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

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14
votes
2answers
141 views

If $M\otimes N=R^n$ need $M$ be projective?

So if over a commutive ring, $R$, we have that $M\otimes N=R^n$, $n\neq 0$, need we have that $M$ and $N$ be finitely generated projective? We have finite generation, because if $M\otimes N$ is ...
3
votes
1answer
38 views

Showing the set of diagonalizable matrices is constructible

Identifying $M_n(k)$ with $k^{n^2}$ with $k$ algebraically closed, I am asked to show that the subset of diagonalizable matrices, $D_n$ is constructible. Constructible is defined as being the finite ...
4
votes
1answer
89 views

is the hilbert polynomial integer-valued everywhere?

Let $R$ be an $\mathbb{N}$-graded Noetherian ring, finitely generated over $R_0$ with $R_0$ local Artinian. Let $M$ be a finite $R$-module of Krull dimension $d$. It is known that the Hilbert function ...
2
votes
0answers
52 views

A certain valuation of $k(X,Y)$ with value group $\mathbb{Z}+\mathbb{Z}\alpha$

Let $k$ be a field, $X$ and $Y$ indeterminate, and suppose that $\alpha$ is a positive irrational number. Then the map $\nu:k[X,Y]\rightarrow \mathbb{R}\cup \{\infty\}$ defined by taking $\sum ...
1
vote
1answer
48 views

Intersection of all maximal ideals containing a given ideal

Let $I$ be a proper ideal in $k[x_1,....,x_n]$, where $k$ is an algebraically closed field. Show that $\sqrt{I}= \cap M$, where $M$ runs through all maximal ideals containing $I$. I am confused ...
7
votes
2answers
94 views

Techniques for showing an ideal in $k[x_1,\ldots,x_n]$ is prime

An affine variety $X$ over a field $k$ is irreducible if and only if its defining ideal $I(X)$ is prime (in this post we use the convention that varieties are not necessarily irreducible). Hence, it ...
2
votes
1answer
39 views

If ring $B$ is integral over $A$, then an element of $A$ which is a unit in $B$ is also a unit in $A$.

Let $A$ be a subring of ring $B$, with $B$ integral over $A$. If $x$ in $A$ is a unit in $B$, then it is a unit in $A$. I know that $f(t) = t - x$ is in $A[t]$ with $f(x) = 0$, and that there ...
1
vote
1answer
57 views

Reference-request for $Monomial\ Ideals$

I newly started to study the book Monomial Ideals by Jürgen Herzog, Takayuki Hibi, but it is difficult in some cases for a beginner like me. Is any other reference which have similar topics (part I) ...
5
votes
1answer
54 views

Topological closure of ideal in $A[[T]]$ - Proposition 1.3.7 in Liu

In Proposition 1.3.7 of Liu's book, one proves that if a ring $A$ is noetherian then so is $A[[T]]$. We take an ideal $I$ of $A[[T]]$ and prove that there exist $F_1,\ldots,F_m\in I$ such that for all ...
0
votes
1answer
41 views

Every irreducible element is prime: always holds under surjective ring homomorphism?

Let $f:R\rightarrow S$ be a surjective ring homomorphism. $R,S$ are both integral domains. If every irreducible element in $R$ is prime, then is it true that every irreducible element in $S$ is ...
2
votes
2answers
41 views

Ring homomorphism with field as image, is the pre-image also a field?

Let $f:R\rightarrow S$ be a surjective ring homomorphism. $R,S$ are both integral domains. Suppose $S$ is a field, then is $R$ also a field? A possible useful fact: A finite integral domain is a ...
1
vote
0answers
25 views

Grading and commutators

If $R$ is a unital associtaitve commutative ring, then can any $R$-algebra $A$ may be filtered as $A_0:=A$ and $A_{i+1}:=(A_i,A_i)$ where $(-,-)$ is the commutator of $A$ with respect to $A$'s ...
0
votes
1answer
47 views

Homogeneous polynomial in a homogeneous ideal

Let $f$ be a non-zero homogeneous polynomial in a homogeneous ideal generated by homogeneous elements $g_1,\ldots, g_s$. Suppose $f= h_1g_1 +\cdots+h_sg_s$. Is it necessary that $\deg(f)=\deg(h_ig_i)$ ...
1
vote
2answers
94 views

Generators for (radical) ideal of product of affine varieties

If $X\subset \mathbb{A}^N$ and $Y\subset\mathbb{A}^M$ are affine varieties with $X=Z(f_1,\dots,f_n)$ and $Y=Z(g_1,\dots,g_m)$ then $X\times Y\subset\mathbb{A}^{N+M}$ is an affine variety with $X\times ...
1
vote
1answer
68 views

When does coprimality carry over to the base ring in an extension of Dedekind domains?

Let $A$ be a Dedekind domain. Let $K$ be the field of fractions of $A$ and $L$ is some finite field extension of $K$. Then let $B$ be the integral closure of $A$ in $L$. (Sorry I don't know how to ...
1
vote
1answer
41 views

Square of tensor product

Let $A$ be an integral domain and $B$ be an $A$ algebra. Let $I$ be and ideal of $B$. Something has been bugging me : Is it true that $$(B\otimes I)/(B\otimes I)^2 \cong B \otimes (I/I^2)$$ We ...
0
votes
1answer
99 views

What is a hypersurface ring and why is it Gorenstein? [duplicate]

My question is about http://mathoverflow.net/questions/131652/canonical-modules and If $R$ is a local ring with $\operatorname{emb dim} (R) = \text{depth}(R) + 1$ then $R$ is Gorenstein In the ...
2
votes
1answer
61 views

Geometric reducedness (integral) versus reducedness (integral)

All the schemes here are over $\mathbb{C}$. Suppose $X \to Y$ is a morphism of varieties, then the geometric reducedness (integral) of the generic fibre implies the geometric reducedness (integral) ...
2
votes
1answer
87 views

Let $I= (X_1,X_2) \cap (X_3,X_4)$. Is $ara(I)≥3$? Is $ara(I)≥4$?

