We investigate compact projective generators in the category of equivariant -modules on a smooth affine variety. For a reductive group G acting on a smooth affine variety X, there is a natural countable set of compact projective generators indexed by finite dimensional representations of G. We show that only finitely many of these objects are required to generate; thus the category has a single compact projective generator. The proof goes via an analogous statement about compact generators in the equivariant derived category, which holds in much greater generality and may be of independent interest.

中文翻译：

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等变模的投影生成

我们研究了平滑仿射变体上的等变模类别中的紧凑投影生成器。对于作用于光滑仿射变体X的归约群G，存在由G 的有限维表示索引的自然可数的紧凑射影生成器集。我们表明只需要有限数量的这些对象来生成；因此该类别具有单个紧凑的投影生成器。证明通过一个关于等变派生类别中紧致生成器的类似陈述进行，该陈述具有更大的普遍性并且可能具有独立意义。