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

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1answer
23 views

Non-finitely generated, non-divisible, non-projective, flat module, over a polynomial ring

(1) Let $R=k[x_1,\ldots,x_n]$. I wish to find an example of a non-finitely generated, non-divisible, non-projective, flat $R$-module. Notice that $k(x_1,\ldots,x_n)$ is NOT an example of what I am ...
3
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1answer
44 views

Non-finitely generated, non-projective flat module, over a polynomial ring

Let $R=k[x_1,\ldots,x_n]$. According to the first answer, every finitely generated flat module over an integral domain is necessarily projective. Therefore, the only hope to find a flat ...
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0answers
27 views

Free and projective modules

I have two simple questions. Let $R$ be a ring. Is every free $R$-Mod of the form $R^{(S)}$ for some set $S$? Can $R^S$ be projective for an infinite set $S$, for suitable, non-trivial $R$? Note that ...
5
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1answer
45 views

Three different notions for “bigness” of a module (length, rank and “mass”).

Let $M$ be an $R-$module where $R$ is an integral domain. I'm trying to understand the relations between these three notions of size: Let $S \subset M$ be a generating set of minimal cardinality. ...
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0answers
34 views

Projective and injective resolutions of cyclic modules

Let $R$ be a ring and $M$ a cyclic $R$-module. It is well-known that always exist projective and injective resolutions of $M$. Is it any method to construct explicitly these resolutions which have the ...
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0answers
22 views

Two short exact sequnce are isomorphic extension but not split [duplicate]

Is there some example of two short exact sequences which are isomorphic but not equivalent? Specially I am looking for a short exact sequence having extension unique up to isomorphism but not split. ...
0
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1answer
21 views

Exact sequences of projective modules

Let $0{\rightarrow} K \stackrel{g}\rightarrow P\stackrel{f}\rightarrow Q \rightarrow 0$ and $0\rightarrow K' \stackrel{g'}\rightarrow P'\stackrel{f'}\rightarrow Q \rightarrow 0$ be exact sequences ...
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1answer
30 views

Ideal as projective module

Let $R$ be a commutative ring without zero divisors. Assume that ideal $a\subset R$ is a projective $R$-module. How to prove that $a$ is finitely generated ? I need only hints.
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1answer
49 views

How many projectives and injectives exist in a path algebra?

I do not know an efficient way to determine whether a quiver representation is projective or injective. The definitions and properties such as "Projectives are summands of free modules", etc do not ...
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0answers
55 views

When a two-generated ideal of a noetherian integral domain have a finite projective resolution?

Let $R$ be a noetherian integral domain, and $I$ a non-zero ideal of $R$ which can be generated by two elements. (We do not know if $I$, considered as an $R$-module, is $R$-projective; maybe yes maybe ...
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0answers
55 views

Projectivity of a (prime) ideal in a noetherian integral domain

Assume $R$ is a noetherian integral domain (and assume $R \neq k[x_1,\ldots,x_n]$), $I$ is a non-zero ideal of $R$ ($I$ is finitely generated, since $R$ is noetherian), and $I$ is not necessarily ...
5
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1answer
145 views

What is $\operatorname{Hom}_R(P,R)$ isomorphic to when $P$ is projective?

Let $R$ be a (possibly noncommutative) ring with $1$. Now, quite clearly we have $$\operatorname{Hom}_R(R^n,R)\cong R^n.$$ I am wondering if there is any similar result for ...
4
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1answer
61 views

Two modules are isomorphic in the stable module category iff they are projectively equivalent

Let $R$ be a (not necessarily commutative) ring. Let ${\text{mod-}R}$ be the category of finitely generated right $R$-modules. Let $\underline{\text{mod-}R}$ be the stable module category, with the ...
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1answer
35 views

Every projective $R$-module $P$ is free

I have come across a theorem which states that if the underlying ring $R$ is a principal ideal domain then every $R$-module $P$ which is projective is free also. But the problem is I have encountered ...
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3answers
36 views

Is there an easy example of a module which is not projective

Is there an easy example of a module which is not projective? I found that direct product of $\mathbb Z$ is not projective but its proof is complicated. Are there any easy examples for this?
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2answers
66 views

Characterization of projective modules?

