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

**13**

votes

**2**answers

524 views

### Does projective imply flat?

Let $\mathcal C$ be an abelian category equipped with a closed symmetric monoidal structure. This implies in particular that the monoidal structure $\otimes$ is right exact in each variable. I care ...

**4**

votes

**1**answer

262 views

### Homomorphisms from projective modules

Let $B$ be a $A$-algebra which is free of finite rank as $A$-module. Let $X$ be a finitely generated projective left $B$ module. (So $X$ is also a f.g. projective $A$ module.) Are these homomorphism ...

**3**

votes

**0**answers

68 views

### Projective dimension of ring over its center

If $A$ is a ring and $Z(A)$ is its center then what is a sufficient condition for the projective dimension of $A$ over $Z(A)$ (ie: $pd_{Z(A)}(A)$) to be finite?
(Assuming that $A\neq Z(A)$).

**2**

votes

**1**answer

255 views

### Example of a Frobenius algebra that is not projective over a Frobenius subalgebra

I'd like to know an example of a Frobenius algebra $A$, with a subalgebra $B$ that is itself a Frobenius algebra, such that $A$ is not projective as a left $B$-module. I don't require any ...

**10**

votes

**1**answer

466 views

### $q$-Deformed Quillen–Suslin Theorem for the Quantum Vector Spaces?

Define n-quantum vector space to be the algebra
$$
{\mathbb C}_q^n := \mathbb{C}\left< x_i \mid i =1, \ldots, N\right>/\left<x_i x_j = q x_j x_i \mid i<j\right>.
$$
For $q=1$, we get ...

**5**

votes

**1**answer

326 views

### Is a projective module of constant finite rank finitely generated?

If $R$ is a (commutative) ring and $P$ is a projective $R$-module, then every localization of $P$ at a prime of $R$ is free by Kaplansky's theorem, and has a well-defined rank. If these ranks are all ...

**6**

votes

**2**answers

325 views

### When is/isn't the monoidal unit compact projective?

I am interested in developing intuition for when the monoidal unit in a closed monoidal abelian category is or isn't compact projective. As such, my question is not looking for a yes/no answer, but ...

**4**

votes

**1**answer

513 views

### A weak version of Bass' conjecture

Let $A$ be a finitely generated $\mathbb{Z}$-algebra which is a UFD. Then (a special case of) the Bass conjecture states that $K_0(A)$ is a finitely generated abelian group. As far as I am aware, this ...

**2**

votes

**0**answers

40 views

### Metabolic vs stably metabolic

Let $A$ be a commutative ring with unit. A non-degenerate symmetric bilinear form $\phi$ on a finitely generated projective $A$-module $P$ is called metabolic if there is a direct summand $L$ of $P$ ...

**7**

votes

**1**answer

223 views

### Why can't one modify Kaplansky's proof to conclude that every projective module is a direct sum of its finitely generated projetive submodules?

Due to the examples given in the answer to this question, I know that the conclusion is of course incorrect. But by reading Kaplansky's proof of theorem 1 in this paper and replacing every occurrence ...

**3**

votes

**3**answers

637 views

### is the tensor product of projective modules again projective?

Let $R$ be a commutative ring and let $A_1$ and $A_2$ be (not necessarily commutative) $R$-algebras. Under which conditions on $A_1$ and $A_2$ is the following true:
For every projective $A_1$-module ...

**7**

votes

**2**answers

376 views

### Projective modules over noncommutative tori?

It is a theorem of Rieffel that for any simple noncommutative tori ($\mathcal{A}$) of dimension $n$, every projective module over it is isomorphic to direct sum of $\mathcal{S}(M)$, Schwartz class ...

**4**

votes

**0**answers

102 views

### Sum of projective submodules of a projective over a semihereditary ring

Sorry in advance if this is too silly. Let $R$ be a right semihereditary ring and $P$ a projective right $R$-module. It is well-known that finitely generated (thus projective) submodules of $P$ form a ...

**2**

votes

**1**answer

264 views

### Projective Modules/Algebras: decomposition of linear functions, and the rank formula

Let $A$ be a ring, $B$ a finite projective $A$-algebra, and $C$ a finite projective $B$-algebra. We can show that $C$ is also finite and projective when regarded as an $A$-algebra (by, for instance, ...

**2**

votes

**1**answer

61 views

### Action of GL(2,O_k) on 1d subspaces of (O_k)^2

Let $\mathcal{O}_k$ be the ring of integers in an algebraic number field $k$. Let $M$ be a rank $1$ projective module over $\mathcal{O}_k$ (in other words, $M$ is a projective module such that $k ...

