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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Unimodular element of a module

Let $A$ be a noetherian ring and $M$ an $A$-module. An element $z \in M$ is said to be unimodular if $Az$ is a direct summand of $M$ and $Ann(z) = \{r \in A | rz=0\} = 0$. The order ideal of $M$ is: $...
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A projective module over a finite direct product of fields

Is a projective module $P$ over a ring $R$ which is a finite direct product of fields free?
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Lifting Property of flips

I am trying to understand the concept of flips before learning the Bass Cancellation theorem. Let $R$ be a Noetherian ring and $P$ be a projective $R$ module. Let $p,q \in P$ , $\phi \in Hom(P,R)$ ...
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Question On Serre's Splitting Theorem

I am learning the splitting theorem from the book F. Ischebeck and Ravi Rao. The statement is as follows: Let $A$ be a commutative Noetherian ring of finite Krull dimension. Let $P$ be a finitely ...
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$\{P\ \text{is a prime of} \ R|\operatorname{rank} M_P=r\}$ is open in $\operatorname{Spec}R$

Let $R$ be a commutative Noetherian ring with identity. $M$ is a finitely generated projective $R$-module. Then for any $r$, the set $\{P\ \text{is a prime of} \ R|\operatorname{rank} M_P=r\}$ is an ...
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Injective projective modules over left artinian ring

When I read Rings and Categories of Modules written by Frank W.Anderson and Kent R.Fuller, I can't understand the proof of Theorem 31.3 (Page 338) Let $R$ be a left or right artinian ring with $J=J(R)...
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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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Computing derived tensor product of bimodules by resolving only one argument?

If $R$ is a ring, $M$ is a right $R$-module, and $N$ is a left $R$-module, the derived tensor product $M \otimes_R^{\mathbf{L}} N$ is computed by choosing projective resolutions $P_* \to M$ and $Q_* \...
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About exactness of $Hom$ functor?

We know that a module $P$ is projective if and only if the functor $Hom(P,-):R-Mod \to Ab$ is exact, i.e it preserves epimorphisms: If $\alpha: M \to N$ is an epimorphism of modules then $Hom(P,\alpha)...
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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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About split module epimorphisms

Let $\alpha:M\to N$ be an R-module epimorphism and $X$ be a class of R-modules. $\alpha$ is said to be a split epimorphism with respect to $X$ if the the induced morphism of abelian groups $Hom(F,M)\...
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When every finitely generated submodule of an R-module is cyclic?

We know that if R is a noetherian ring then every submodule of a finitely generated module is finitely generated. In case R is absolutely flat, we have every finitely generated ideal is principal, can ...
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Change of rings of scalars and projecivity?

Let $f:R\to S$ be a ring homomorphism and M be a left S-module. We can consider M as an R-module via $ r.m := f(r)m $. I know that if M is a flat S-module and S is flat as R-module then M is a flat R-...
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Are projections of non projective modules interesting?

While reading about semisimple modules, the property every quotient is a subobject hence every submodule is projective gave me an idea to introduce projective modules directly as projections of free ...
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If $A$ is representation finite, then $\{pd(M)|pd(M)<\infty, M\in mod-A\}=max\{pd(M)|M$ indecomposable $\}$?

Let $A$ be finite dimensional $\mathbb{C}$ algebra. Suppose $A$ is representation finite.(i.e. $A$'s has a finite list of isomorphism classes of indecomposables.) Set $mod-A$ to be category of ...
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Do images and kernels of invariant modules have the form $Mr$ and $\text{Ann}(r)$ respectively for some $r\in R$?

Let $M$ be an $R$-module and $N$ a submodule of $M$. $N$ is said to be a fully invariant submodule of $M$ if $f(N)\subseteq N$ for all $f\in \text{End}_R(M)$. We call an $R$-module $M$ invariant if ...
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Projective resolution of $\mathbb{R}$ as an abelian group

I had a sense that this should be easy to find but googling many different versions of the question I couldn't find anything. My question is how does some projective resolution of the additive group ...
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Non-finitely generated R-module

Currently studying for qualifying exams and came across the following problem: Give an explicit example of a ring $R$ (commutative with identity) and a surjection $\psi: M \rightarrow N$ of finitely ...
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Show that the homology groups of different projective resolutions of the same R-module are isomorphic to one another

I am taking a course on commutative algebra, and we just defined the Tor functor using projective resolutions of a module. The definition we have is: Let $R$ be a commutative ring with unity. Let $M$ ...
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Being direct summand of free module implies having dual basis.

