# Questions tagged [projective-module]

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

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### Is $C$ projective as $C[x]$ module?

I'm asking myself if $C = C[X]/(x)$ is a projective/flat $C[X]$ module. I found on the internet that this isn't the case, but on the other hand we have $C[X] = C \oplus xC[X]$, so $C$ is a direct ...
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### Prove that $\mathbb{Z}/ 3\mathbb{Z}$ is a projective $\mathbb{Z} / 6\mathbb{Z}$ module which is not free.

I am a student of a masters course and this question was asked in my quiz of commutative algebra. Question: Prove that $\mathbb{Z}/ 3\mathbb{Z}$ is a projective $\mathbb{Z} / 6\mathbb{Z}$ module ...
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### Functors and projective covers

I'm looking to understand how a covariant functor that is an equivalences of categories preserves projective covers, and how a contravariant functor that is a dual equivalence of categories maps ...
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### The functor $\mathrm{Hom}(A,-)$ cannot commute with arbitrary direct sums for infinitely generated projective module $A$

It is easy to see the functor $\mathrm{Hom}(A,-)$ commutes with every arbitrary direct sum (i.e. $\mathrm{Hom} (A,\oplus_{i\in I} N_i)=\oplus_{i\in I}\mathrm{Hom}(A,N_i)$) for finitely generated ...
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### Examples of rings by their relation to projective coverings

A projective covering of an $R$-module $M$ is an epimorphism $\pi:P\rightarrow M$ s.t. $P$ is a projective $R$-module and $\textrm{Ker}(\pi)$ is co-essential in $P.$ The existence theorem for ...
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### $A/I$ is a projective $A$-module iff $I$ is a principal ideal generated by an idempotent element

The following question was asked in my assignment of modules and I could not solve this question despite thinking a lot. Question: Show that $A/I$ is a projective $A$-module if and only if $I$ is a ...
33 views

### Existence of non-trivial homomorphism from projective module to ring

Let $R$ be a commutative ring and let $P$ be a nonzero projective $R$-module. I want to show that there exists a non-trivial homomorphism from $P$ to $R$. I don't see how can I start. Can someone give ...
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### Description of projective $\mathrm{SL}_{2}(\mathbb{Z})$-modules

I am working my way through Ken Browns book on the cohomology of groups, and in particular chapters 8 and 9 on finiteness conditions, and Euler characteristics. Most of the concepts in chapter 9 (such ...
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1 vote
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### Prove that if $\mathcal{A}$ has enough projectives, then so does $Ch(\mathcal{A})$

This is exercise 2.2.2 in Weibel's AIHA. We already know that a chain complex $P_{\bullet}$ is projective in $Ch(\mathcal{A})$ iff it is a split exact complex of projectives. Here's my proof, but it ...
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### Why does tensor product satisfy pushout-product axiom?

Example 11.4 in this paper claims that the tensor product of chain complexes of bimodules (over not-necessarily-commutative rings) satisfies the pushout-product axiom (the first condition of a Quillen ...
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### To show 'dual' of a Projective is projective

I am reading Chapter IV, section 2 in Assem's book Elements of the Representation Theory of Associative Algebras and I am stuck at a claim: Let $A$ be a finite dimensional algebra over $\mathbb{C}$. ...
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1 vote
### $0\to \text{Hom}(P_0,G)\to \text{Hom}(P_1,G)\to \text{Hom}(P_2,G)\to\dots$ is an exact sequence. [duplicate]
I am reading ''Algebraic number theory'' by Cassels and Fröhlich and in the chapter IV.4 it says the following: If $\dots\to P_2\to P_1\to P_0\to\mathbb{Z}\to 0$ is a projective resolution and $G$ is ...
$\newcommand{\Tor}{\operatorname{Tor}}$In the category of R-modules, a morphism $f:M\to N$ is called a phantom morphism if for every finitely presented module $F$ and every morphism $g:F\to M$, $fg$ ...