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

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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?
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1answer
28 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 ...
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2answers
36 views

Proving module homomorphism has right inverse [duplicate]

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 ...
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1answer
41 views

Prove that every $\mathbb{Z}/6\mathbb{Z}$-module is projective and injective. Find a $\mathbb{Z}/4\mathbb{Z}$-module that is neither.

I want to show that every $\mathbb{Z}/6\mathbb{Z}$-module is a direct sum of projective modules. I know that $\mathbb{Z}/6\mathbb{Z}$ is the direct sum of $\mathbb{Z}/2\mathbb{Z}$ and ...
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0answers
32 views

projective resolution of finitely generated modules

I am in the condition where I have a noetherian ring $R$ of finite global dimension. Consider the category of finitely generated (right) modules over $R$. Then I want to show that every module admits ...
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2answers
47 views

Prove that $\operatorname{Ext}^{d+1}(A, B)\cong \operatorname{Ext}^1(M_d,B)$

So, given a resolution, with $P_{i}$ projective modules: $$0\longrightarrow M_d\longrightarrow P_{d-1} \longrightarrow \cdots \longrightarrow P_0 \longrightarrow A\longrightarrow 0,$$ I'm trying to ...
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31 views

Projective objects of chain category

I am tempted to think that the projective objects in the chain category $\text{Ch}(\mathcal C)$ for $\mathcal C$ abelian are exactly the complexes $P_\bullet$ for which each $P_i$ is projective. Is ...
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2answers
62 views

Can you prove that over a division ring , every module is both projective and injective?

I have problem with this : over a division ring or a ring with identity every module is both projective and injective . Can you help me? thank you .
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11 views

$\text{Hom}$ to a projective $D[G]$-module for a complete DVR $D$

Suppose you have a complete DVR $D$ and a finite group $G$ with $D[G]$-modules $A$ and $B$. Does $B$ being projective imply that $\text{Hom}_{D[G]}(A,B)$ is $D$-free? Or should it be $A$ that's ...
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1answer
76 views

$k[x]\otimes k[x]$ as a right $k[x]$-module

Let $k$ be a commutative ring. Consider the ring map $\varphi:k[x]\to k[x]\otimes_k k[x]$ given by $\varphi(x)=x\otimes 1-1\otimes x$. Now consider $k[x]\otimes_k k[x]$ as a right module over itself. ...
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Projective covers for simple Lie algebras in characteristic zero

I want to use the projective cover for the trivial module over a finite-dimensional simple Lie algebra in characteristic zero. Is there a reference that I can quote to assert its existence?
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1answer
57 views

When is the localization of a commutative ring a finitely generated projective module?

Let $R$ be a commutative ring and $M$ an $R$-module. The tensor product $(-)\otimes M$ has a left adjoint $(-)\otimes M^\ast$ for $M^\ast =\mathsf{hom}(M,R)$ iff $M$ is finitely generated projective. ...
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1answer
39 views

$P\cong P^\ast$ iff $P$ is a f.g projective module?

Is it true that for a noncommutative $R$, a module $P$ is f.g projective iff $\mathsf{hom}(P,R)=P^\ast \cong P$? Here's what I thought of as a proof: Since $(-)^\ast$ is additive, it preserves ...
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24 views

Verification of Ext groups and projective resolution for S3 over F3

So I've been looking at Ext groups of irreducible representations of $S_3$ over $\mathbb{F_3}$. Specifically, I'm doing a project where I'll be looking at extensions themselves, so am really only ...
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1answer
33 views

Are finitely generated projective models of an algebraic theory always finitely presented?

I know that for modules over rings, a finitely generated projective module is finitely presented. I was wondering whether this holds in full generality for algebraic theories, and if not, which parts ...
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1answer
46 views

Dual, Projective Modules, and Isomorphisms

Given a finitely generated right $R$-module $M$ with a generating set $\{m_i\}$, can one define a map $M$ to its dual left-module of right module functions by $m_i \mapsto m_i^*$ in analogy with the ...
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0answers
26 views

Weakest condition on module such that free prime localizations imply projective?

I know that for a finitely presented commutative module $M$, if all prime localizations $M_\mathfrak{p}$ are free, then $M$ is projective. Is there a weakening or a converse of this statement? What ...
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1answer
21 views

If $M$ and $N$ are simple $R$-modules, and there is nonzero map from projective cover $P_N\to M$, is $N\cong M$?

