On cohomology and support varieties for Lie superalgebras

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Support varieties for Lie superalgebras over the complex numbers were introduced in BKN1 using the relative cohomology. In this paper we discuss finite generation of the relative cohomology rings for Lie superalgebras, we formulate a definition for subalgebras which detect the cohomology, also discuss realizability of support varieties. In the last section as an application we compute the relative cohomology ring of the Lie superalgebra S(n) relative to the graded zero component S(n)₀ and show that this ring is finitely generated. We also compute support varieties of all simple modules in the category of finite dimensional S(n)-modules which are completely reducible over S(n)₀.