Tagged Questions
4
votes
2answers
88 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
59 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
77 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
56 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
125 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
132 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
47 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
59 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
142 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
179 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 ...
