Cohomological dimension, Lyubeznik numbers, and connectedness properties   in mixed characteristic
  We establish a "second vanishing theorem" for local cohomology modules over regular rings of unramified mixed characteristic, which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology. Applying this, and new results on the mixed characteristic Lyubeznik numbers, we further study connectedness properties of the spectra of a certain class of rings. 
Daniel J. Hern\'andez, Luis N\'u\~nez-Betancourt, Felipe P\'erez,   Emily E. Witt
math.AC math.AG