I'm having a hard time with a characterization of projective modules: A $R$-module $P$ is projective if and only if for every epimorphism $f:I\longrightarrow I^{\prime\prime}$ with $I$ injective and ...
2
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0answers
21 views

How to show that direct product of $\mathbb Z\times\mathbb Z\times\mathbb Z\times…$ is not projective as a $\mathbb Z$ module? [duplicate]

How to show that direct product of $\mathbb Z\times\mathbb Z\times\mathbb Z\times...$ is not projective as a $\mathbb Z$ module? I know that $\mathbb Z$ is a free $\mathbb Z$ module since it has ...
1
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1answer
31 views

Finitely generated projective modules over a simple algebraic ring extension of a polynomial ring

The well-known theorem of Quillen-Suslin says that a finitely generated projective module over $k[x_1,\ldots,x_n]$ is free, See ...
3
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1answer
134 views

Projectivity of $B$ over $C$, given $A \subset C \subset B$

I have found a result concerning projectivity of a certain ring extension: Lemma 2.64. This says the following: Let $A$ be an integral domain or a noetherian ring, $B$ an $A$-algebra, $C$ an ...
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0answers
22 views

Dual of Osofsky Theorem

A theorem of Osofsky reads: "A ring $R$ is semisimple iff the intersection of two injective submodules of any right $R$-module is injective" (Exercises in Modules and Rings, T.Y.Lam, Ex.3.11). Does a ...
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0answers
22 views

Right projective but not bimodule projective

Let $R$ be a ring with $1$. What is an example of a right $R$-module $M$ (and a ring $R$) such that $M$ is projective as a right $R$-module but not projective as an $R$-bimodule (assuming $M$ has ...
1
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1answer
24 views

There exists $h:P\longrightarrow M$ such that $fh=g\Leftrightarrow \textrm{Im}(g)\subseteq \textrm{Im}(f)$?

Let $R$ be a ring with identity and $P$, $M$ and $N$ three left modules over $R$. Futhermore, suppose $P$ is projetive. Let $f\in \textrm{Hom}_R(M, N)$ and $g\in\textrm{Hom}_R(P, N)$. How can I show ...
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2answers
122 views

Is a specific ring extension $B$ of $K[x,y]$ integrally closed? separable?

Let $A=K[x,y] \subset K[x,y][w]=B$, $K$ is a field of characteristic zero, $w$ is integral over $A$ (so $B$ is a f.g. $A$-module), but $w$ is not in the field of fractions of $A$, and $B$ is an ...
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1answer
47 views

Need for projective modules

I wanted to ask why we require projective modules. After studying all the essential ingredients my guess is - Firstly, we worked with vector spaces (say modules over field $F$) (which are free ...
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0answers
9 views

simple question about dimension of moduli space

If $M_g$ is the moduli space of a curve $C$ with positive genus $g$, the dimension of $M_g$ is $3g-3$ with $g >3$. Where can i find this proof?
2
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0answers
34 views

Number of parameter of a quadric

Suppose for example that $S$ is an algebraic complex surface contained in $\mathbb{P}^6$. $S$ is the complete intersection of four quadrics in the six dimensional projective space. If i take a quadric ...
5
votes
1answer
44 views

Is $K[x_1,\ldots,x_{n+1}]$ separable over $K[x_1,\ldots,x_n]$?

Let $R \subseteq S$ be commutative rings. $S$ is separable over $R$ if $S$ is a projective $S \otimes_R S$-module (under $\mu: S \otimes_R S \to S$ defined by $\mu(s_1 \otimes s_2)=s_1s_2$). Let ...
3
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0answers
115 views

Moduli space of algebraic surfaces Vs moduli space of curves

Define the surface $S$ as the complete intersection of four quadrics $Q_i$ with $i=1,2,3,4$ in $\mathbb{P}^6$ (complex six dimensional projective space) i.e. $$S=Q_1 \cap Q_2 \cap Q_3 \cap Q_4$$. Put ...
5
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1answer
66 views

Separability of $A \subseteq C$ implies separability of $B \subseteq C$, where $A \subseteq B \subseteq C$

For commutative rings $R \subseteq S$, recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module. (via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$). My ...
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1answer
24 views

Prove that if $A$ is $R$-projective and $C$ is $S$-injective then $\operatorname{Hom}_R(A,C)$ is $S$-injective

In the situation $(_RA,_RC_S)$, prove that if $A$ is $R$-projective and $C$ is $S$-injective then $\operatorname{Hom}_R(A,C)$ is $S$-injective. I appreciate your help.
2
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1answer
27 views

If $A \times B$ is the direct product of two rings, must $A$ necessarily be a projective $A \times B$-module?

If $A \times B$ is the direct product of two rings, must $A$ necessarily be a projective $A \times B$-module? I understand I am supposed to think of $A$ as an $A \times B$-module by identifying ...
1
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1answer
41 views

If M and N are nonzero, finitely generated R-modules with M projective, then $M\otimes N$ is nonzero

I am trying to work through the following problem: If M and N are nonzero, finitely generated R-modules with M projective, then $M\otimes N$ is nonzero. My thought on how to approach this problem is ...
1
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1answer
31 views

Does the nilpotent extension of a $1$-dimensional algebra always give a projective module?

