  We study homological properties of a locally complete intersection ring by
importing facts from homological algebra over exterior algebras. One
application is showing that the thick subcategories of the bounded derived
category of a locally complete intersection ring are self-dual under
Grothendieck duality. This was proved by Stevenson when the ring is a quotient
of a regular ring modulo a regular sequence; we offer two independent proofs in
the more general setting. Second, we use these techniques to supply new proofs
that complete intersections possess symmetry of complexity.
