Questions tagged [projective-module]
For questions related to projective modules, their structures, and properties.
844 questions
1
vote
2
answers
84
views
Complement direct summand to a non-principal ideal in a number ring
Let $K$ be a number field, that is, a finite extension of ${\mathbb Q}$. Let $R={\mathcal O}_K$ be the Dedekind domain of algebraic integers in $K$.
It is well-known that an ideal $I\subset R$ is a ...
- 5,566
1
vote
1
answer
74
views
How can we show that $- \otimes M$ preserves all limits if and only if $M$ is finitely generated projective?
Let $R$ be a commutative ring with unity and let $R \operatorname{-Mod}$ be the category of $R$-modules.
Question: How can we show that for an $R$-module $M$, the following are equivalent?
The ...
- 4,970
2
votes
1
answer
52
views
Finitistic projective dimension of commutative rings of finite weak global dimension
For a commutative (not necessarily Noetherian) ring $R$, define the following two quantities
$$fpd(R):=\sup\{\operatorname{pd}_R(M) | M \text{ is a finitely presented }R\text{-module of finite ...
- 1,901
1
vote
0
answers
117
views
Is trace ideal of $P$ equal to $P \otimes_E P^*$?
Let $R$ be an associative ring, $P$ is a finitely generated projective right $R$-module. Denote $E=\operatorname{End}_R(P,P)^{\operatorname{op}}$ the opposite of the ring of endomorhisms, and $P^*=\...
- 6,826
1
vote
1
answer
69
views
Variation on finitely-generated projective module over local Noetherian ring being free
I'm familiar with the proof that finitely-generated projective modules over a local Noetherian ring are free, but I'm a little stuck on directly showing a small variation of that result:
Let $(A, \...
- 333
-2
votes
1
answer
78
views
Projective Module Example (I need help) [closed]
The question is:
Show that if we have a diagram
in which the row is exact, $P$ is projective, and $h\circ f = 0$, then there exists a $k:P → M$ such that $f = g\circ k$.
I know that $\text{Im}g=\ker ...
- 23
2
votes
1
answer
77
views
Motivations behind the terminology "factor through projectives"
in Exercise 8.10 from Module Theory: An Approach to Linear Algebra by T. S. Blyth, the author defined a terminology as follows:
If $M$ and $N$ are $R$-modules, an $R$-linear map $f:M\to N$ factor ...
- 3,175
1
vote
1
answer
84
views
When is a projective module is projective bimodule over self-injective algebra?
We know the following fact:
Let $A$ and $B$ are finitely self-injective $k$-algebras whose semi-simple quotients are separable, $Z$ is a finitely generated $A$-$B$-bimodule, where $k$ is a field. ...
- 169
0
votes
0
answers
40
views
Lifting Morphisms Through Resolutions and Tensor Products
We encountered the following problem related to tensoring and lifting morphisms:
Let $\cdots \to A_n \to \cdots \to A_0 \to A \to 0$ be a projective resolution of $A$. Tensoring this complex with a ...
- 21
0
votes
0
answers
86
views
When is every two-sided maximal ideal idempotent?
Let $R$ be a (possibly noncommutative) ring.
Recall that an ideal $I \subseteq R$ is called idempotent if $I^2=I$, meaning that every element of $I$ can be written as a finite sum of products of ...
- 451
3
votes
1
answer
69
views
Is there a left and right SF-ring that is not von Neumann regular?
A ring $R$ is called a left SF-ring (resp. right SF-ring) if every simple left (resp. right) $R$-module is flat. It is well-known that if $R$ is a von Neumann regular ring, then $R$ is both a left and ...
- 451
1
vote
1
answer
166
views
A strange flat but not projective module
This question is from the 14th Yau-Contest.
Let $R$ be a commutative ring with identity. Let $S=\prod_{\mathbb{N}} R$ the product of countably many copies of $R$, and $I=\oplus_{\mathbb{N}} R \subset ...
- 57
0
votes
1
answer
108
views
Product of modules is projective over $\mathbb{Z}G$
Let $G$ be any fixed group (possibly infinite). Suppose there is a projective $\mathbb{Z}G$-module $P$ and a $\mathbb{Z}G$-module $F$ which is free over $\mathbb{Z}$. Then I want to show that the ...
- 73
3
votes
2
answers
150
views
$\mathbb{Q}G$ is a projective module over $\mathbb{Z}G$ for $G=\mathbb{Z}/n$
I'm new to homological algebra. I was studying projective modules from Brown's book and was trying to work out some examples. I read here that $\mathbb{Q}$ is not a projective module over $\mathbb{Z}$,...
- 31
0
votes
1
answer
55
views
What is $R_{k}(C_{2})$?
For $C_{2}$ (cyclic group of order $2$) there are two irreducible representations. Let $E_{0}$ be the trivial one and $E$ the sign one. They form the basis of $R_{\mathbb{C}}(C_{2})$. Do I understand ...
- 994