The monoid of fractional ideals of a one-dimensional domain William Heinzer and dl in honor of Robert Gilmer and Joe Mott

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Notation G(R) = Q(R)*/U(R) F(R) = monoid of fractional ideals of R S(R) = F(R)/G(R) (I(R) = (invertible fractionals)/G(R) ) [F&S] A semigroup is “Clifford” iff it is a union of groups, i.e., iff, for all a , there is an x for which a = a2x .

Example: R = k[[x3,x4,x5]] G(R’) = G(k[[x]]) = Z G(R) = Z x ({a + bx + cx2 : a not 0} / k*) Prop: V = k + M , N = Mn , R = k + N : G(R) = Z x ({a0 + a1t + … + an-1tn-1 : a0 not 0}/ k*)

Example (ctnd): R = k[[x3,x4,x5]] S(R[x2]) = { R[x2] , R[x] } , both idempotent, so Clifford. S(R) = { R[x2] , R[x] , R, I = (x3,x4)R } , I2 isomorphic to M or R[x] but S(R) not is Clifford: IR[x] = R[x] , not I .

Example: R = k[[x3,x4]] G(R) = Z x ((U(k[x])/ (x6))/(U(k[x3,x4]/(x6)))) S(R) = { R , R[x5] , R[x2] , R[x], (x3,x4)R, (…???) } R + R(x + ax2) ? ( a = 0: (x3,x4)R )

Example: R = k[t(t-1),t2(t-1)] , localized G(R’) = Z2 G(R) = Z2 x ({ at + b(1-t) : a,b not 0} / k*) Prop: Vj = k + Mj birational DVRs, j = 1,…,n , M = intersection of Mj’s , R = k + M : G(R) = Zn x (U(R’/M)/k*) = Zn x (k* x … x k*)/k*

Conj: Vj birational rk 1 val rings, j = 1,…,n , D = intersection of Vj’s M = intersection of m(Vj)’s k common subfield of k(Vj)’s R pullback of k (diagonal) in D/M : G(R) = G(V1) x … x G(Vn) x (U(D/M)/k*)