0
votes
1answer
106 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, ...
9
votes
1answer
181 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 ...
5
votes
1answer
370 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 ...
11
votes
2answers
788 views

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

Is every finitely generated projective $\mathbf{Z}[x]$-module free?
1
vote
2answers
385 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 ...
2
votes
1answer
466 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 ...
2
votes
1answer
332 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
1answer
231 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 ...
8
votes
3answers
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
1answer
331 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 ...
15
votes
6answers
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
3answers
475 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
1answer
573 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.
6
votes
3answers
504 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 $M\neq 0$: ...
37
votes
2answers
3k 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 ...
17
votes
2answers
2k 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
3answers
1k 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 ...