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

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4
votes
2answers
158 views

Question on an isomorphism in the proof that $k[V \times W] \cong k[V] \otimes_k k[W]$

First I should say that I am aware of the existence of this question here and this question here. My question is a little different from these two because I am asking about a certain detail in the ...
8
votes
1answer
194 views

What does projective space classify?

Let $A$ be a ring and let $\mathbb{P}^n = \operatorname{Proj} \mathbb{Z} [x_0, \ldots, x_n]$. Question. What does $\mathbb{P}^n$ classify? In other words, is there some kind of algebraic structure ...
19
votes
1answer
477 views

Are finitely generated projective modules free over the total ring of fractions?

Let $Q(A)$ be the total ring of fractions of a commutative reduced non-noetherian ring $A$. Let $P$ be a finitely generated projective module over $Q(A)$ which is of constant rank (i.e. locally free ...
1
vote
0answers
45 views

Lorenzen embedding theorem for an $\ell$-group

The Lorenzen embedding theorem for an lattice-ordered group says that any lattice-ordered group can be embedded into a product of totally ordered group. What condition on lattice-ordered group makes ...
9
votes
2answers
370 views

Intersection of finitely generated ideals

Let $I$, $J$ be finitely generated ideals in a ring $A$ (commutative with identity). I know that the intersection need not be finitely generated: can somebody give me an example? Thanks.
1
vote
0answers
90 views

Minimal syzygies for polynomial ideals

Let $I$ be an ideal of $S=k[x_1,\dots,x_n]$. I am asked to find a minimal free graded resolution of $I$, by means of syzygy matrices. I suppose there has to be an algoritmic approach to it, provided ...
1
vote
1answer
76 views

Ideals / Direct sum decomposition

Let $u = (u_1 , \ldots , u_n ) \in \mathbb{A}^n$. Let $I$ be the ideal of $A = \mathbb{C}[x_1 , \ldots x_n ]$ generated by the elements $x_1 - u_1 , \ldots , x_n - u_n$. (i) Show that as a ...
-2
votes
4answers
140 views

Radical of an ideal I

Let $I$ be the ideal of $\mathbb{C}[x,y]$ generated by $x^8$, $x^2y^3$, $x^7 - y^5$, $y^{42}$. Find a simple expression for the radical $\sqrt{I} = \{ f \in \mathbb{C}[x,y] : f^n \in I\;\text{for ...
3
votes
1answer
137 views

vector bundles on the affine line over a PID

Let $R$ be a PID. Is every finitely generated projective $R[T]$-module free? In other words, is every vector bundle on $\mathbb{A}^1_R$ trivial? For $R=k[X]$ this is true by the Theorem of ...
3
votes
1answer
219 views

Coordinate ring of a cartesian product

I am considering the coordinate ring $k[X \times\mathbb{A}^n]$, where $X$ is an algebraic variety in $\mathbb{A}^n$. I want an isomorphism between this and the polynomial ring $k[X][y_1,\ldots, y_n]$. ...
-1
votes
1answer
90 views

Noetherian modules

Question: Let $R$ be a Noetherian ring, and $M$ be an $R-$module, show that $M$ is Noetherian if and only if $M$ is finitely generated. This is a question on my homework, I'm really confused about ...
2
votes
1answer
49 views

practical condition for minimality in primary decomposition

Situation: $I$ is an ideal in a polynomial ring with a primary decomposition, not necessarily minimal (minimal=irredundant). I want to minimal-ize it. For any couple of primary ideals with the same ...
6
votes
2answers
185 views

Dimension of the total ring of fractions of a reduced ring.

