2   Valuation Rings  Definition 2 1   Valuation    Let K be a field  A valuation on K is a function v   K       such that   i  v   x y     v   x     v   y     ii  v   x   y     min   v   x     v   y      Fix 0       1  If v is a valuation  on K  then        x         v   x   x   0 0 x   0     determines a non archimedean absolute value on K   Conversely a non archimedean absolute value determines a valuation  v   x     log     x     Remark   Ignore the trivial valuation v   x     0   Say v 1   v 2 are equivalent if there exists c       0 such that v 1   x     c v 2   x   for all x   K     Example  K       v p   x       log p   x   p is known as the p adic valuation   If k is a field  consider K   k   t     Frac   k   t     the rational function field  Then define      v   t n f   t   g   t       n     for f   g   k   t   with f   0     g   0     0  We call this the t adic valuation   K   k     t       Frac   k     t             i   n   a i t i   a i   k   n        known as the field of formal Laurent series over k  Then we can define      v     I a i t i     min   i   a i   0       is the t adic valuation on K   Definition 2 2  Let    K           be a non archimedean valued field  The valuation ring of K is defined to be     O K     x   K     x     1       B     0   1           x   K     v   x     0       0         Proposition 2 3   i  O K is an open subring of K   ii  The subsets   x   K     x     r   and   x   K     x     r   for r   1 are open ideals in  O K    iii  O K       x   K     x     1     Proof   i    0     0     1     1 so  0   1   O K  If  x   O K  then      x       x   hence    x   O K  If  x   y   O K  then        x   y     max     x       y       1       Hence  x   y   O K  If  x   y   O K  then     x y       x     y     1  hence  x y   O K  Thus  O K is a ring  Since  O K   B     0   1    it is open    ii  Similar to  i     iii  Note that     x     x   1       x x   1     1  Thus       x     1     x   1     1   x   x   1   O K   x   O K         Notation  m       x   O K     x     1   is a max ideal of  O K   k     O K   m is the residue field   Corollary 2 4  O K is a local ring with unique maximal ideal m  a local ring is a ring with a unique maximal ideal     Proof  Let m   be a maximal ideal  Suppose m     m  Then there exists x   m     m  Using part  iii  of Proposition 2 3  we get that x is a unit  hence  m     O K  a contradiction     Example  K     with        p  Then      O K       p       a b       p   b         and m   p     p    k     p   Definition 2 5  Let v   K       be a valuation  If v   K          we say v is a discrete valuation  K is said to be a discretely valued field  An element      O K is uniformiser if v         0 and v       generates v   K       Example  K     with p adic valuation is a discrete valuation ring   K   k   t   with t adic valuation is a discrete valuation ring   K   k   t     t 1   2   t 1   4   t 1   8        Here  the t adic valuation is not discrete   Remark   If v is a discrete valuation  can replace with equivalent one such that v   K          Call such a v normalised  valuations   then v         1 if and only if   is a unit     Lemma 2 6  Assuming that   v is a valuation on K  Then  the following are equivalent    i  v is discrete   ii  O K is a PID   iii  O K is Noetherian   iv  m is principal  Proof   i     ii    O K is an integral domain since it is a subset of K  which is an integral domain  Let  I   O K be a non zero ideal  Let x   I such that v   x     min   v   a     a   I    which exists since v is discrete  Then we claim      x O K     a   O K   v   a     v   x         is equal to I        I is an ideal       Let y   I  Then v   x   1 y     0  Hence  y   x   x   1 y     x O K    ii     iii    Clear    iii     iv    Write   m   x 1 O K       x n O K  Without loss of generality       v   x 1     v   x 2         v   x n         Then  x 2       x n   x 1 O K  Hence  m   x 1 O K    iv     i    Let  m     O K for some      O K and let c   v        Then if v   x     0  x   m hence v   x     c  Thus  v   K         0   c        Since v   K     is a subgroup of            we deduce v   K             Suppose v is a discrete valuation on K       O K a uniformiser  For x   K    let n     such that v   x     n v        Then  u       n x   O K   and x   u   n  In particular   K   O K   1     and hence  K   Frac   O K     Definition 2 7   Discrete valuation ring    A ring R is called a discrete valuation ring   DVR  if it is a PID with exactly one non zero prime ideal  necessarily maximal    Lemma 2 8   i  Let v be a discrete valuation on K  Then  O K is a discrete valuation ring    ii  Let R be a discrete valuation ring  Then there exists a valuation on K     Frac   R   such that  R   O K   Proof   i  O K is a PID by Lemma 2 6  Hence any non zero prime ideal is maximal and hence  O K is a discrete valuation ring since it is a local ring    ii  Let R be a discrete valuation ring  with maximal ideal m  Then m         for some     R  Since PIDs are UFDs  we may write any x   R     0   uniquely as   n u with n   0  u   R    Then any y   K   can be written uniquely as   m u  with u   R    m      Define v     m u     m  check v is a valuation and  O K   R     Example      p    k     t      k a field  are discrete valuation rings