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The Derived Category of a Locally Complete Intersection Ring
Let R be a commutative noetherian ring. A well-known theorem in commutative algebra states that R is regular if and only if every complex with finitely generated homology is a perfect complex. This homological and derived category characterization of a regular ring yields important ring theoretic information; for example, this characterization solved the well-known ``localization problem" for regular local rings. The main result of this thesis is establishing an analogous characterization for when R is locally a complete intersection. Namely, R is locally a complete intersection if and only if each nontrivial complex with finitely generated homology can build a nontrivial perfect complex in the derived category using finitely many cones and retracts. This answers a question of Dwyer, Greenlees and Iyengar posed in 2006 and yields a completely triangulated category characterization of locally complete intersection rings. Moreover, this work gives a new proof that a complete intersection localizes.
Pollitz, Joshua, "The Derived Category of a Locally Complete Intersection Ring" (2019). ETD collection for University of Nebraska - Lincoln. AAI22587026.