Questions tagged [projective-modules]

For questions about projective modules over a ring and projective objects in related categories.

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Factoring through projective modules is an equivalence relation

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PHom{PHom}$I'm reading about stable module categories, and I have a question about the definition of the maps. Let $R$ be a ring, and take (left) ...
StuckInTheFridge's user avatar
-3 votes
0 answers
133 views

Extending the proof of Maschke's Theorem from finite groups to algebras

In the theory of representations of a finite group there is Maschke's Theorem that any finite-dimensional representation of a finite group $G$ can be decomposed into a direct sum of irreducible ...
Dale's user avatar
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3 votes
1 answer
225 views

Hattori-Stallings trace

Let $R$ be a (possibly non-commutative) unital ring and $M$ be a left $R$-module. If $M$ is finitely generated and projective, the natural map $$\iota:\mathrm{Hom}_R(M,R)\otimes_R M\to \mathrm{Hom}_R(...
Qwert Otto's user avatar
3 votes
1 answer
282 views

Is every graded hereditary ring hereditary?

Let R be a graded (associative, unital) ring. If R is left graded hereditary (i.e. its left graded global dimension is 0 or 1), does it follow that R is left hereditary (i.e. its left global dimension ...
Ralle's user avatar
  • 471
3 votes
0 answers
172 views

The monoid of stably-free modules over integral group rings

Fix a torsion-free group G, let $M_G$ be the monoid of stably-free $\mathbb{Z}G$-modules under operation $\oplus$, the direct sum of modules. In studying objects related to Wall’s D2 problem on CW-...
William Thomas's user avatar
3 votes
1 answer
212 views

On Flat and Projective Modules over integral domain

Is this true that finitely generated flat module over an integral domain is projective. If Yes, please provide a proof.
Amit Phogat's user avatar
1 vote
0 answers
83 views

Quotient sheaf obtained from a quotient of $\mathrm{SL}_2$ having a section

$\DeclareMathOperator\SL{SL}$Let $K$ be a field and let $G$ be a $K$-defined, closed, algebraic subgroup of $\SL_2$. Denote by $\mathcal{C}$ the site whose objects are $K$-schemes, with your favorite $...
kindasorta's user avatar
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3 votes
1 answer
182 views

Flatness over regular local rings of dimension 3

Let $R$ be a regular local ring of dimension $n$. Let $i:\text{Flat}\to\text{Fin}$ be the fully faithful inclusion of the category of flat finitely generated type $R$-modules into all finitely ...
Matthieu Romagny's user avatar
5 votes
0 answers
109 views

Homological characterization of perfect resolutions

Suppose that $R$ is a left Noetherian associative ring with unit and $M$ a finitely generated left $R$-module. It is a standard fact that if the $\mathrm{Ext}$-groups $\mathrm{Ext}^i_R(M,N)$ vanishe ...
Yonatan Harpaz's user avatar
2 votes
0 answers
88 views

Is there a standard reference for: taking projective covers of simple modules commutes with finite Galois field extensions?

Let $\Lambda$ be an Artin algebra over a finite field $k$. Is there a standard reference for the fact that taking projective covers of simple $\Lambda$-modules commutes with finite Galois field ...
Stein Chen's user avatar
0 votes
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228 views

The coevaluation map for a projective module and its dual

$\DeclareMathOperator\coev{coev}$Let $R$ be a noncommutative ring and let $P$ be a bimodule over $R$, that is finitely generated and projective as a left module. It is "well-known" that any ...
Adam Bondal's user avatar
1 vote
0 answers
65 views

Prove that $f(M)=f^2(M)$ implies $f(M)$ is a direct summand of $M$ whenever $\text{End}_R(M)$ is a reduced ring

Let $M$ be a right $R$-module with the property that every homomorphism $\gamma:Sf\to M, f\in S=\text{End}_R(M)$, extends to $S\to M$. If $S$ has the property $f^2=0$ implies $f=0$ for every $f\in S$...
mariam's user avatar
  • 111
2 votes
1 answer
318 views

