4
votes
2answers
120 views

Proving that tensoring a projective module with a flat module gives a projective module?

If $P$ is a projective module and $M$ is a flat module, both over some commutative ring $R$, then how do you prove that $M\otimes_R P$ is flat? I tried using the fact that $P$ is the direct ...
2
votes
1answer
83 views

Finite projective dimension may lead to projectiveness!

Assume a ring $R$ is injective as an $R$-module. If the projective dimension of an $R$-module $P$ is finite could one conclude that $P$ is a projective $R$-module? Probably one should start with ...
1
vote
1answer
55 views

Injective modules in a short exact sequence

Let $0→A→B→C→0$ be an exact sequence in the category of $R$ modules, where $R$ is commutative with $1$, and $B$ be injective. In a text book it is said that all three modules are injective, or the ...
0
votes
2answers
55 views

Injective resolution for an integral domain

How could one write an injective resolution for an arbitrary commutative integral domain $R$? Thanks in advance!
1
vote
1answer
65 views

if $R$ is a noetherian local ring, then every 2-generated ideal has finite projective dimension iff $R$ is a UFD

This question is about m zcn's comment on my question Projective dimension of all principal ideals is finite. Is R an integral domain?. It's a good point. so i ask it for use of everybody: if ...
2
votes
1answer
91 views

Global dimension of $\mathbb Q [x]$

I'm trying to show that the global dimension of $\mathbb Q [x]$ is $1$. I have shown that $D(\mathbb Q [x]) \leq 1$ as follows. One can reduce to the case of showing that ...
0
votes
0answers
88 views

Regular sequence and projective module

Let $R$ be commutative ring and $x,y$ an $R$-regular sequence. Then I know that $ R/(x)$ is not a projective $R$-module. My question: Is $R^{2}/(x,y)R$ a projective $R$-module?
3
votes
0answers
61 views

Projective modules over semilocal rings having constant rank are free

I'm starting to study algebraic K-theory by myself and I need a hint how to prove $R$ is a semilocal ring with maximal ideals $\mathfrak m_1,\ldots, \mathfrak m_n$, $P$ is a projective module and ...
0
votes
1answer
78 views

The ideal $I=(3,2+\sqrt {-5})$ is a projective module

Let $R=\mathbb Z[\sqrt{-5}]$ and $I=(3,2+\sqrt {-5})$ be the ideal generated by $3$ and $2+\sqrt{-5}$. I'm trying to prove that $I$ is a projective $R$-module. I'm using the lifting property ...
4
votes
0answers
81 views

Existence of finite projective resolution

The situation I'm considering is quite involved. All rings are noetherian commutative with $1$. All modules are finitely generated. First of all we fix a non reduced local ring $A$ where all zero ...
3
votes
2answers
35 views

Requirements on ring for injective-projectiveness

What requirements could be asked (minimal) of a ring R, so that any module M on R which is injective must also be projective? Is this possible?
9
votes
2answers
358 views

Projective module over a PID is free? [duplicate]

A common result is that finitely generated modules over a PID $R$ are projective iff they are free. Is the same true that an arbitrary projective module over a PID is free? I can't find this fact ...
1
vote
1answer
63 views

Annihilator of a projective module is always a projective idempotent ideal?

I found as a remark in a book that if $M$ is a projective module over a ring $R$, $\mathfrak a$ is the ideal generated by all $f(m)$ where $f\in \text{Hom}(M,R)$ and $m\in M$, and $\text{Ann}(M)$ is ...
3
votes
1answer
73 views

Question on Nakayama?

In reading a certain proof on the stacks project "http://stacks.math.columbia.edu/tag/00NV", I can't see how Nakayama's lemma is used to make the following conclusion: "Assume M is finitely presented ...
3
votes
1answer
49 views

Why is $M=\{(x_i)\in R^n: \sum r_ix_i=0\}$ a projective module?

Suppose $R$ is a commutative unital ring with generators $r_1,\dots, r_n$. How can we see that the submodule $$ M=\{(x_1,\dots,x_n)\in R^n:\sum_{i=1}^n r_ix_i=0\} $$ is a projective submodule? I ...
3
votes
1answer
117 views

Torsion-free and projective modules over a Dedekind domain

Suppose that $A$ is a Dedekind domain (and integral domain). I am trying to prove that if $M$ is a torsion-free $A$-module, then it is projective and vice versa. Suppose that $M$ is projective. Then ...
2
votes
1answer
95 views

Prove that if $P$ and $Q$ are projective and finitely generated $R$-modules then $\operatorname{Hom}_{R}(P,Q)$ is projective and finitely generated.

