vote
1answer
23 views

Maximal element of $(I : x)$, where $x$ is in $A - I$, is prime belonging to $I$

Given that $I$ is decomposable, I am supposed to prove that any maximal element $P$ of the set {$(I : x) | x \in A - I$} must belong to $I$, i.e., $P$ is prime and for every reduced primary ...
2
votes
2answers
126 views

Help in this proof in Lang's Algebra book

I'm trying to understand this part of the proof: I didn't understand why not all coefficients of $f_2,\ldots,f_n$ can lie in the maximal ideal, maybe I'm forgetting something, it should be a very ...
1
vote
1answer
62 views

A question about the depth of a ring with respect to some ideal

So here is my question: I want to compute the depth of $k[x,y]$ with respect to the ideal $(x,y^2)$ where $k$ is a field. The depth $t_{(x,y^2)}(k[x,y])$ is defined as follows, $$ ...
3
votes
1answer
44 views

Is the Derivation Algebra functorial

Suppose $A$ is a commutative, associative $k$-algebra with unit and $Der(A)\subset End_k(A,A)$ is the algebra of derivations on $A$, that is the subalgebra of endomorphisms, such that ...
2
votes
1answer
63 views

Why is this projective curve in $\mathbf{P}^3_k$ nonsingular?

Consider $C$ in $\mathbf{P}^3_k = \mathrm{Proj}[x_0,...,x_3]$ defined by $$x_0x_3 - x_1^2 = 0$$ and $$x_0^2 + x_2^2 - x_3^2 = 0$$ where $k$ is an algebraically closed field. Why is this curve ...
1
vote
2answers
38 views

In $\mathbb{Z}[t]$, $Q = (4, t)$ is not a power of $M = (2, t)$

The problem of showing that Q, as above, is not a power of M, as above, rises as part of a larger problem. I'm confident about my response to the other parts, but the best justification I can come up ...
2
votes
3answers
328 views

There are infinitely many monomial orders

Show that if $n ≥ 2$ there are infinitely many monomial orders on $k[x_1, \ldots , x_n]$. I think it is Robbiano theorem (with the exception $n>2$) at the link below but i can't understand ...
0
votes
1answer
46 views

Finitely generated ideal in boolean ring [duplicate]

A boolean ring is a commutative ring where $x^{2} = x$ for every $x$. Why in such a ring a finitely generated ideal is principal ?
5
votes
5answers
664 views

Idempotents in a local ring

Is it true that a local ring, i.e., a commutative ring with a unique maximal ideal, doesn't contain idempotent elements $\neq 0, 1$ ? Why ? Any hint ?
1
vote
2answers
147 views

Zero dimensional local ring with maximal ideal not principal.

Probably it is well known. I am looking for a zero dimensional local ring with maximal ideal not principal.
1
vote
1answer
70 views

Are Ideals and Varieties Inclusion Reversing?

Let $S_1$, $S_2$ be sets or varieties (I don't think it matters, does it?). Then if $S_1 \subset S_2$, is it always the case that $I(S_2) \subset I(S_1)$ (where I is an ideal)? Also, is it always the ...
1
vote
1answer
153 views

Height unmixed homogeneous ideal and a non-zero divisor

Let $R=k[x_1,\ldots,x_n]$ be a standard graded polynomial over field $k$ and $I$ an unmixed homogeneous ideal of $R$. Let $x\in R$ be an $R/I$-regular element. Can we conclude that $x+I$ is an ...
4
votes
1answer
127 views

Localization of Coordinate Rings: $\mathbb C[V_f] = \mathbb C[V]_f$.

Let $V\subseteq\mathbb C^n$ be an irreducible affine variety, then the coordinate ring $$\mathbb C[V] = \mathbb C[x_1,\dots,x_n]\big/\mathbf I(V)$$ is an integral domain. Let $f\in\mathbb ...
6
votes
1answer
76 views

$S^{-1}(\mathbb{Z}[i])$, where $S=\{x\in \mathbb{Z}|5\nmid x\}$.