Suppose $M$ and $N$ are simple modules over a commutative ring $R$, and $P_N$ is the projective cover of $N$. If there is a nonzero morphism $P_N\to M$, does this imply $N\cong M$? I'm not too ...
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1answer
27 views

Different definition of $K_0(R)$. Prove equivalence

I need to prove that the following two definitions of the zero-th $K$-theory group of a ring $R$ (with unit) are equivalent. def 1 the abelian group generated by the isomorphism classes $[P]$ of ...
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25 views

Help/Verification of Ext group calculation

I have been looking at extensions of irreducible representation over fields of positive characteristic. Specifically at the moment, to get the hang of things, I'm looking at $S_3$ over $\mathbb F_3$, ...
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1answer
60 views

If $P$ is projective and $A,B$ are direct summands of $P$ then $A\cap B$ is a direct summand of $P$

I need to prove that if $P$ is a projective module over $\mathbb Z$ and $A,B$ are direct summands of $P$ then $A\cap B$ is a direct summand of $P$. This in turn would imply if $A$ and $B$ are ...
3
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1answer
38 views

Proving $\Bbb Z_p$ has no projective cover.

I need to prove that $\mathbb Z_p$ has no projective cover when seen as a module over $\mathbb Z$. I would appreciate your help. A projective cover of $M$ is a projective module $P$ along with a ...
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15 views

Two quotients of projective modules are equal, prove the crossed direct sums of the projective modules and kernels are isomorphic. (Schanuel's Lemma)

We have the following: $$\alpha_1:P_1\rightarrow M$$ $$\alpha_2:P_2\rightarrow M$$ with $\alpha_1,\alpha_2$ surjective, and $P_1,P_2$ projective. Prove $$P_1\oplus \ker(\alpha_2) \cong P_2\oplus ...
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1answer
35 views

Direct sums of modules versus products

In my module theory course, we recently proved the following theorems: For a fixed ring $R$ and index set $I$, let $P_i$ and $Q_i$ be $R$-modules for all $i\in I$. Then $$P=\bigoplus _{i\in ...
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1answer
52 views

Trouble calculating projective modules

I've been trying to calculate projective modules in an effort to eventually classify extensions of modules of $S_3$ over a field of three elements. So I know that projective modules can be found by ...
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0answers
29 views

A projective module which is not free [duplicate]

Why is $R=\mathbb{Z}/2\mathbb{Z}$ a projective module over $\mathbb{Z}/6\mathbb{Z}$, but not free? I'm not sure if the question makes sense, but I'm searching for a projective module which is not ...
2
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2answers
74 views

Projective but not free module over groupring

Let $G$ be a nontrivial finite group and consider the groupring $\Bbb QG$. My question is whether we can find a module over $\Bbb QG$ that is projective but not free?
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77 views

In which algebraic theories do 'free' and 'projective' coincide?

Free models of algebraic theories are always projective objects in the category of models, but the converse is not always true. For instance, some (actually, all) projective modules are direct ...
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1answer
48 views

Projective resolutions of modules in a short exact sequence (Dummit & Foote Proposition 17.1.7)

This proposition is as follows: Let $$ 0\rightarrow L\rightarrow M\rightarrow N\rightarrow 0 $$ be a short exact sequence of R-modules. Let the following be projective resolutions of L and N ...
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2answers
52 views

Projective module over $\mathbb{Z}_8$

I am working on some homology and I just want to check if my thoughts are correct, I am working on projective modules over the ring $\mathbb{Z}_8$ and I want to show that $\mathbb{Z}_4$ is projective ...
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48 views

Question on projective module

Let $R$ be a ring an $M$ be a projective $R$-module. Show that there exists a free $R$-module $F$ such that $$M\oplus F\cong F.$$ Any hints?
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1answer
29 views

Why is $0\to\ker\varepsilon\to P_0\xrightarrow\varepsilon N\to 0$ a projective resolution?