**9**

votes

**1**answer

262 views

### An analogue of the Bass-Quillen conjecture with power or Laurent series

The famous Quillen-Suslin theorem (formerly known as Serre's problem/conjecture) states that every projective module over $k[x_1,\dots, x_n]$ is free for $k$ a field. Replacing $k$ by a more general ...

**3**

votes

**0**answers

241 views

### Rejects and injectives

Let $A$ be any ring and consider modules on the left. For $M$ $A$-module, the trace $Tr(M,A)$ is a two-sided ideal of $A$. If $A$ is a unitary ring then:
$Tr(P,A)P=P$, for $P$ projective;
...

**0**

votes

**0**answers

103 views

### When does the rank of a module behave sub-multiplicatively under tensoring?

Let $\cal{E}$ be a finitely generated projective bimodule over a (noncommutative) algebra $A$. Moreover, let us assume that $\cal{E}$ is of finite rank $n$. The tensor product
$
\cal{E} \otimes_A ...

**5**

votes

**1**answer

463 views

### Baer's criterion for projective modules

Let $R$ be a commutative ring. If necessary, assume that $R$ has any convenient properties you like.
Is there some $R$-module $Q$ such that an $R$-module $P$ is projective if and only if ...

**2**

votes

**2**answers

419 views

### Let $M$ be a $R$-Bimodule that happens to be projective, is its associated left $R \otimes R^{op}$-module projective too?

Let $k$ be a commutative ring, a $R$-Bimodule $M$ over a $k$-algebra $R$ is a $k$-module with two actions of $R$ on $M$, on the left and on the right, the classical example of this being $R$ itself ...

**11**

votes

**2**answers

924 views

### Is every projective $\mathbf{Z}[x]$-module free?

Is every finitely generated projective $\mathbf{Z}[x]$-module free?

**1**

vote

**2**answers

282 views

### For finite group G and field k of char=p, if P,P′ are projective k[G]-modules with [P]=[P′], is it true that P=P′ ?

That is: is it true that if projective k[G]-modules have same composition factors then they are isomorphic?
This is easy to see for char(k)=0, or if G is a composition of a p-group and a p′-group. ...

**4**

votes

**3**answers

471 views

### Are the global sections of a vector bundle a projective module?

Given a scheme $X$ with structure sheaf $\mathcal{O}_X$, we can associate to each $\mathcal{O}_X$-module $\mathcal{F}$ its global sections $\Gamma(\mathcal{F})$, which gets the structure of a ...

**0**

votes

**1**answer

290 views

### why a projective module is a projective cover for its largest semisimple quotient?

Why a projective module is a projective cover for its largest semisimple quotient? That is - why the projection on the quotient is an essential morphism in this case?

**4**

votes

**0**answers

397 views

### Kaplansky's theorem and Axiom of choice

Kaplansky in his paper titled by Projective Modules gave an important and essential theorem as follow:
Theorem: Let $R$ be a ring, $M$ an $R$-module which is a direct sum of (any number of) countably ...

**7**

votes

**2**answers

1k views

### On the difference between a projective chain complex and a level-wise projective chain complex

Let R be an associative ring with a unit, and consider the standard projective model structure of non-negatively graded (left) R-module, $Ch_R$. A map $f:M\to N$ in $Ch_R$ is a weak equivalence if it ...

**1**

vote

**2**answers

454 views

### Projective Modules and their Determinants, Extended or not?

Let $A$ be a commutative noetherian ring, and let $P$ be a projective $A[T]$-module with constant rank $n$. Let $L$ be the determinant of $P$, $\wedge^n(P)$. We say that $P$ (resp. $L$) is extended ...

**0**

votes

**1**answer

300 views

### Nakayama algebra

Let A be a self-injective connected Nakayama algebra. What is the Loewy length of any indecomposable projective A-module?

**5**

votes

**2**answers

659 views

### Indecomposable projectives correspond to irreducibles - reference

Hello,
We have the following assertion:
In an abelian category that has enough projectives and in which every object has finite length, indecomposable projectives correspond bijectively to ...

**2**

votes

**1**answer

487 views

### When are two projective modules of equal rank isomorphic?

Let $R$ be a commutative ring and let $M,N$ be two finitely generated projective $R$-modules which have equal rank (not necessarily constant). What kind of general results are there concerning the ...

**0**

votes

**1**answer

549 views

### Projectivity and faithfully flatness (module theory) [closed]

Is it true that every projective module is faithfully flat, if not what is a counter example.
Thanks!

**2**

votes

**2**answers

846 views

### Projective modules and tensor products

My question(s) relate(s) to pp51-52 of Local Representation Theory by JL Alperin -- the relevant pages are contained in the Google Books preview ...