Let $A$ be a commutative reduced ring (need not be noetherian). Let $S$ be the set of all non-zerodivisors of $A$. What is the Krull dimension of $S^{-1}A$ ?
4
votes
3answers
315 views

Noetherian Local Ring

I came across this old exam problem. If $R$ is a local Noetherian ring and $I$ is an ideal in $R$ such that $I^2=I$ then $I =0$. Any hint would be appreciated. I'm only familiar with what the ...
0
votes
1answer
87 views

Hypotheses of the Conormal Exact Sequence

On Wikipedia, in the description of the conormal exact sequence, it is described as arising from a closed immersion, which corresponds in the affine case to a surjection of algebras. However, in ...
5
votes
3answers
385 views

The vanishing ideal $I_{K[x,y]}(A\!\times\!B)$ is generated by $I_{K[x]}(A) \cup I_{K[y]}(B)$?

Let $K$ be a field, $x=(x_1,\ldots,x_m)$, $y=(y_1,\ldots,y_n)$, $A\!\subseteq\!\mathbb{A}^m_K$, $B\!\subseteq\!\mathbb{A}^n_K$. Does there hold $$I_{K[x,y]}(A\!\times\!B)=\langle\langle I_{K[x]}(A) ...
0
votes
3answers
269 views

Localization of Noetherian and Artinian Modules

Theorem: Let $R$ be a commutative ring with unity, and $S\subset R$ be a multiplicatively closed subset. If $M$ is a Noetherian (Artinian) $R$-module then $S^{-1}M$ is Noetherian (Artinian) ...
5
votes
2answers
309 views

Companion Lecture Notes to Atiyah-MacDonald?

Is there a set of lecture notes that follow Atiyah-MacDonald and expand on the dense passages, point out typos and so forth?
-2
votes
1answer
136 views

Prime ideal in a Dedekind domain

Let $\mathfrak p $ be a prime ideal in a Dedekind domain $O$ with field of fractions $K$. Define $$\mathfrak p^{-1}= \{x \in K: x\mathfrak p \subset O\}.$$ Let $\mathfrak a \subset \mathfrak p$ and ...
4
votes
2answers
79 views

Zerodivisors and nilpotents in $A/I$

I'm studying primary decomposition in the case of polynomial rings with coefficients in a field. I have defined associate prime ideals of an ideal I as the radicals of the primary ideals appearing in ...
3
votes
1answer
291 views

Finite + surjective + projective implies flat?

Let $f: X \rightarrow Y$ be a morphism of irreducible projective varieties, that is both finite and surjective. Does this mean that it is flat? I have tried the following: By finiteness, the map is ...
1
vote
1answer
87 views

Generators for this ideal

I have a ring $R$ unitary and commutative with four elements and characteristic $2$. I have $$I=\{f \in R[X,Y]; f(t,t^2)=0\ \forall t \in R\}.$$ I have to find a finite number of generators for this ...
6
votes
1answer
155 views

A valuation ring

In Qing Liu, Algebraic Geometry and Arithmetic Curves, page 116, exemple 4.1.8, one has $\mathcal{O}_K$ a discrete valuation ring with uniformizing parameter $t$, $P\in\mathcal{O}_K[S]$ an Eisenstein ...
4
votes
4answers
1k views

A proof that shows surjective homomorphic image of prime ideal is prime

Let $A, B$ be commutative rings with $1_{A}, 1_{B}$. Suppose that $\mathfrak{p} \neq (1)$ is a prime ideal in $A$ with $\mathfrak{p} \supseteq \ker{\varphi}$ where $\varphi: A \rightarrow B$ is a ...
4
votes
4answers
147 views

Irreducible element of the ring.

Element $X_1 X_2 \cdots X_n - 1$ is irreducible in $K[X_{1},\ldots,X_{n}]$ for $n\ge 1$, where $K$ is a field. For $n=2,3$ it is easy to see that the element is irreducible but for higher value of $n$ ...
3
votes
2answers
198 views

Elements of the square of a prime ideal

Let $R$ be a commutative ring with unity, and let $\mathfrak{p} \subset R$ be a prime ideal. If $ab \in \mathfrak{p}^2$, does one of the following hold? $a \in \mathfrak{p}^2$; $b \in ...
1
vote
2answers
517 views

Tensor products of fields

Let $K/F$ be a field extension. I am interested in the situation where there exists a field extension $L/F$ such that the ring $L \otimes_FK$ is not a field. If there exists $z\in K \setminus F$ ...
4
votes
2answers
145 views

Epic maps in the category of commutative rings with identity.