Let $S=\{x\in \mathbb{Z}|5\nmid x\}$. I would like to know all the prime ideals of $S^{-1}(\mathbb{Z}[i])$. My attempt: Since $S\subset \mathbb{Z}$, the given question can be rewritten as ...
-1
votes
1answer
87 views

A big list of examples that a power of a prime ideal is not primary in an algebra of finite type over a field

Let $k$ be a field. Let $A$ be an integral domain which is a $k$-algebra of finite type. I would like to know examples that a power of prime ideal of $A$ is not primary. The more example, the better. ...
3
votes
0answers
249 views

Algorithm to determine whether a power of a prime ideal is primary in a polynomial ring over a field

Let $P$ be a prime ideal of a polynomial ring $k[X_1,\cdots, X_m]$ over a field $k$. Suppose a finite basis of $P$ is given. Is there an algorithm to determine whether $P^n$ is primary for a given ...
1
vote
1answer
176 views

Primary ideals in Noetherian rings

For an $R$-module $M$ I have the following definition for a submodule $N\subset M$ to be $\mathfrak{p}$-primary: this is the case when $\text{Ass}(M/N) = \{\mathfrak{p}\}$, that is, $M/N$ is coprimary ...
1
vote
0answers
21 views

Examples of d-extensions in realisation of $\operatorname{Ext}^d$

If $R$ is a commutative unital associative ring and $A$ is an $R$-algebra of dimension $d$, which is local as a ring, then from dimension theory we know that the global dimension of $A$ must be at ...
1
vote
1answer
75 views

Maximal (among non-principal ideals) Ideal Must be Prime

If I is an ideal which is maximal among the ones that are not principal, then I is prime. This would mean that for all $f \in R$, $(f) \subset I$. Could I then use column ideals? I was thinking ...
0
votes
1answer
72 views

Maximal Ideal Must be Prime

I am trying to prove that an ideal that is maximal with respect to not being finitely generated must be prime. What does it mean to be an ideal that is maximal with respect to not being finitely ...
0
votes
0answers
82 views

Zero dimensional ideals and their primary decomposition

Let $S=k[x_1,\dots,x_n]$ be a polynomial ring over a field $k$, and $I$ a zero dimensional ideal with a primary decomposition $I=\cap Q_i$. Why is $\sum \dim_k S/Q_i = \dim_k S/I$?
1
vote
1answer
97 views

Local Cohomology - Theorem 3.5.8 in Bruns and Herzog, Cohen-Macaulay Rings

This question arises in the context of Theorem 3.5.8 in Bruns and Herzog, Cohen-Macaulay Rings. Let $(R,m)$ be a local complete Cohen-Macaulay ring of dimension $d$. Denote by $H_m^d(-),\omega_R$ ...
0
votes
2answers
64 views

The image of an ideal under an automorphism is a subset of the same ideal

Let $A$ be a commutative ring with an identity element $1$, let $a$ be an ideal of $A$, and $f: A \rightarrow A$ be any automorphism. Is it true that $ f(a)\subseteq a$.
0
votes
1answer
96 views

Is the residue field of an algebraically closed field with respect to a non-trivial valuation infinite?

Let $K$ be an algebraically closed field with a non-trivial valuation. Let $\Bbb k$ be the residue field, i.e. $\Bbb k = R/\mathfrak m$ where $R$ is the valuation ring $R:=\{a\in K\mid \text{val} ...
7
votes
0answers
167 views

Module of Kähler differentials for a formal power series ring

Let $A$ be a ring and $A[[T]]$ the formal power series over $A$. Then, one can show that $\Omega^1_{A[[T]]/A}$ is not finitely generated over $A[[t]]$. Now, in $\Omega^1_{A[[T]]/A}$ I am trying to ...
1
vote
1answer
89 views

Socle degrees and last shift of free resolution

I have seen in several references that the degrees of the socle of an Artinian graded algebra $k[x_1,\ldots,x_d]/I$ can be computed by looking at the shifts of the end of its graded free resolution. ...
4
votes
2answers
120 views

$Z(I:J)$ is the Zariski closure of $Z(I)-Z(J)$

Let $(I:J)$ denote the colon ideal (or ideal quotient). It is pretty clear that the Zariski closure of $Z(I)-Z(J)$ is contained in $Z(I:J)$. How can we prove that the the Zariski closure of ...
9
votes
2answers
446 views

A proof using Yoneda lemma

Martin Brandenburg pointed out elsewhere in the comments that he could give a one line proof, using the Yoneda lemma, of $$\frac{\mathbf{C}[x_1,\ldots,x_{n+m}]}{I(X)^e+I(Y)^e} \cong ...
2
votes
0answers
72 views

Find the projective closure of the ideal $I=\langle y-x^2,z-x^3\rangle$

When I looked at this example, my first instinct was to homogenize only the generators of $I=\langle f_1 := y-x^2,f_2:=z-x^3\rangle$ in a new variable $w$. But then, I realized that I would miss some ...
3
votes
1answer
96 views

Proposition 3.5.1 in Bruns and Herzog, Cohen-Macaulay Rings

Bruns and Herzog in their book Cohen-Macaulay Rings, page 128 consider a local Noetherian ring $(R,m,k)$, an $R$-module $M$ and they define the functor $\Gamma_m(\cdot)$ as $\Gamma_m(M) = \varinjlim ...
1
vote
0answers
49 views

How can we show that $\operatorname{Tor}_1^A(\hat{A},G_f/G)=0$?

