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We introduce higher-order support varieties for pairs of modules over acommutative local complete intersection ring, and give a complete descriptionof which varieties occur as such support varieties. In the context of a groupalgebra of a finite elementary abelian group, we also prove a higher-orderAvrunin-Scott-type theorem, linking higher-order support varieties andhigher-order rank varieties for pairs of modules.

Realizability and the Avrunin-Scott theorem for higher-order support varieties