Here all rings are assumed to lie in the category $\cal C$ of commutative rings with identity, and ${\cal C} (\ R\ ,\ S\ )$ is the set of all ring homomorphisms $F$ from $R$ to $S$ for which ...
1
vote
1answer
42 views

Nice closed form for polynomial defined as an antiderivative

Let $n$ be an integer $\geq 1$, and let $f_n(t)=(t(1-t))^n$. Let $F_n(t)$ denote the antiderivative of $f_n(t)$ satisfying $F_n(0)=0$. Of course, using Newton’s binomial formula we have an expansion ...
0
votes
0answers
112 views

Does finitely generated associated graded module imply stable filtration?

Let $R$ be a Noetherian ring and $\mathfrak a$ be an ideal of $R$. Then (i) $G_{\mathfrak a}(R)$ is Noetherian. (ii) If $M$ is a finitely generated $R$-module and $\mathcal F=\{M_n\}$ is a stable ...
1
vote
1answer
183 views

torsion free sheaves

I have a stupid question. Let $X$ be a singular curve over a field. Let $F$ be a torsion free sheaf generically (outside singularities) of rank one. Is $F$ coherent? Thanks
1
vote
1answer
95 views

Homogeneous Ideal in Laurent Polynomial Ring

This question is exercise 13.2 from Matsumura, Commutative Ring Theory. Let $R = R_0 + R_1 + \cdots$ be a graded ring, $I$ an ideal of $R$ and $t$ an indeterminate over $R$. Set $R' = R[t, ...
3
votes
1answer
110 views

Irredundant primary decomposition of a submodule transferred to irredundant primary decomposition of an ideal

Let $N=\bigcap_{i=1}^nN_i$ be an irredundant primary decomposition of a submodule $N$ of the $R$-module $M,$ where $(N_i:M)$ is a $P_i$-primary ideal of $R$. (Irredundant means that all the $P_i$'s ...
3
votes
0answers
52 views

Direct Sum Decomposition of Certain Quotient Algebras

Consider a weighted homogeneous polynomial $f \colon (\mathbb{C}^{n}, \mathbf{0}) \to (\mathbb{C},0)$ with an isolated critical point at the origin and satisfying $\lambda f(z_1, \dots, z_n) = ...
6
votes
2answers
226 views

Restriction of scalars and tensor product

All rings I'll consider will be commutative with identity. Given a homomorphism $f:R \to S$ we can give an $S$-module an $R$-module structure via restriction of scalars. In particular, $S$ can be ...
2
votes
1answer
167 views

Relation between associated primes and primary decomposition for non-finite modules

Theorem 6.8(ii) p.41 in Matsumura's Commutative Ring Theory, says that if $A$ is a Noetherian ring, $M$ a finite $A$-module and $N=N_1 \cap \cdots \cap N_s$ an irredundant primary decomposition of a ...
3
votes
1answer
197 views

Localization and Radicals

Let $R$ be a commutative ring with unity, $S \subset R$ a multiplicative closed subset and $I \subseteq R$ be an ideal. Show that$$S^{-1} \sqrt{I}=\sqrt{S^{-1}I}$$ and $$S^{-1}J(R)=J(S^{-1}R),$$ where ...
5
votes
2answers
232 views

$A$ PID, $M$ flat (i.e., torsion-free). Then $\operatorname{Ext}_A^1(M,N)$ is injective, for all $N$.

