Cohomological support varieties under local homomorphisms
Ryan Watson

TL;DR
This paper investigates how cohomological support varieties of modules over local rings behave under local homomorphisms, revealing new invariance properties and recovering classical results in the context of complete intersections.
Contribution
It extends the understanding of support varieties under local maps, especially for complete intersections and finite flat dimension maps, connecting to known theorems.
Findings
Support variety dimension is preserved or increases under certain local maps.
Recovers Bergh and Jorgensen's theorem for complete intersection surjective maps.
Shows the complete intersection property is stable under localization.
Abstract
Given a bounded complex of finitely generated modules over a commutative noetherian local ring , one assigns to it a variety, , called the cohomological support variety of over . The variety holds important homological information about the complex and the ring. In this paper, we study the behavior of cohomological support varieties under restriction of scalars along local maps. In the case where the rings involved are complete intersections and the map is a surjective complete intersection, this recovers a theorem of Bergh and Jorgensen. Additionally, we show that if is a local map of finite flat dimension, then the dimension of is less than or equal to that of . This allows us to recover Avramov's result that the complete intersection property is preserved under localization.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Commutative Algebra and Its Applications · Algebraic structures and combinatorial models