Example of non vanishing Ext

Let $R$ be a commutative Noetherian ring and $I$ is a proper ideal of $R$. suppose that $M$ is a f.g. $R$-module. $\DeclareMathOperator\Ext{Ext}$I'm looking for an example that has this property: $$\...
pink floyd's user avatar
1 vote
0 answers
130 views

Existence of a finite resolution

I have tried to formulate a question in which I was very curious, any hints suggestions are also welcomed. Thanks in advance. Let $M$ be an $R$ module ($R$ commutative ring with unity). It is given ...
user443060's user avatar
1 vote
0 answers
114 views

Composition of faithfully flat ring extensions

Let $R$ be a not necessarily commutative, unital, ring, and for simplicity let module always mean right module. We say that a unital ring extension $R \hookrightarrow S$ is flat, or faithfully flat, ...
Tim Montegue's user avatar
3 votes
1 answer
242 views

Faithful flatness for rings

Let $R$ be a ring and let $M$ be a right module over $R$. We say that $M$ is faithfully flat as a right module if the functor $M \otimes_R -$ from left $R$-modules to abelian groups that preserves ...
Tim Montegue's user avatar
4 votes
1 answer
312 views

In a category with a projective generator, do morphisms from the generator determine the object?

I have a cocomplete abelian category $\mathcal C$ and two objects $X$, $Y$ in $\mathcal C$. Further, $\mathcal C$ has a projective generator $P$. I have an isomorphism $$ \mathcal C(P,X) \cong \...
Nombres's user avatar
  • 131
2 votes
2 answers
239 views

An algebra which is a direct sum of simple sub-bimodules over a subalgebra

Let $A$ be an infinite-dimensional noncommutative algebra over a field, let $B$ be an infinite-dimensional subalgebra of $A$, and let $A$ be a direct sum of projective simple $B$-sub-bimodules. Then ...
Boris Henriques's user avatar
6 votes
2 answers
314 views

Are finite projective modules over $R[t]$ free when $R$ is DVR?

Let $R$ be a discrete valuation ring (DVR) and let $M$ be a projective module of finite type over the polynomial ring $R[t]$. Is $M$ free over $R[t]$? As far as I understand, this should be a ...
Stabilo's user avatar
  • 1,479
2 votes
0 answers
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Profinite projective and free modules

I am studying cohomology of profinite groups and the following question came to my mind: suppose we have $G$ a pro-$p$ group which is Poincaré Dual of dimension $d$. This means that $\mathbb{Z}_p$ as ...
N.B.'s user avatar
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4 votes
1 answer
141 views

Are finitely presented algebras over VNRs projective?

Question: Let $A$ be a commutative von Neumann regular ring, and $B$ an $A$-algebra of finite presentation, i.e. $B = A[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$. Is $B$ a projective $A$-module? In the ...
Badam Baplan's user avatar
5 votes
0 answers
611 views

Rings such that torsion-free/flat/projective modules are flat/projective/free

While thinking about this question (and specifically YCor's remarks), I tried to remember what can be said about rings such that every torsion-free module is free, and I could not; and such things, ...
2 votes
2 answers
847 views

Characterization of projective modules in terms of Ext groups

This is from Hartshrone exercise 6.6 part (a). Let $A$ be a regular local ring and $M$ be a finitely generated $A$-module, prove the following $M$ is projective $\iff$ $\operatorname{Ext}^{i}(M,A)=\{...
John117's user avatar
  • 395
4 votes
2 answers
307 views

Projective modules restricted to smooth curves

I asked this question on Stack Exchange, but no one answered this. I want to prove a coherent sheaf $M$ on $X$ is locally free if and only if this is true for $M|_{X'}$ , for all smooth curves $X'$ ...
FunctionOfX's user avatar
3 votes
1 answer
202 views

Is a tower of locally-free modules locally a tower of free modules?

