Decent intersection and Tor-rigidity for modules over local hypersurfaces

We study two properties of modules over a local hypersurface $R$: decency and rigidity. We show that the vanishing of Hochster's function $θ^R(M,N)$, known to imply decent intersection, also implies rigidity. We investigate the vanishing of $θ^R(M,N)$ to obtain new results about decency and rigidity over hypersurfaces. We employ a mixture of techniques from Commutative Algebra and Intersection Theory of algebraic cycles.

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