Are projective modules over $\mathbb{Z}[x_1,...,x_m]$ free?

Finitely generated projective modules are free over $R[x_1,\dots,x_m]$ for any PID $R$. This was proved by Quillen in his original proof; I'm not sure about Suslin's proof. See Lam's Springer monograph "Serre's Problem on Projective Modules". (In fact all projective modules are free by a 1963 result of Bass.)

Tags:

Abstract Algebra

Commutative Algebra

Ring Theory

Projective Module

Free Modules

Related

Is the radius of convergence related to the ratio limit or half of the interval of convergence? Given positive real numbers $a$, $b$, $c$, $d$, $e$ with $\sum_{\text{cyc}}\,\frac{1}{4+a}=1$, prove that $\sum_{\text{cyc}}\,\frac{a}{4+a^2}\le1$. A weak cancellation property for monoids How can I find real, symmetric matrices $A,B$ such that $Ax =\lambda B x $ has no real eigenvalues and eigenvectors? Is it possible to solve $\frac{dy}{dx}=y^x$? Is Fodor's Lemma necessary for the $\omega_1$ train station puzzle? Completing the proof using strong induction for $E = \bigcap_{n=1}^\infty E_n $ What are all possible positive integers $k$ such that $k=\frac{a^2+b^2+c^2}{bc+ca+ab}$ for some positive integers $a$, $b$, and $c$? Struggling to Understand Algorithm for Displaying Polynomial Matings with Julia Sets Why is $ \frac{5}{64}((161+72\sqrt{5})^{-n}+(161+72\sqrt{5})^{n}-2)$ always a perfect square? What's a fair way to share fees in a group road trip with a personal and a rental car? Explain why the characteristic curves generate the solution surface

Recent Posts