**3**

votes

**1**answer

575 views

### Perfect complexes and RGamma(X,F) without mentioning derived categories

Let $A$ be a commutative noetherian ring.
Let $K_{parf}(A)$ be the full subcategory of the homotopy category $K(A)$ of $A$-modules whose objects are bounded complexes of finitely generated ...

**0**

votes

**1**answer

275 views

### Completion of unimodular rows

Is the unimodular row $(x,y,z)$ completable over the ring
$({\mathbb Z}/2{\mathbb Z})[x,y,z,y',z']/\langle x^2+yy'+zz'-1 \rangle$ ?

**2**

votes

**1**answer

343 views

### Commutator tensors and submodules

Let $k$ be a commutative ring with $1$, and let $B$ be a submodule of a $k$-module $A$.
For every $n\in\mathbb N$ and every $k$-module $V$, let $K^n\left(V\right)$ denote the kernel of the canonical ...

**3**

votes

**1**answer

233 views

### Freeness of modules along ring homomorphisms

This question arises from my discussion with a Master student. It concerns with the following situation: let $\phi: R \to S$ be a homomorphism between Noetherian commutative rings. Suppose the ...

**2**

votes

**1**answer

297 views

### Descriptions of projectives (injectives) in category of D-modules

Is there any work describing (indecomposable)projectives, injectives in category of D-modules on some flag variety?
I wonder whether someone has used quivers(say Auslander-Reiten sequences)to ...

**8**

votes

**3**answers

1k views

### Example of a projective module which is not a direct sum of f.g. submodules?

This semester I am teaching a graduate course in commutative algebra, and I have been taking the occasion to try to look at the proofs of some the results in my basic source material (Matsumura, ...

**1**

vote

**1**answer

382 views

### Unimodular column property

Hi, I know that if $R$ is a ring such that every projective $R$-module finitely generated is free then $R$ has the unimodular column property.
I would like to know if there is a ring $R$ that doesn't ...

**16**

votes

**6**answers

1k views

### Nonfree projective module over a regular UFD?

What is the simplest example of a domain $R$ which is regular (in particular Noetherian) and factorial which admits a finitely generated projective module that is not free?
In fact I'll be at least ...

**1**

vote

**3**answers

538 views

### Stably free module not finitely generated is free

Hi. I have read that stably free modules not finitely generated are free; this is proved in
M.R. Gabel, stably free projectives over commutative rings, Thesis, Brandeis Univ., Waltham, MA 1972.
But ...

**2**

votes

**1**answer

667 views

### Projective modules over semi-local rings

Let $R$ be a semi-local ring, and $M$ a finite projective $R$-module. If the localizations $M_m$ have the same rank for all maximal ideals $m$ of $R$ then $M$ is free.

**7**

votes

**2**answers

417 views

### Is there any transitivity for separable algebras?

If $R$ is a commutative ring (with $1$), then an $R$-algebra $A$ is said to be separable if $A$ is projective as an $A$-$A$-bimodule. (The notion of an "$A$-$A$-bimodule" includes the requirement that ...

**8**

votes

**3**answers

649 views

### Projective dimension of zero module

Is there any consensus on what the projective dimension of the zero module should be? Here are three statements one commonly encounters in textbooks, sometimes with or without the condition $M\neq 0$:
...

**3**

votes

**4**answers

2k views

### projective module

Is it true that if $\mbox{Ext}^{1}_{A}(P,A/I)=0 \ \forall I$ then P is projective? Similar statements are true for flat and injective modules, but I'm beginning to suspect that projective modules ...

**7**

votes

**1**answer

661 views

### A ring on which all finitely generated projectives modules are free but not all projectives are free?

Dear all,
Sorry if the question is naive: any nice example of such a ring or, better, of a class of such rings?

**41**

votes

**2**answers

4k views

### What is the insight of Quillen's proof that all projective modules over a polynomial ring are free?

One of the more misleadingly difficult theorems in mathematics is that all finitely generated projective modules over a polynomial ring are free. It involves some of the most basic notions in ...

**19**

votes

**2**answers

3k views

### A finitely generated, locally free module over a domain which is not projective?

This is a followup to a previous question
What is the right definition of the Picard group of a commutative ring?
where I was worried about the distinction between invertible modules and rank one ...

**13**

votes

**3**answers

2k views

### What is the right definition of the Picard group of a commutative ring?

This is a rather technical question with no particular importance in any case of actual interest to me, but I've been writing up some notes on commutative algebra and flailing on this point for some ...

**4**

votes

**1**answer

256 views

### Does the non-commutative Chern class depend on the choice of connection?

In classical geometry the calculation of the Chern classes of a vector bundle using a connection is independent of the choice of connection. Does any such result hold for projective modules in ...