# Tagged Questions

For questions related to projective modules, their structures, and properties.

36 views

### Question on projective modules, a course in homological algebra

Show that if $0\rightarrow N\xrightarrow{\tau} P\xrightarrow{\epsilon} A\rightarrow 0$ and $0\rightarrow M\xrightarrow{\tau'} Q\xrightarrow{\epsilon'} A\rightarrow 0$ are exact sequences with $P,Q$ ...
43 views

52 views

33 views

### Analogue of the trivial extension for higher Ext.

I've been doing some homological algebra and some work on showing some extensions are equivalent, and a thought just came to me, which is that I didn't know how to write down what the analogue of the ...
60 views

### How to prove the following sequence is exact?

Let R be a ring and $F',F,F'',G',G,G''$ left R-modules. Assume we are given R-module homomorphism $i:F'\to F,p:G'\to G,p':G\to G''$ and $a:F'\to G',b:F\to G,c:F''\to G''$ such that the following ...
31 views

### definitition of projective resolution of $R$-modules (with homology)

Let $R$ be a commutative ring with unit $1_R$, $M$ be a $R-$module. I have a small question about different definitions of projective resolutions of $M$ (and I'm confused with the degrees of the ...
23 views

### Are torsion-free modules over principal ideal domains/Dedekind domains projective

An exercise in "Commutative algebra with a view towards Algebraic geometry" by Eisenbud states that a torsion-free module over a Dedekind domain is a projective module (see page $484$, Exercise $19.6$)...
212 views

### Grouping Problem

Suppose there are 9 strangers. We will assign them into 3 groups and each group has exactly 3 people. For each grouping, the strangers who were assigned into the same group will get to know each other ...
21 views

### On the equivalence condition of p.p. rings

A ring is known to be left p.p. if every principal left ideal is projective. It is well known that this condition is equivalent to the fact that every annihilator of each element is generated by an ...
23 views

### If an R-module P is free and A and B are direct summands of P then A∩B is isomorphic to a direct summand of P? is it true? I could not prove it?

If an R-module, P, is free and A and B are direct summands of P, then $A\cap B$ is isomorphic to a direct summand of P. I couldn't prove this proposition on my own. Is it true?
36 views

### Left vs right projective resolutions and homology of monoids

Let me use the ad hoc notation $\mathbb Z^l$ and $\mathbb Z^r$ to distinguish between left and right modules. These are trivial modules. The homology of a (discrete) finite monoid $M$ with ...
I have met this as part of a problem in module theory which states Let $f : M \to U$ be a surjective module homomorphism over ring $R$ where $M$ is finitely generated and $U$ is free. How would I ...