Suppose we have a (commutative, unital) ring $R$ and a (commutative, unital) $R$-algebra $A$ such that $A$ is projective of constant rank $n$ as an $R$-module. This condition is equivalent to there ...
Owen Biesel's user avatar
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8 votes
2 answers
439 views

tangent bundle on noncommutative manifold

Using the Serre-Swan's theorem, one can do vector bundle theory on noncommutative manifold $(A,H,D)$, by replacing vector bundle by finitely generated projectve module $M$. For the construction of ...
Ken.Wong's user avatar
  • 493
1 vote
2 answers
346 views

When splitting of short exact sequence preserves the kernels

This is a problem that I thought at first was obvious but that became less clear the more I thought about it. Assume we have a finitely generated algebra $A$ over a field $k$, and a short exact ...
user127776's user avatar
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6 votes
1 answer
125 views

On the finiteness of an Auslander-Reiten component

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is Theorem 2.7: And this is part of it's proof, in which the direction (2) $\Rightarrow $ ...
mathStudent's user avatar
-1 votes
1 answer
140 views

infinite left degrees

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is a part of the paper: Definition: Let $f: X \rightarrow Y$ be an irreducible morphism ...
mathStudent's user avatar
1 vote
1 answer
171 views

About composition factors [closed]

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is part of the proof of Lemma 2.3 $A$ is assumed to be an Artin algebra and mod(A) the ...
mathStudent's user avatar
1 vote
1 answer
325 views

Question on simple modules and projective covers

I have the following question: Let $A$ be an Artin algebra. Let $S_1$ and $S_2$ be simple modules in $\text{mod}(A)$ and let $P(S_1)$ be the projective cover of $S_1$. Let $f: P(S_1) \rightarrow S_2$ ...
mathStudent's user avatar
3 votes
2 answers
187 views

Question on injective hulls

How can I show the following: Let $f: M \rightarrow N$ be a morphism in $\text{mod}(A)$, where $A$ is an Artin algebra. Suppose $f \neq 0$. Then there exists a simple module $S$ with its injective ...
mathStudent's user avatar
1 vote
1 answer
359 views

injective hull and projective cover of simple modules are indecomposable

Let $A$ be an Artinian algebra. Let $S$ be a simple module over $A$. Let $\pi: S \rightarrow I$ be the injective hull and $\tau: P \rightarrow S$ be the projective cover of $S$. Then $I$ and $P$ must ...
mathStudent's user avatar
6 votes
1 answer
167 views

Finitely presented modules admitting projective covers

A ring $R$ is called semi-perfect if every finitely generated $R$-module has a projective cover, and it can be proved that this is equivalent to say that the category consisting of the finitely ...
user avatar
0 votes
1 answer
425 views

Separability of $\mathbb{C}[x]$ over its $\mathbb{C}$-subalgebras

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: $f(s_1 \otimes_R s_2)=s_1s_2$. ...
user237522's user avatar
  • 2,783
5 votes
1 answer
374 views

Trace ideal of a projective module

In his 1969 paper "On projective modules of finite rank", Wolmer Vasconcelos writes Let $M$ be a projective $R$-module... The trace of $M$ is defined to be the image of the map $M \otimes_R ...
Andrea Ferretti's user avatar
6 votes
0 answers
148 views

Ring with different graded and ungraded global dimensions

Let $A$ be a $\mathbb N$-graded ring. One can consider the two categories $M_A^g$ and $M_A^u$ of graded and ungraded modules over $A$. Both have, say, enough projectives, hence one can compute various ...
Andrea Ferretti's user avatar
1 vote
1 answer
203 views

Second summand to make projective module free

Suppose there's a projective $R$-module $P$ (non-free). We know that there is another $R$-module $M$ such that $P\oplus M$ is free over $R$. Is there a way to write down such an $M$ in terms of $P$? ...
zjs's user avatar
  • 465
4 votes
1 answer
380 views