This question is quite closely related to my last question: Is it always true that $\hat{A} \otimes_A A_f/A \cong \hat{A}_f/ \hat{A}$? Let $A$ a commutative ring, $f \in A$ a non zerodivisor. Let ...
2
votes
0answers
100 views

Relation between closed subschemes and saturated ideals

Let $A=\mathbb{C}[x_0,x_1,\dots,x_n]$ and $X=\operatorname{Proj}A$. For any homogeneous ideal $I\subset A$, define the saturation $I^{\rm sat}:=\{f\in A\mid (x_0,\dots,x_n)^mf\subset I$ for some ...
3
votes
0answers
65 views

proving an isomorphism of direct limits

Let $(R,m,k)$ be a local Noetherian ring and $M$ an $R$-module. Let $\left\{I_s\right\}_s$ be a directed system of ideals whose induced topology is equivalent to the $m$-adic topology. Using the ...
2
votes
1answer
161 views

Hilbert-Burch theorem characterizes perfect ideals of grade $2$

Bruns and Herzog in their book Cohen-Macaulay Rings, page 120 write: "The Hilbert-Burch theorem 1.4.17 identifies perfect ideals of grade $2$ as the ideals of maximal minors of certain matrices". ...
1
vote
1answer
50 views

Showing two thing are isomorphic as $k$-algebras

Let $k$ be a finite field of order $q$. Let $x$ be a closed point in $\mathbb{P}^1_k$ and let $D$ be a divisor $D = (e+1) x$, where $e \in \mathbb{N}$. We define few things, $$ P_N = \{ x \in k[u,v]: ...
0
votes
1answer
47 views

proving that the height of $(X)$ in $R[X]$ is equal to 1

Let $R$ be a Noetherian ring. I want to prove that $\operatorname{height}(X) = 1$. Here is how i do it: It is not hard to show that $\operatorname{height}(X) = \operatorname{height}(p,X)$, where $p$ ...
0
votes
1answer
48 views

Could you please explain the detail of the proof

Proposition: Proof: Question: Why it's isomorphism?
1
vote
0answers
52 views

Describing a locally free sheaf sitting between two locally free sheaves which are given as extensions

Assume $X$ is a two dimensional scheme with $Pic(X)=\mathbb{Z}$ such that every rank two locally free sheaf $\mathcal{E}$ is given by an exact sequence $0\rightarrow \mathcal{O}_X(n)\rightarrow ...
1
vote
2answers
63 views

Problem related to prime ideals of B and A where B is integral over A

Let $ A $ be an entire ring, integrally closed. Let $ B $ be entire, integral over $A$. Let $ Q_1, Q_2$ be prime ideals of $B$ with $Q_1 \supseteq Q_2$ but $Q_1 \neq Q_2$. Let $P_i=Q_i \bigcap A$. ...
2
votes
1answer
82 views

Why is $f'(x)$ the annihilator of $dx$?

Let $B=A[x]$ be an integral extension of a Dedekind ring $A$ where $x$ has minimal (monic) polynomial $f(x)$. Then the module of Kahler differentials $\Omega_A^1 (B)$ is generated by $dx$. Why is its ...
4
votes
5answers
521 views

Every module over a field is free. Is every commutative ring whose modules are all free a field?

Let $A$ denote a commutative ring. Then if $A$ is a field, we may deduce that every $A$-module is free. Does the converse hold? i.e. If every $A$-module is free, can we deduce that $A$ is a field?
2
votes
0answers
51 views

Bounds dimension, scheme and projective dimension

Is the dimension of a (commutative unital associative) algebra always bounded above by its protective (injective) dimension? If not is it always bounded above by its global dimension?
1
vote
1answer
82 views

Basis of a subset of finitely generated torsion free module

Based on the comments of rschwieb's answer in this question asked recently: Can we contruct a basis in a finitely generated module. If $M=\langle e_1,\ldots,e_n\rangle$ is a finitely generated ...