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Suppose you have a scheme $X$ that is acted on by a torus $T$. Then the action induces a grading on the functions on $X$ by the character lattice of $T$. So for a fixed character $\lambda$, we can consider $\mathcal{O}_{X,\lambda}$, the $\lambda$ graded part. Assuming the quotient $X/T$ exists, these graded parts should descend to quasicoherent sheaves on the quotient.

My question is, when are these sheaves line bundles?

In the basic examples I know, they are always line bundles. For example, if you take $X= \mathbb{A}^{n+1} - 0$ and $T = \mathbb{C}^*$, then on $X$ you get the ordinary grading by homogeneous degree. When you descend to the quotient, you get the line bundles $\mathcal{O}(k)$ on $\mathbb{P}^n$. You can also take $G$ a complex semi-simple group, $B$ a Borel subgroup, $U$ the maximal unipotent. Then $G/U \rightarrow G/B$ is a torus quotient, and the graded pieces descend to line bundles. I think, a similar story is true for all homogeneous spaces, but I'm having a little trouble phrasing it in terms of torus quotients.

In fact, in these situations, these are all the line bundles.

So more generally, my question is, what properties can you require of the general $X$ so it behaves like the two examples above?

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State hypotheses that gp is over alg closed field, or tori are split. Over general field, conn'd ss groups need not contain Borels and tori need not be split. Linear rep of non-split torus may not be direct sum of rank-1 reps over base. If $T$ splits then $T$-torsor over base $S$ is $S$-affine with relative coord ring projective over affine opens in $S$ (etale-local property, Raynaud-Gruson) so decomposes into a direct sum of qcoh sheaves projective over open affines with action through chars. Over noetherian ring, proj module is finite loc free or is free (Bass' "big proj mods are free"). –  BCnrd Mar 27 '10 at 2:46
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up vote 2 down vote accepted

(EDIT: thought I had added this, guess I was wrong. As Brian points out, you definitely want all your tori to be split, or you won't have enough 1-d representations (for example, $S^1$ has no 1-d real representations); over an algebraically closed field, this is automatic.)

If $X/T$ is actually a nice scheme, and $T$ acts freely, then you can think of this as follows: the pushforward of the structure sheaf on $X$ is etale locally isomorphic to the pushforward of the structure sheaf on $X/T \times T$, so proving that your sheaf is line bundle can be reduced to the case of a trivial bundle, in which case it's obvious (you always get the structure sheaf). This actually proves it's a line bundle in the etale topology, but that's the same as being a line bundle in the Zariski topology.

This actually all works in the land of Artin stacks. In more abstract language, if you have any character of the torus, there's a line bundle on the stack $pt/T$ corresponding to this character (since a quasi-coherent sheaf on $pt/T$ is a vector bundle with $T$ action). The line bundles you're talking about are the pullback by the map $X/T\to pt/T$. This is essentially tautological; the stack $X/T$ is specifically designed so that a quasi-coherent sheaf on $X/T$ is a quasi-coherent sheaf on $X$ (of the same rank) which is $T$-equivariant. You're seeing that the structure sheaf can be $T$-equivariant in a bunch of different ways, since you can always tensor the action with a character of $T$.

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