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If $R=k[t,t^{-1}]$ is a graded ring where $R_0=k$ is a field and $t\in R$ is a homogeneous element of positive degree which is transcendental over $k$, how can I prove that every graded $R$-module is free?

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2 Answers

up vote 3 down vote accepted

A low-tech way:

Let $M$ be a graded $k[t^{\pm1}]$-module, so that in particular $M=\bigoplus_{n\in\mathbb Z}M_i$ as a vector space. For each $n\in\mathbb Z$ the map $M_n\to M_{n+1}$ given by multiplication by $t$ is a linear bijection, with inverse given by multiplication by $t^{-1}$, of course. It follows that for all $n\in\mathbb Z$ we have $M_n=t^nM_0$. Moreover, it is easy to see that the map $k[t^{\pm1}]\otimes_k M_0\to M$ obtained by restricting the multiplication map $k[t^{\pm1}]\otimes_k M\to M$ to the subspace $k[t^{\pm1}]\otimes_k M_0$ is in fact an isomorphism of $k[t^{\pm1}]$-modules, if we see its domain as a $k[t^{\pm1}]$-module in the obvious way. But this obvious module is free: any basis of $M_0$ gives a basis.

N.B.: This is, I guess, what the high-tech proof amounts to in this concrete situation...

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(I am assuming the degree of $t$ is $1$; if it is not, the same argument works, mutatis mutandi) –  Mariano Suárez-Alvarez Jan 26 '11 at 3:12
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One way:

A graded $H=k[t^{\pm1}]$-module $M$ is the same thing as Hopf $H$-module (so it is an $H$-module, and $H$-comodule, and both structures are compatible). The Fundamental Theorem of Hopf modules tells you, then, that such a beast is free as an $H$-module.

The best reference for this is, in my opinion, Sweedler's book [Sweedler, Moss E. Hopf algebras. Mathematics Lecture Note Series W. A. Benjamin, Inc., New York 1969 vii+336 pp. MR0252485]

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Extraordinarily, Wikipedia does not have a page on Hopf modules! There's a nice little project for some enterprising soul! –  Mariano Suárez-Alvarez Jan 26 '11 at 3:00
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