This question is related to Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $k[X_1,X_2,X_3,X_4]$ Let $R=k[X_1,X_2,X_3,X_4]$ and $I= (X_1,X_2) \cap (X_3,X_4)$. I know that ...
4
votes
1answer
35 views

If $I\leq K[X_0,\dots,X_n]$ for $K$ a field is an ideal whose radical is homogeneous, is it always the case that $I$ is homogeneous?

If $I\leq K[X_0,\dots,X_n]$ (for $K$ a field, let's say algebraically closed) is an ideal whose radical is homogeneous, is it always the case that $I$ is homogeneous? I'm trying to understand ...
0
votes
1answer
112 views

$\operatorname{inj.dim}_R N= \operatorname{inj.dim}_R \widehat{N}$?

$(R,m)$ is a local ring. For an $R$-module $N$, we know that $\operatorname{inj.dim}_R N= \operatorname{inj.dim}_\widehat{R} \widehat{N}$. Is it true that $\operatorname{inj.dim}_R N= ...
2
votes
1answer
60 views

Map factor through quotient map?

Affine coordinate axes in $\mathbb{A}^2(\mathbb{C})$ (place where $x=0$ or $y=0$), so $\simeq \mathbb{C}[x,y]/\langle xy\rangle=:A$. $\forall a,n\in\mathbb{N}-\{0\}$: Do all ring homomorphism $f:A\to ...
1
vote
1answer
131 views

I need help in this proof in Lang's algebra book

I have a simple doubt in this proof in Lang's Algebra book: I can understand that there are some $g_1, g_2$ such that $f_1g_1+f_2g_2$ has leading coefficient 1 and degree $\le d-1$ but why by row ...
2
votes
0answers
91 views

Associated primes and their heights

Let $(S,m)$ be a commutative Gorenstein local ring, $I$ an ideal of $S$ such that $\operatorname{ht} I=t$, and $R=S/I$. Let $a \in m$ be an $R$-regular element such that for any prime ideal ...
1
vote
1answer
43 views

Prime ideals in formal power series

Let $A$ be a commutative ring with unit. If $\mathfrak{p} \subset A $ is a prime ideal, then $\mathfrak{p}$ is the contraction of a prime ideal of $A[[x]]$, the ring of formal power series. Why is ...
6
votes
0answers
95 views

Construct a minimal free resolution, Bruns and Herzog, Exercise 2.3.18(a)

Here is a question form Bruns-Herzog, Cohen-Macaulay Rings, exercise 2.3.18(a). Let $S$ be a regular local ring of dimension $4$, and $y_1$, $y_2$, $y_3$, $y_4$ a regular system of parameters. ...
7
votes
1answer
114 views

What's the motivation of definition of primary?

Primary ideal can be regard as the generalization of prime ideal and radical. But Why it's defined like that?It's not symmetry. Why not define like that:
1
vote
1answer
47 views

an isomorphism of extension functors

Let $(R,m)$ be a Noetherian *local ring and suppose that $m$ is maximal in the ordinary sense. Then why is it true that $\operatorname{Ext}^i_R(R/m^j,M) \cong ...
2
votes
1answer
86 views

Question about some details of a proof

i) Why it's a unit can prove this proposition ii)see picture
0
votes
0answers
37 views

Bruns and Herzog, Cohen-Macaulay Rings, Proposition 3.6.16

Let $(R,m)$ be a graded Noetherian *local *complete ring, i.e. $(R_0,m_0)$ is complete, with $m_0 = m \cap R_0$. For a graded $R$-module $M$, define the functor $M^\vee = ...
0
votes
1answer
49 views

Solved exercises in commutative algebra

I'm looking for books or teaching material with solved exercises in commutative algebra, where can I find them ?
1
vote
1answer
57 views

Prime, Maximal, and Radical Ideals in $\mathbb{C}$[x] and $\mathbb{R}$[x]

What are the prime, maximal, and radical ideals in $\mathbb{C}$[x] and $\mathbb{R}$[x]? My gut feeling is that the prime are ideals in $\mathbb{C}$[x] are those which are generated by linear terms, so ...
2
votes
1answer
40 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. ...
1
vote
0answers
39 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
24 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
82 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 ...
-2
votes
1answer
67 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
42 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
84 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
19 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 ...
0
votes
0answers
20 views

Sections of the twisting sheaf over distinguished opens

For a ($\mathbb{Z}-$)graded commutative ring $S$ and a graded $S$-module $M$ we obtain a sheaf $\widetilde{M}$ on $\operatorname{Proj}(S)$ satisfying ...
2
votes
2answers
114 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
41 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, $$ ...
2
votes
1answer
30 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
54 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
36 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 ...
1
vote
1answer
91 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
31 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 ?
4
votes
3answers
200 views

Idempotent 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
64 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.