Tagged Questions

2
votes
0answers
6 views

Finitely generated flat modules that are not projective

Over left noetherian rings and over semiperfect rings, every finitely generated flat module is projective. What are some examples of finitely generated flat modules that are not projective? Compare ...
2
votes
0answers
15 views

Partial cycles in projective resolutions of square-free algebra

Short version: Over a square-free algebra must every projective resolution of a simple module eventually terminate or contain a shift of itself as a direct summand? I suspect not, but have not ...
7
votes
1answer
44 views

Specific projective dimension of a module over bound quiver

Suppose $K$ is an algebraically closed field, and $A$ is the algebra presented by the quiver $$\require{AMScd} \begin{CD} 1 @>>> 2\\ @V{}VV @V{}VV \\ 3 @>>> 4 @>>> 5 ...
4
votes
2answers
93 views

Injective and Projective module

Ok, this problem is driving me nuts. At first, I thought I did it. But when reading another textbook (having a similar proposition, they (the problem in my textbook, and the proposition in the other ...
0
votes
1answer
27 views

A module which is not singular

Suppose that M is a projective R-module and that it's simple, isomorphic to $R/I$ where $I$ is a maximal left ideal of $R$ such that $I$ is not a direct summand of $R$. How to get to a contradiction?
2
votes
2answers
32 views

Exact sequences and finitely generated projective modules

Let $R$ be a ring and $A,B,C$ be $R$-modules in an exact sequence $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0.$$ Submodules or quotients of finitely generated projective modules are not ...
2
votes
1answer
61 views

Resolutions of bimodules as $R^e$-modules.

Let $k$ be a commutative ring, let $R$ be a $k$-algebra, a $R$-Bimodule $M$ over $R$ is a $k$-module with two actions of $R$ on $M$, on the left and on the right, the classical example of this being ...
3
votes
1answer
78 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 ...
2
votes
0answers
64 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 ...
1
vote
1answer
32 views

Cancellation law of morphisms?

The title is probably misleading, but I wasn't quite sure how to boil my question down to one line. My problem is this: Assume I have two projective $R$-modules, $P$ and $Q$, and epimorphisms ...
2
votes
1answer
61 views

The indecomposable projective $\mathbb{F}_pG$-module with $UJ/J\cong \mathbb{F}_p$

Let: $G$ be a finite group; $p$ be prime; $J$ be the Jacobson radical of $\mathbb{F}_pG$. A paper I'm trying to read mentions the following object: The indecomposable projective ...
2
votes
1answer
129 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 $R$.
1
vote
0answers
138 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 ...
0
votes
0answers
51 views

The projective dimension of a module over a hypersurface

Let $A$ be a commutative ring and $M$ be an $A$-module. Assume that there is an element $x\in Ann(M)$ such that $x$ is non-zero divisor of $A$. Then $M$ is naturally $A/(x)$-module. I have seen the ...
2
votes
0answers
60 views

Projective dimension of simple module

Let $R$ be a commutative ring and $M$ a simple $R$-module. Then $\mathfrak{m}=Ann(M)$ is a maximal ideal of $R$. Then it is known that $$ \mathrm{pdim}_{R}(M)=\mathrm{pdim}_{R_{\mathfrak{m}}}(M), $$ ...
3
votes
2answers
152 views

About Rim's theorem for projective modules over group rings

Let $G$ be a finite group. A theorem of Rim (Proposition 4.9 here) states that a $\mathbb{Z}G$-module $M$ is projective if and only if $M$ is $\mathbb{Z}P$-projective for all Sylow subgroups $P$ of ...
8
votes
4answers
180 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} ...
5
votes
2answers
150 views

How can I find an element $x\not\in\mathfrak mM_{\mathfrak m}$ for every maximal ideal $\mathfrak m$

Let $R$ be a commutative ring with finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_n$. Let $M$ be a finitely generated projective module such that $M_{\mathfrak m_i}$ has the same ...
1
vote
1answer
149 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?
2
votes
1answer
117 views

Is the kernel of a linear form $f : A^n \to A$ a projective module?

Let $R$ be a ring (not necessarily commutative, but if you have interesting things to say in the commutative case, please say them!) and $M$, $M''$ two finitely generated projective ...
1
vote
1answer
90 views

Projective module (The lifting property)

I'm looking for an example for a left R-module that doesn't have the lifting property, From theorems I read, $Z/2Z$ as a $Z$ module should be an example since it' not a direct sum (there isn't a ...
10
votes
1answer
226 views

Lemma on infinitely generated projective modules

Is it true that every finitely generated submodule of a non-finitely generated projective over a (not necessarily commutative!) ring is contained in a proper summand? (Ideally there's a standard ...
5
votes
1answer
202 views

Find a f.g. $F[x]$-module which is not projective

I was working through a problem that was showing that for a field $F$, $F[x]$-modules correspond to pairs $(V,T)$ of vector spaces over $F$ and linear transformations on that vector space. The last ...
3
votes
3answers
271 views

A module is projective iff it has a projective basis

I have a question related to this: Projective modules I'm trying to understand the "philosophy" of the statement, because it seems too similar to the statement "a module is free iff every element can ...
2
votes
1answer
264 views

Projective modules

I have to prove that if $P$ is a $R$-module , $P$ is projective $\Leftrightarrow$ there is a family $\{x_i\}$ in $P$ and morphisms $f_i\colon P\to R$ such that for all $x\in P$ $$ x= \sum_{i\in I} ...