$\Pi$-supports for modules for finite group schemes
$\Pi$-supports for modules for finite group schemes
复制标题
$Pi$-支持有限群方案的模块
DOI：
10.1215/s0012-7094-07-13923-1
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发表时间：
2007
影响因子：
2.5
通讯作者：
J. Pevtsova
中科院分区：
文献类型：
--
作者：
E. Friedlander;J. Pevtsova
引入有限群格式G的Π点等价类的空间Π(G)，并将子空间Π(G)M与任意G模M关联。我们的结果推广到任意域k上的任意有限群方案G和任意大G模上的“上同调支持变”的基本结果及其表示理论解释。特别地，我们证明了任意（可能是无限维的）G模的投影可以通过其沿G的Π点的限制来检测。与G模M的上同调支持变化不同，不变量m7→Π(G)M满足所有模的良好性质，从而使我们能够确定有限维G模的稳定模范畴的厚的、张量理想的子范畴。最后，利用G的稳定模范畴，给出了Π(G)与方案ProjH•（G, k）同构的环空间结构。
We introduce the space Π(G) of equivalence classes of π-points of a finite group scheme G, and associate a subspace Π(G)M to any G-module M . Our results extend to arbitrary finite group schemes G over arbitrary fields k of positive characteristic and to arbitrarily large G-modules the basic results about “cohomological support varieties” and their interpretation in terms of representation theory. In particular, we prove that the projectivity of any (possibly infinite dimensional) G-module can be detected by its restriction along π-points of G. Unlike the cohomological support variety of a G-module M , the invariant M 7→ Π(G)M satisfies good properties for all modules, thereby enabling us to determine the thick, tensor-ideal subcategories of the stable module category of finite dimensional G-modules. Finally, using the stable module category of G, we provide Π(G) with the structure of a ringed space which we show to be isomorphic to the scheme ProjH•(G, k).