# Questions tagged [projective-module]

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

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### Finitness conditions under change of rings?

Let $R \to S$ be a homomorphism of commutative rings, such that $S$ is finitely generated and projective $R$-module. Proof that if $A$ is a finitely presented $S$-module then $A$ is a finitely ...
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### If every homomorphic image of an injective module is also injective, then every submodule of a projective module is projective

Let $R$ be a ring with $1$. All modules considered in this problem are unitary right $R$-module. Assume that every homomorphic image of an injective module is also injective. I need to prove that ...
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### If $R$ is a right hereditary ring, then any submodule of a right projective $R$-module is again projective.

Recall that a ring $R$ is called right hereditary if every right right ideal $I\subset R$ is projective as a right $R$-module. I need to prove that if $M_R$ is projective module and $N\leq M$ is any ...
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### Indecomposable modules with simple socle

Proposition: Suppose $I$ is an indecomposable $R$-module. Then every $R$-submodule $S \subseteq I$ is also indecomposable if and only if $I$ has a simple socle (that is, $I$ only has one simple ...
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### Example of invertible modules?

I am trying to find an example of a non-trivial invertible module (let's say over $\mathbb Z$). This seems to be very simple, but after trying and searching around, I do not find any examples. (Many ...
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### Projective dimension of $K[x,y]$ over $K[xy]$

Let $K$ be a field and $K[x,y]$ the polynomial ring in two variables $x$ and $y$ over $K$. Let $R = K[xy]$ be the subring generated as a $K$-algebra by the monomial $xy$. My question is: What is the ...
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### Direct limit of n-presented modules?

A module $M$ is said to be $n$-presented If there exist an exact sequence $$F_{n}\to F_{n-1}\to \cdots \to F_{1}\to F_{0}\to M$$ with each $F_{i}$ is free finitely generated. For example $M$ is $0$-...
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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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### $0\to \text{Hom}(P_0,G)\to \text{Hom}(P_1,G)\to \text{Hom}(P_2,G)\to\dots$ 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 ...
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### Phantom morphisms and Tor functor?

$\newcommand{\Tor}{\operatorname{Tor}}$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 $g:F\to M$, $fg$ ...
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### Is a module a limit of n-presented modules?

Let n be an integer. A module M is said to be n-presented if there exist an exact sequence of the form $$F_{n}\to F_{n-1}\to ...\to F_{1}\to F_{0}\to M \to 0$$ with every $F_{i}$ is a finitely ...
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### On splitness of epimorphisms?

Let $\alpha: M \to N$ be an epimorphism of left R-modules. Let $Q$ be a left R-module of finite projective dimension d with a morphism $\beta : Q \to N$. We know that if d=0 i.e, $Q$ is projective ...
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### Bruns & Herzog 1.4.9: Rank of projective modules

I'm reading Cohen-Macaulay Rings by Bruns & Herzog. The lemma below which is used in the proof of the Proposition 1.4.9 and whose proof is left to the reader is difficult for me to prove: Lemma. ...
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### Why does the isomorphism $A^e\otimes_A M\simeq A\otimes_k M$ imply that $A^e\otimes M$ is $A$-projective?

Let $k$ be a field, $A$ an associative $k$-algebra, $M$ a left $A$-module and $A^e := A\otimes_k A^{op}$ the enveloping algebra of $A$. I have an $A^e$-projective resolution of $A$  \cdots \...
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### Exercise 2.2.2. from Weibel's Homological Algebra. If $A$ has enough projectives, then so does $\operatorname{Ch}(A)$.

$\newcommand{\A}{\mathcal{A}}\newcommand{\Ch}{\mathbf{Ch}}$The following is an exercise from Weibel's An Introduction to Homological Algebra. Show that if $\A$ has enough projectives, then so does ...
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### Prove a full subcategory of $\mathsf{K^-(P)}$ is equivalent to $\mathsf{A}$
This is exercise 6.2 in Paolo Aluffi's Algebra: Chapter 0 (the second print). Assuming that $\mathsf{A}$ has enough projectives, prove that the full subcategory $\mathsf{\hat{A}}$ of \$\mathsf{K^{-}(P)...