Let $A$ be a $1$-dimensional reduced Noetherian algebra over an algebraic closed field $k$ with characteristic zero. Let $(B,N)$ be a nilpotent extension of $A$, i.e. $B$ is a Noetherian $k$-algebra, ...
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1answer
89 views

example of inverse limit and direct limit

Does a direct limit of projective need to be projective? And is the inverse limit of injectives injective? I guess they need not, but I can't find an example. Can you help please?
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1answer
26 views

generalized inverse in the theory of projective module

A module P over a ring R is projective which is an important topic in the theory of commutative ring due to its structural property of being a direct summand of free module. But my question is why ...
4
votes
2answers
47 views

Is $m$ a projective $A$-module?

$A$ is a Noetherian local ring and $m$ be its maximal ideal. Then is $m$ a projective $A$-module? I got this problem while solving another problem. Can anyone please help me to figure it out?
1
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1answer
108 views

Short exact sequence of modules over a Noetherian local ring of depth $1$.

I am reading an article in algebraic geometry and am having trouble understanding a particular point that reduces to a problem in commutative algebra. I'm not familiar with the concepts involved so am ...
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1answer
45 views

Why is the $\mathbb Z$-module $\mathbb Q$ projective in category $\sigma[\mathbb Q]$?

We know that $\mathbb Q$ is not a projective $\mathbb Z$-module, but, I've read this paper and it says in Example 2.11 that it's easy to check that "$\mathbb Q$ as $\mathbb Z$-module is projective ...
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1answer
23 views

$P$ is projective . Show that $P$ is a direct summand of a free $R-$ module

$P$ is projective . Show that $P$ is a direct summand of a free $R-$ module I am using the definition of a projective module as $P$ is projective if every exact sequence $M\rightarrow P\rightarrow ...
2
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1answer
44 views

“Direct sums of injective modules over Noetherian ring is injective” and its analogue

I have a commutative algebra class and I heard the theorem from the professor: Let $R$ be a Noetherian ring and $\{E_i : i\in I\}$ be a collection of injective $R$-modules then $\bigoplus_{i\in I} ...
3
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1answer
33 views

Two short exact sequences with projective objects in the middle

Problem: Prove that for two short exact sequences $$ 0\rightarrow A \xrightarrow{f} B \xrightarrow{g} C \rightarrow 0 $$ $$ 0\rightarrow A' \xrightarrow{f'} B' \xrightarrow{g'} C \rightarrow 0, $$ ...
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1answer
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A question about the quotient of two chain complexes of $R$-modules

Let $P$ be an acyclic object of $Ch_R$, let $P^{(k)}$ be the chain subcomplex of $P$ which agrees with $P$ above the degree $k-1$, contains $Bd_{k-1}P$ in degree $k-1$, and vanishes below degree ...
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vote
1answer
25 views

$\text{Coker}(f),\text{Coker}(g)$ projective modules $\Rightarrow$ $\text{Coker}(gf)$ projective

Let $\textbf{Ch}_R$ the category of chain complexes of $R-$modules ($R$ is an associative ring with unit). I want to prove that this cat. satisfies the model category axioms. In particular we want to ...
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0answers
26 views

Connection between Chladni Plates and Projective Geometry?

Does anybody know of a connection between Chladni Plates and Projective Geometry? Any published research on the subject? I am interested to know if Chladni Plates (or surfaces) can be ...
3
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1answer
34 views

Equivalent properties of projective modules

For a left $R$ module $P$, the following are equivalent: Given $M\xrightarrow{\psi} N \to 0$ exact and $\omega: P \to N$, there exists $\tilde{\omega}: P \to M$ such that $\psi \circ ...
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0answers
16 views

can we decompose every module as a direct sum of projective submodule P and a submodule C such that C has no nonzero projective submodule

In the noetherian rings, we can write every module as a direct sum of injective submodule E and a submodule D where D has no injective submodule.
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1answer
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Projective dimension of module over local ring

This question arose reading the well known article by Buchsbaum Lectures on regular local rings. He states without proof that, given $(R,m)$ a local ring and an $R$-module $M$ over $R$, we have the ...
4
votes
1answer
65 views

Semisimplicity is equivalent to each simple left module is projective?

As it is well-known, a ring with unity $R$ is semisimple if and only if each left $R$-module is projective. My question: Is semisimplicity of $R$ equivalent to each simple left $R$-module being ...
2
votes
1answer
68 views

Characterization of projective and injective modules

In Theorems 4.7 and 8.4 Hilton & Stammbach give two lists of 5 different characterizations of projective and injective modules, respectively. Even though I can follow the proofs they give, I'd ...
2
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1answer
83 views

Hartshorne II Ex 5.9(a) or R. Vakil Ex 15.4.D(b): Saturated modules

The question is basically like this: Prove that if $S_{\cdot}$ is a finitely generated (in degree 1) graded ring over a field $k$ and $M_{\cdot}$ is finitely generated, then the saturation ...