  Given an effective Cartier divisor D with simple normal crossing support on a
smooth and proper scheme X over a perfect field of positive characteristic p,
there is a natural notion of de Rham-Witt sheaves on X with zeros along D. We
show that these sheaves correspond under Grothendieck duality for coherent
sheaves to de Rham-Witt sheaves on X with modulus (X,D), as defined in the
theory of cube invariant modulus sheaves with transfers developed by
Kahn-Miyazaki-Saito-Yamazaki. From this we deduce refined versions of Ekedahl -
and Poincar\'e duality for crystalline cohomology generalizing results of
Mokrane and Nakkajima for reduced D, and a modulus version of Milne-Kato
duality for \'etale motivic cohomology with p-primary torsion coefficients,
which refines a result of Jannsen-Saito-Zhao. We furthermore get new integral
models for rigid cohomology with compact supports on the complement of D and a
modulus version of Milne's perfect Brauer group pairing for smooth projective
surfaces over finite fields.
