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\(\Pi\)-supports for modules for finite group schemes. (English) Zbl 1128.20031

Over the past 25 years, the study of cohomological support varieties and representation-theoretic rank varieties has led to numerous results in the modular representation theory of finite groups, restricted Lie algebras, and other related structures. This work follows previous work of the authors [Am. J. Math. 127, No. 2, 379-420 (2005; Zbl 1072.20009), Erratum ibid. 128, No. 4, 1067-1068 (2006; Zbl 1098.20500)] in attempting to formulate a unifying theory for arbitrary finite group schemes (equivalently finite-dimensional cocommutative Hopf algebras) over arbitrary fields of prime characteristic. The paper contains several foundational results as well as a number of illuminating examples.
Let \(G\) be a finite group scheme over a field \(k\) of prime characteristic \(p\). Extending their previous notion of a \(p\)-point (defined over algebraically closed fields), the authors introduce the notion of a \(\pi\)-point of \(G\) which is a flat map of \(K\)-algebras \(K[t]/t^p\to KG\) which factors through the group algebra of a unipotent Abelian subgroup scheme for a field extension \(K/k\). The \(\Pi\)-points of \(G\), denoted \(\Pi(G)\), is the set of equivalence classes of such \(\pi\)-points under a certain specialization relation.
The first of several important results is that \(\Pi(G)\) is homeomorphic to the projectivized prime ideal spectrum of the (even-dimensional) cohomology ring of \(G\) over \(k\). For a finite-dimensional \(G\)-module \(M\), the \(\Pi\)-support of \(M\) is defined as a certain subset of \(\Pi(G)\) and can be identified cohomologically. Moreover, the authors extend this definition to arbitrary (i.e., even infinite-dimensional) \(G\)-modules. For an arbitrary module, the \(\Pi\)-support does not have a direct cohomological interpretation. The authors show that it does satisfy a number of nice properties and that every subset of \(\Pi(G)\) can be identified with the \(\Pi\)-support of some module. Another fundamental result is that the projectivity of a module can be detected by restriction to \(\pi\)-points which extends several known results in special cases. Further, the \(\Pi\)-support is used to determine the tensor-ideal thick subcategories of the stable module category of finite-dimensional \(G\)-modules, thus verifying a conjecture of M. Hovey, J. H. Palmieri, and N. P. Strickland [Mem. Am. Math. Soc. 610 (1997; Zbl 0881.55001)]. Using this stable module category information, the authors give a scheme structure to \(\Pi(G)\) and show that the aforementioned homeomorphism of varieties can be extended to an isomorphism of schemes.

MSC:

20G05 Representation theory for linear algebraic groups
14L15 Group schemes
20C20 Modular representations and characters
20G10 Cohomology theory for linear algebraic groups
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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