0. A PREHISTOR Y O F TIGH T CLOSUR E 9 0.3. Le t R b e a regular loca l ring of positive characteristi c an d le t x G R, I C R, I a n ideal . Prov e tha t ( I : R x)M = I [q] : R x q fo r al l q = p e . 0.4. Le t R be a complete local domain containing a coefficient fiel d k. Le t x\,.., Xd be any system of parameters, and set A k[[xi,...,£*]], the complete subring of R generate d ove r k b y th e X{. A i s isomorphi c t o a powe r serie s rin g i n the Xi an d R i s module-finite ove r A. Choos e a maxima l A-hee submodul e of R, sa y G , an d a nonzer o elemen t c G A suc h tha t cR C G. Prov e tha t for al l t 1 , c((#i,...,#£.) # : x fc+i) S ( x i •••5 x fc)^' i- e - thes e ar e amiabl e parameters.
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