Let $R$ be a commutative ring with $1$. My professor defined $\operatorname{Tor}_i^R(M,N)$ as follows: Tensor a projective resolution $\dots\to P_1\to P_0\to M\to 0$ with $N$ and set ...
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1answer
153 views

Every finitely generated flat module over a ring with a finite number of minimal primes is projective

Over a commutative ring $R$, a finite type locally free (weak sense) module for which the rank function is locally constant is projective. If we notice that for each minimal prime $p$ of the ring, ...
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1answer
58 views

Isomorphic modules

Let $R$ be a commutative ring with identity such that $R^{n}\simeq R^{n}\bigoplus M$ ($R$-module isomorphism), $n$ is fixed, and $n$ is a natural number. Then $M=\lbrace 0 \rbrace$ ?
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31 views

Is this module over this group algebra projective?

Assume that $G$ is a finite group. Let $k$ be a field. Let $\varepsilon$ be the augmentation $kG\rightarrow k$. Consider the following map $\varepsilon\otimes id:k[G]\otimes_k k[G]\rightarrow ...
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1answer
39 views

Projective dimension of direct summand

I want to prove that if $M$ is an $A$-module with finite projective dimension and $N$ is an $A$-module that is a direct summand of $M$ then the projective dimension of $N$ is less or equal to that ...
0
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1answer
54 views

A Direct Summand of Finite-Free Module over a Local Ring is Again Free. [duplicate]

Let $R$ be a local ring (commutative with identity) and suppose we have $M$ and $N$ submodules of $R^n$ such that $M\oplus N\cong R^n$. Then $M\cong R^s$ and $N\cong R^t$ for some $s$ and $t$ such ...
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1answer
62 views

Projective objects in BGG category $\mathcal{O}$ are projective $U(\mathfrak{g})$-modules?

Let $\mathfrak{g}$ be a finite dimensional semi-simple complex Lie algebra. Then, BGG category $\mathcal{O}$ is defined to be the full subcategory of finitely generated $U(\mathfrak{g})$-modules of ...
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Projective dimension over a factor ring

$\newcommand{\pdim}{\operatorname{pdim}}$If $\pdim_A M$ is the projective dimension of $M$ as an $A$-module how can i prove that if $A/I=A'$ then $$\pdim_A M\leq \pdim_A A' + \pdim_{A'} M$$ If the ...
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0answers
30 views

If $M$ is an $A$-module via $\varphi\colon A\to\operatorname{End}(M)$, $\mathrm{coker}(\varphi)\otimes M$ is projective?

I have a brief passage I don't understand. Suppose $R$ is a commutative ring, $A$ is an $R$-algebra, which is projective and finitely generated as an $R$-module. Let $M$ be a progenerator for $A$, so ...
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2answers
66 views

Can we construct a homomorphism from a projective module into a free module?

In short, I have a projective module and a free module, and want to construct a module homomorphism between the two. Is this always possible, at least in some way? Let me go into more detail. Suppose ...
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0answers
32 views

Deciding whether a non-f.g. non-divisible flat module is projective or not.

Assume $S= R[T]/(f)= R[w]$ is a flat non-divisible $R$-module, where $R$ is a noetherian UFD, $T$ is an indeterminate over $R$, and $f\in R[T]$ is a non-monic polynomial of positive degree. Can we ...
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1answer
88 views

Constant Dimension for Localization of Projective Modules

It is a well known fact that the localization of a projective module over a commutative ring is free. However, I don't know anything about the dynamics of how the dimension of the resultant free ...
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1answer
60 views

$A \subseteq B \subseteq C$ with $A \subseteq C$ separable and $pd_{B \otimes_A B}(B) = \infty$

Assume $A \subseteq B \subseteq C$ are commutative rings such that $C$ is separable over $A$, namely $C$ is a projective $C \otimes_A C$-module. Separability of $C$ over $A$ does not imply ...
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1answer
48 views

Injectivity, Projectivity, and $P$-injectivity of Localization [closed]

Let $R$ be a commutative ring with unity. I have read that if $M$ is an injective $R$-module, then $S^{-1}M$ is not necessarily an injective $S^{-1}R$-module. I need an example... Does last ...
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112 views

Milnor patching for general modules

The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings $$ \begin{array}{} R & \xrightarrow{f_2} & R_2 \\ \downarrow{f_1} & ...
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1answer
35 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 ...
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1answer
96 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
36 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 ...
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1answer
68 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. ...