Let $A$ be a PID and $M$ a flat (i.e., torsion-free) $A$-module. Then, for every $A$-module $N$, $\text{Ext}_A^1(M, N)$ is injective in $A\text{-}\mathbf{Mod}$. It is easy when $M$ is finitely ...
6
votes
1answer
44 views

Is there an ideal decomposition that counts the number of monomial generators?

Consider the ideal $I\subseteq S[x,y,z]$ where $S$ is some field of characteristic 0 (probably any field will do) and $I=\langle x^9-y^4z^4,y^9-x^5z^4,z^8-x^4y^5,x^6\rangle$. Notice that because the ...
4
votes
2answers
128 views

Homomorphism between finitely generated algebra

Let $A,B,C$ are the coordinate ring of three affine varieties in affine space over an algebraically closed field $k$, $B$ and $C$ are normal, and $i: A\longrightarrow B$ is an injective $k$-algebra ...
3
votes
1answer
59 views

Why $\dim \operatorname{supp} M_{m}\ge \dim \operatorname{supp} M_{m'}$?

Jacob Lurie made the following claim during his lecture: If $R\rightarrow R'$ is a morphism that makes $R'$ a finitely generated $R$-module (in particular, integral over $R$). Let $m'\subset R'$ be ...
2
votes
0answers
176 views

Example on non-projectively normal variety

This question orginally comes from the exercise 3.18 of Hartshorne, Algebraic Geometry. If $Y$ is a projective variety in $\mathbb{P}^n$ then $Y$ is projectively normal (w.r.t the embedding) if its ...
2
votes
2answers
206 views

Direct limit in category theory

$\newcommand{\al}{\alpha}$Let $(M_\alpha)_\alpha$ be a direct system of abelian groups, and $\varinjlim M_\alpha$ its direct limit. Then one can show that every element of $\varinjlim M_\alpha$ can be ...
4
votes
0answers
140 views

Definition of analytically unramified rings

A noetherian local ring A is said to be analytically unramified if the complete local ring $\hat{A}$ is reduced. I don't see why it makes sense to call such a ring analytically unramified. The ...
6
votes
2answers
248 views

Is there an irreducible polynomial vanishing on two components? (In the Zariski sense)

The polynomial $$f(x,y) = (x^2 − 1)^2 + (y^2 − 1)^2$$ is an example of an irreducible polynomial in $\mathbf{R}[x,y]$ which is irreducible but whose zero set has multiple components in the Zariski ...
8
votes
4answers
293 views

Subrings of fraction fields

Let $R$ be an integral domain and let $S$ be a ring with $R \le S \le \text{Frac}(R)$ (fraction field). Question: Is there a multiplicatively closed subset $U \subseteq R\setminus \{0\}$ such that ...
1
vote
1answer
103 views

An example of a ring of formal power series with a certain property

On page 118 of Matsumura it is said that it is not true in general that $A[[X]]\otimes_A k(\mathfrak p)$ is isomorphic to $k(\mathfrak p)[[X]]$ where $\mathfrak p$ is a prime ideal of $A$ and ...
2
votes
1answer
153 views

On theorem 15.4 of Matsumura

Matsumura is proving that $\dim A[X]=\dim A+1$. Using a theorem proved previously he proves that if $P$ is a prime ideal of $A[X]$ and $p=P\cap A$ then $\mathrm{ht}\;P=\mathrm{ht}\;p+1$, but it ...
5
votes
2answers
520 views

$R[[x]]$ for a Noetherian ring $R$?

Let $R$ be a Noetherian ring. How can one prove that the ring of the formal power series $R[[x]]$ is again a Noetherian ring? It is well-known that the ring of polynomials $R[x]$ is Noetherian. I ...
8
votes
1answer
201 views

When is the pushforward / direct image of a reflexive sheaf locally free?

I have seen a number of theorems that guarantee the direct image of a reflexive sheaf to be reflexive again, or for the direct image of a locally-free sheaf to be locally free again. This makes me ...