Projective (or injective) object in a subcategory

Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be a full subcategory of $\mathcal{A}$. Suppose that $\mathcal{B}$ is abelian and that the inclusion of $\mathcal{B}$ in $\mathcal{A}$ is ...
user avatar
5 votes
1 answer
161 views

Projective module which splits off sequence of submodules, but not the sum

Does there exist an example of a module $X$ over some ring $R$ together with submodules $T_i$ such that: $X$ is projective, $X$ splits as an internal direct sum $X\cong T_1\oplus T_2\oplus \ldots \...
nikola karabatic's user avatar
2 votes
0 answers
54 views

Non-singular rings which are Rickart

A ring $R$ is said to be a right Rickart ring if the right annihilator of any element in $R$ is of the form $eR$ for some idempotent $e \in R$. It turns out that a ring $R$ is right Rickart iff every ...
Johan Öinert's user avatar
12 votes
2 answers
1k views

Is every locally free module of rank $1$ over a commutative ring concretely invertible?

Since this subject is full of misunderstandings (see here, here, here, and here) let us fix a precise terminology. Let $A$ be a commutative ring and $P$ an $A$-module. I) We'll say that $P$ is a ...
Georges Elencwajg's user avatar
9 votes
1 answer
536 views

Bézout ring with non-trivial Picard group?

[I asked this on stackexchange here a few weeks ago to no response] A ring is called Bézout when its finitely generated ideals are principal. Q: Is there a nice example of a Bézout ring $R$ with ...
Badam Baplan's user avatar
3 votes
1 answer
441 views

Hom functor and arbitrary coproducts

Let $M$ be a finitely generated $R$-module. It's easy to check that, in this case, $\mathbf{Hom}(M,-)$ preserves infinite sums. Now suppose that $M$ is projective. Is the reciprocal true? That is, if ...
user_12345's user avatar
8 votes
2 answers
535 views

Basis for free modules over an affine domain

Let $A=k[x_1,\cdots,x_r]/I$ for some prime ideal $I$ and some field $k$. Consider the free $A$-module $A^n$. Question 1. Given an element $e\in A^n$, is there a method to tell whether $e$ can be ...
user127776's user avatar
  • 5,851
4 votes
0 answers
84 views

Weyl theorem - possible corollary - alternative characterization of projective representation of $Z_N\times Z_N$

For an integer $N$, let $\omega=e^{2i\pi/N}$ and $A$, $B$ be the clock and shift operators: $A=\left(\begin{matrix} 1 & 0 & \cdots & 0 \\ 0 & \omega & \cdots & 0 \\ \vdots &...
MarcO's user avatar
  • 553
8 votes
1 answer
611 views

Property of the trace on finitely generated projective modules

Let $A$ be a commutative ring with unit and let $P$ be a projective $A$-module finitely generated. By definition, there exists an $A$-module $P'$ such that $P\oplus P'$ is free of finite rank $r$. If $...
Stabilo's user avatar
  • 1,479
2 votes
1 answer
139 views

A weak Schur's lemma for non-semisimple finite dimensional algebras

Let $B \subseteq C$ be an inclusion of finite dimensional (associative) algebras over a field $k$. Assume that $C$ is a free $B$-module. Let $\bigoplus_i U_i$ be a decomposition of $B$ into ...
Sebastien Palcoux's user avatar
1 vote
0 answers
135 views

Lifting idempotents and projective coverings --- reference request

Let $R$ be a (non-commutative, unital) ring with Jacobson radical $J$, write $\overline{R}=R/J$ and denote the quotient map $R\to \overline{R}$ by $a\mapsto \overline{a}$. It is easy to see that if ...
Uriya First's user avatar
  • 2,846
2 votes
1 answer
253 views

A module associated to an endomorphism of a vector bundle

Let $E$ be a vector bundle over a compact connected Hausdorff space $X$. To an endomorphism $\alpha \in End(E)$, we associate a $C(X)-$ module $\Gamma(E,\alpha)$ consisting of all $\beta\in End(E)$ ...
Ali Taghavi's user avatar