The vanishing of a higher codimension analog of Hochster's theta   invariant
  We study H. Dao's invariant $\eta_c^R$ of pairs of modules defined over a complete intersection ring $R$ of codimension $c$ having an isolated singularity.   Our main result is that $\eta_c^R$ vanishes for all pairs of modules when $R$ is a {\em graded} complete intersection ring of codimension $c > 1$ having an isolated singularity. A consequence of this result is that all pairs of modules over such a ring are $c$-$\Tor$-rigid. 
W. Frank Moore, Greg Piepmeyer, Sandra Spiroff and Mark E. Walker
math.AC math.AG