5   Teichm ller lifts  Definition 5 1   Perfect    A ring R of characteristic p   0  prime  is a perfect ring  if the Frobenius x   x p is a bijection  A field of characteristic p is a perfect field if it is perfect as a ring   Remark   Since characteristic R   p    x   y   p   x p   y p  so Frobenius is a ring homomorphism    Example   i    p n and   p   are perfect fields    ii    p   t   is not perfect  because t   Im   Frob      iii    p   t 1 p           p   t   t 1 p   t 1 p 2       is a perfect field  called the perfection of   p   t      Fact   A field of characteristic p   0 is perfect if and only if any finite extension of k is separable   Theorem 5 2  Assuming that     K           is a complete discretely valued field  such that  k     O K   m is a perfect field of characterist p  Then  there exists a unique map          k   O K such that   i  a     a   mod m for all a   k   ii    a b       a     b   for all a   b   k  Moreover if  characteristic O K   p  then       is a ring homomorphism   Definition 5 3  The element    a     O K constructed in Theorem 5 2 is the Teichm   ller lift  of a   Lemma 5 4  Assuming that     K           is a complete discretely valued field  such that  k     O K   m is a perfect field of characterist p      O K a fixed uniformiser  x   y   O K such that x   y mod   k  k   1   Then  x p   y p mod   k   1   Proof  Let x   y   u   k with  u   O K  Then     x p     i   0 p p i y p   i   u   k   i   y p     i   1 p p i y p   1   u   k   i     Since   O K     O K has characteristic p  we have  p     O K  Thus      p i   u   k   i y p   i     k   1 O K   i   1       hence x p   y p mod   k   1     Proof of Theorem  5 2     Let a   k  For each i   0 we choose a lift  y i   O K of a 1 p i  and we define      x i     y i p i       We claim that   x i   i   1   is a Cauchy sequence and its limit is independent of the choice of y i   By construction  y i   y i   1 p mod    By Lemma 5 4 and induction on k  we have y i p k   y i   1 p k   1 and hence x i   x i   1 mod   i   1  take i   p   Hence   x i   i   1   is Cauchy  so  x i   x   O K   Suppose   x i     i   1   arises from another choice of y i   lifting a i 1 p i   Then   x i     i   1   is Cauchy  and  x i     x     O K  Let      x i       x i i even x i   i odd       Then x i   arises from lifting      y i       y i i even y i i odd       Then x i   is Cauchy and x i     x  x i     x    So x   x   and hence x is independet of the choice of y i  So we may define    a     x   Then x i   y i p i     a 1 p i   p i   a mod    Hence x   a mod    So  i  is satisfied   We let b   k and we choose  u i   O K a lift of b 1 p i  and let z i     u i p i t  Then  lim i     z i     b     Now u i y i is a lift of   a b   1 p i  hence        a b     lim i     x i z i     lim i     x i     lim i     z i       a     b         So  ii  is satisfied   If characteristic K   p  y i   u i is a lift of a 1 p i   b 1 p i     a   b   1 p i  Then       a   b     lim i       y i   u i   p i   lim i     y i p i   u i p i   lim i     x i   z i     a       b       Easy to check that    0     0     1     1  and hence        is a ring homomorphism   Uniqueness  let      k   O K be another such map  Then for a   k      a 1 p i   is a lift of a 1 p i  It follows that       a     lim i         a 1 p i   p i   lim i         a         a         Example  K     p             p     p  a     p       a   p   1     a p   1       1     1  So    a   is a   p   1   th root of unity    Lemma 5 5  Assuming that     K           complete discretely valued field  k   O K   m     p    a   k    Then     a   is a root of unity   Proof     a   k     a     p n   for some n     a   p n   1     a p n   1       1     1       Theorem 5 6  Assuming that     K           complete discretely valued field  characteristic   K     p   0  k is perfect  Then  K   k     t       k   O K   m    Proof  Since  K   Frac   O K    it suffices to show  O K   k     t    Fix      O K a uniformiser  and let           k   O K be the Teichm ller map and define         k     t       O K       i   0   a i t i       i   0     a i     i     Then   is a ring homomorphism since        is  and it is a bijection by Proposition 3 4  ii