# Questions tagged [projective-modules]

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

148 questions
Filter by
Sorted by
Tagged with
260 views

### 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) ...
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 ...
• 431
225 views

• 238
1 vote
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 ...
• 167
1 vote
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, ...
• 435
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 ...
• 435
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 \...
• 131
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 ...
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 ...
• 1,479
135 views

### 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 ...
• 757
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 ...
• 938
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, ...
847 views

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 $\iff$ $\operatorname{Ext}^{i}(M,A)=\{... • 395 4 votes 2 answers 307 views ### Projective modules restricted to smooth curves I asked this question on Stack Exchange, but no one answered this. I want to prove a coherent sheaf$M$on$X$is locally free if and only if this is true for$M|_{X'}$, for all smooth curves$X'$... • 367 3 votes 1 answer 202 views ### Is a tower of locally-free modules locally a tower of free modules? Suppose we have a (commutative, unital) ring$R$and a (commutative, unital)$R$-algebra$A$such that$A$is projective of constant rank$n$as an$R$-module. This condition is equivalent to there ... • 2,336 8 votes 2 answers 439 views ### tangent bundle on noncommutative manifold Using the Serre-Swan's theorem, one can do vector bundle theory on noncommutative manifold$(A,H,D)$, by replacing vector bundle by finitely generated projectve module$M$. For the construction of ... • 493 1 vote 2 answers 346 views ### When splitting of short exact sequence preserves the kernels This is a problem that I thought at first was obvious but that became less clear the more I thought about it. Assume we have a finitely generated algebra$A$over a field$k$, and a short exact ... • 5,851 6 votes 1 answer 125 views ### On the finiteness of an Auslander-Reiten component I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is Theorem 2.7: And this is part of it's proof, in which the direction (2)$\Rightarrow $... • 251 -1 votes 1 answer 140 views ### infinite left degrees I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is a part of the paper: Definition: Let$f: X \rightarrow Y$be an irreducible morphism ... • 251 1 vote 1 answer 171 views ### About composition factors [closed] I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is part of the proof of Lemma 2.3$A$is assumed to be an Artin algebra and mod(A) the ... • 251 1 vote 1 answer 325 views ### Question on simple modules and projective covers I have the following question: Let$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$... • 251 3 votes 2 answers 187 views ### Question on injective hulls How can I show the following: Let$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$with its injective ... • 251 1 vote 1 answer 359 views ### injective hull and projective cover of simple modules are indecomposable Let$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$must ... • 251 6 votes 1 answer 167 views ### Finitely presented modules admitting projective covers A ring$R$is called semi-perfect if every finitely generated$R$-module has a projective cover, and it can be proved that this is equivalent to say that the category consisting of the finitely ... 0 votes 1 answer 425 views ### Separability of$\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$. ... • 2,783 5 votes 1 answer 374 views ### Trace ideal of a projective module In his 1969 paper "On projective modules of finite rank", Wolmer Vasconcelos writes Let$M$be a projective$R$-module... The trace of$M$is defined to be the image of the map$M \otimes_R ...
• 14.5k
148 views

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 ...
• 14.5k
1 vote
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 $P$? ...
• 465
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 ...
161 views

• 553