I need to provet equality of two definitions of projective module: being direct summand of free module (or equally: having embedding into free module) and having dual basis. Lets use Wikipedia ...
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Milnor Squares and Milnor Patching: Examples?

In Weibel's book on K-theory, he introduces Milnor squares and Milnor patching as follows: I was wondering if someone might be able and willing to help me a little by constructing a nice friendly ...
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Let $R$ be an integral domain and let $X$ be a torsion module over $R$. Then $Tor_n(X,Y)$ is a torsion module for every $n≥0$.

Let $R$ be an integral domain and let $X$ be a torsion module over $R$. Then $Tor_n(X,Y)$ is a torsion module for every $n≥0$. I attempted proving this fact using the definition of what it means to ...
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Three ways to to prove that projective modules are flat

I am trying to show that projective modules are flat using their defining property that $Hom(P,-)$ is an exact functor when $P$ is projective. The two ways I know of come down to the fact that ...
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Simple objects with isomorphic projective covers

Let $X$ and $Y$ be two simple objects of an abelian category. Assume that they have projective covers $P(X)$ and $P(Y)$. Question: If $P(X)$ and $P(Y)$ are isomorphic, is it true that $X$ and $Y$ are ...
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2 answers
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Simple, Cyclic and Projective Modules

I have been taking a course in Homological Algebra and revisiting the lecture notes for some reason. There was a non-answered question about showing two non zero modules A,B over a non trivial unital ...
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Powers of the Auslander-Reiten translations

Let $A$ be a finite dimensional algebra over a field $k$, and let $\tau$ be the Auslander-Reiten translation functor vanishing on projective $A$-modules. I'm studing the book "An introduction to ...
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Is a nonzero projective module over a noncommutative domain faithfully flat?

Let $R$ be a (noncommutative) unital ring which is a domain and let $\mathcal{N}$ be a non-zero projective (right) module. Projectivity of course implies that $\mathcal{N}$ is flat, but does the fact ...
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Find indecomposable projective/injective module over $k[x]/ \langle x^n \rangle$

Problem: find indecomposable injective modules and indecomposable projective modules over $k[x]/ \langle x^n \rangle. (n\geq 2)$. From this post, I learned that any indecomposable module over $k[x]/ \...
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Indecomposable objects in stable category of modules

Let $A$ be a finite-dimensional algebra over a field. Define the stable category of right $A$-modules $\underline{\text{mod}}-A$ as a category with objects the finitely generated modules over $A$ and $...
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$\mathbb{Q} $ is not a projective $\mathbb{Z} $ module [duplicate]

Prove that $\mathbb{Q} $ is not a projective $\mathbb{Z} $ module Let on the contrary it is projective. Then $P=\mathbb{Q}$. Then bottom row is given to be exact. Now h is given to be an $\mathbb{Z}$...
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Vector Space over a Division Ring [duplicate]

Prove that every vector space over a division ring D is both a projective and an injective D-module. This question is from an assignment in Module Theory. I am familiar with the definitions and are ...
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2 answers
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Is the internal sum of projective submodules projective.

Let $R$ be a unital ring, and $M$ a (right) $R$-module. Assume that $M$ admits a (possibly infinite) family of projective submodules $N_i$, for $i \in I$, such that $$ M = \sum_{i \in I} N_i. $$ Does ...
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Flat ring extensions and flat modules

Let $R$ be a not necessarily commutative unital ring, and $R \hookrightarrow S$ a unital ring extension such that $S$ is flat as a (right) module over $S$ (i.e. a flat extension). See here for a ...
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4 votes
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Exercise 2.4.3. (Dimension shifting) from Weibel's Homological Algebra.

$\newcommand{\F}[1]{\mathcal{#1}}\DeclareMathOperator{\im}{im}$The following is an exercise from the book mentioned in the title: If $0 \to M \to P \to A \to 0$ is exact with $P$ projective (or $\F{F}...
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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)...
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