Suppose $R$ is a commutative ring and $P$ and $Q$ are projective and finitly generated $R$-modules. Prove that $\operatorname{Hom}_{R}(P,Q)$ is projective and finitely generated. suppose I proved ...
3
votes
1answer
118 views

Module has finitely generated projective resolution

Let $M$ be a finitely generated module (over a local noetherian ring $(R,\mathfrak m))$ such that the projective dimension of $M$ is finite $(pd\ M=n<\infty)$. We know that i) There is a free ...
5
votes
1answer
172 views

Modules which are isomorphic to their tensor product.

Suppose that we have a commutative ring $R$. I am interested in finding the (finitely generated and projective, if you want) $R$-modules $M,$ such that $M\cong M\otimes_R M$ as $R$-modules. I know ...
8
votes
2answers
172 views

How does this step in the proof of the structure theorem for f.g. modules over a Dedekind domain work?

I am trying to show that every finitely generated projective module $P$ over a Dedekind domain $D$ is a direct sum of (fractional) ideals. May's notes on Dedekind domains claim the result can be ...
0
votes
2answers
104 views

Projectivity of $\mathbb Q$ over $\mathbb Q\otimes_{\mathbb Z}\mathbb Q$

Consider $\mathbb Q\otimes \mathbb Q$, where $\mathbb Q$ is considered as $\mathbb Z$-algebra and consider $\mathbb Q$ as a right $\mathbb Q\otimes\mathbb Q$ module. Then is it true that $\mathbb Q$ ...
2
votes
3answers
238 views

Showing an ideal is a projective module via a split exact sequence

Let $R=\mathbb{Z}[\sqrt{-6}]$ and $I=(2,\sqrt{-6})$ the ideal generated by $2$ and $\sqrt{-6}$. I want to show that $I$ is a projective $R$-module by producing a short exact sequence that splits, ...
3
votes
1answer
139 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 ...
4
votes
0answers
75 views

Separability of finitely generated projectives over commutative ring

A class $\mathcal{C}$ of $R$-modules is called -separative if $A \oplus A \simeq A \oplus B \simeq B \oplus B$ implies $A \simeq B$ for each $A,B \in \mathcal{C}$ -cancelative if $A \oplus C \simeq ...
8
votes
1answer
147 views

Projective Modules over the Ring of Trigonometric Functions

Let $ R = \mathbb{R}[ \cos x, \sin x] $ and consider the ideal $ \langle 1 - \cos x, \sin x\rangle $. Is this ideal a projective module over $R$ ?
2
votes
1answer
60 views

$R$-linear maps of projective fractional ideals

Let $R$ be an integral domain with field of fractions $K$, and let $I$ be a fractional ideal. If $I$ is projective then every $R$-linear maps $I\to R$ is multiplication by an element of $K$. The ...
1
vote
1answer
193 views

Projective module over a ring

If $R$ is domain, as a projective module always exist over R. But how to produce such a module over $R$.
2
votes
0answers
332 views

Is the module quotient of projective modules projective?

Let $R$ be a commutative ring, let $M$, $N$ and $P$ be $R$-modules, and let $N' \subseteq N$ and $P' \subseteq P$ be submodules. Let $\mu:M\times N \to P$ be a surjective bilinear map. Define the ...
3
votes
2answers
305 views

Are projective modules over an artinian ring free?

Quoting a comment to this question: By a theorem of Serre, if $R$ is a commutative artinian ring, every projective module [over $R$] is free. (The theorem states that for any commutative ...
4
votes
1answer
154 views

Is the intersection of two f.g. projective submodules f.g.?

Let $R$ be a commutative unital ring and $M$ a finitely generated projective $R$-module. My question is: if $N_1$ and $N_2$ are f.g. projective submodules of $M$, is $N_1 \cap N_2$ f.g.? Is it ...
10
votes
4answers
251 views

Show $\mathbb{Q}[x,y]/\langle x,y \rangle$ is Not Projective as a $\mathbb{Q}[x,y]$-Module.

Disclaimer: Though I have been re-reading my notes, and have scanned the relevant texts, my commutative algebra is quite rusty, so I may be overlooking something basic. I want to show $\mathbb{Q} ...
7
votes
2answers
162 views

Relation between projective modules over $R$ and $R[T]$

Let $R$ be a commutative ring and $R[U]$ the polynomial ring in one variable. What is the relation between projective modules over $R$ and projective modules over $R[U]$? Is every projective module ...
5
votes
2answers
1k views

Finitely generated projective module

Would anyone can help me how to show that a finitely generated projective module over a local ring and PID are free? What I know about a finitely generated projective module $M$ over a PID $R$ ...
4
votes
1answer
335 views

Why is this ideal projective but not free?

Let $R=\mathbb{Z}[\sqrt{-5}]$ and $I=(2,1+\sqrt{-5})$. How can I prove that $I$ is projective but not free?
4
votes
1answer
194 views

Projective modules over a semi-local ring

I need a little bit of help, I found that theorem, but the book doesn't prove it and gives a reference to another book that I don't have; does anyone have an idea? Let $R$ be a semi-local ring, ...