An étale cohomology duality theorem for number fields with a real embedding

The restriction on $2$-primary components in the Artin-Verdier duality theorem [2] has been eliminated by Zink [9], who has shown that the sheaf of units for the étale topology over the ring of integers of any number field acts as a dualizing sheaf for a modified cohomology of sheaves. The present paper provides an alternate means of removing the $2$-primary restriction. Like Zink's, it involves a topology which includes infinite primes, but it avoids modified cohomology and will be more directly applicable in the proof of a theorem of Lichtenbaum regarding zeta- and $L$-functions [4, 5]. Related results--including the cohomology of units sheaves, the norm theorem, and punctual duality theorem of Mazur [6]--are also affected by the use of a topology including the infinite primes. The corresponding results in the new setting are included here.