William HEINZER, Craig HUNEKE and Judith D. SALLY*. A local ring ... is Rees' criterion, [Re], for a local ring to be analytically unramified, namely : a reduced ...
J . Math. Kyoto Univ. (JMKYAZ) 26-4 (1986) 667-671
A criterion for spots By William
H E IN Z E R ,
C raig
HUNEKE
a n d Ju d ith D. SALLY*
A lo cal ring (S, n) is a sp o t over a lo c a l rin g (R , m ) if S is the localization at a prime ideal o f a finitely generated rin g over R . I f R is a o n e dimensional an alytically unramified dom ain then, a s is w e ll k n o w n , e v e ry lo c a l domain birationally dominating R is a sp o t. T h e m a in result in this paper is a criterion f o r s p o ts w h ich g en eralizes th is classical resu lt. Onoda a n d Yoshida [0-Y] use similar techniques to give criteria for a subring of an affine domain over a field to be affine. Recall that a local domain (R , n i) is said to be analytically unramified (respectively analytically irreducible, respectively analytically n o r m a l) if its madic completion P is reduced (respectively a domain, respectively normal domain). (R , m ) is said to be quasi-unmixed if dim ension P/15 = dimension P f o r every m in im a l p r im e o f P . Recall also, [C , Theorem 1 ], that a local domain (R , ni) satisfies the dim ension inequality :
dim R ± tr. d. S: R
dim S ± tr. d. S/n : Rim
fo r all quasi-local domains (S, n) dominating R . I f equality holds, S is said to sa tisfy th e d im e n sio n equality w ith respect to R . I f every S w hich is a spot over R 13 , fo r some prim e ideal 43 o f R , satisfies th e d im e n sio n equality with re sp e c t t o R13, th en R is said to satisfy th e d im e n sio n fo rm u la . Ratliff [R] proved that R is quasi-unmixed if a n d only if R satisfies the dimension formula. Consequently, quasi-unmixedness im plies that all m axim al ideals in R ', th e integral closure o f R , h a v e th e sam e height. (W e w ill consistently u se th e notation R ' fo r th e integral closure o f a r in g R in its to ta l q u o tie n t rin g .) We will say th at a lo c a l domain (S, n) birationally dominates a lo c a l domain (R , n i) if R S , r tn R = m a n d S is contained in the quotient field of R. F o r th e proof o f th e theorem we rely o n two well-known results. The first is Rees' criterion, [R e ], fo r a lo c a l rin g to be analytically unramified, namely : a reduced lo c a l rin g (R , ni) is analytically unramified if a n d only if f o r every finitely generated rin g S= R [s,, ••• , s t ] inside th e to ta l q u o tie n t rin g o f R , the integral closure S ' i s a finitely generated S m o d u le . T h e se c o n d result is a local form o f Zariski's M a in Theorem [Z; c f . also [N (3 7 .4 )]] which, when combined with a result of C ohen [C, Theorem 3 ], states that if (R , in) is a Communicated by Prof. N agata, A ugust 9, 1985 * The authors are partially supported by grants from the NSF.
668
W illiam H ein z er, C raig H unek e and Judith D. S ally
dimensional, normal, analytically irreducible local domain a n d (S , n) is a d-dimensio n al lo c a l dom ain b iratio n ally dom inating R w i t h m S prim ary fo r n , then S= R. T h e sp o t criterion involves th e following property. D e fin itio n . A d-dimensional local dom ain (R , n i) h a s p r o p e r ty L , respect iv e ly L , i f every d-dim ensional n o rm a l s p o t b iratio n ally dom inating R is analytically irreducible, respectively analytically normal. A ll one dimensional local domains h a v e L, as do all excellent local domains. Lipman [L , Proposition, pg. 160] proved that a norm al spot birationally dominatin g a 2-dimensional analytically normal dom ain is analytically n o rm a l. Thus all 2-dimensional analytically norm al domains h a v e L . A n exam ple o f a 2-dimensional analytically unramified dom ain not having property L is given at the end o f th is paper. Theorem 1. L et (R , m) be a d-dimensional analytically u n ra m ified local domain w ith p ro p e rty L . L et (S , It) be a d-dim ensional local dom ain w hich b ira tio n a lly d o m in ate s R . I f S is norm al or quasi-unm ix ed, then S is a spot ov er R . P ro o f . W e w ill give th e proof for the case where S is quasi-unm ixed. The case w here S is norm al is analogous a n d e a s ie r . Set T = R [t i , ••• , trin n R ct
i
, ,1 ,],
w here t,, ••• , 4 is a s e t o f elements o f S w hich generate n. Since R is a n a ly tic a lly unram ified T ' is a finitely generated T -m odule. L e t O , ••• , Z k b e the m axim al ideals—all o f height d — in S E T 'lg S / E ach ch,= 0 ,( 1 T ' is a h e ig h t d m a x im a l id e a l o f T ' and (t 1 , • • , 4 )T / Ç A ,. T h u s q ‘S [T ']ia , is prim ary for the maximal ideal of S [T '] e and the local form o f ZMT im plies that T = S E T ' i . F ro m th is w e c o n c lu d e th a t th e re is a t m o st one CI, over each m axim al ideal i
t
t
o f T ' . It also follow s that S [ T '] = S ', fo r S [ T '] = r )S [ T '] 0 ., =TÇ,v , w here W=
L e t cr k + i , •••, ci b e t h e rem aining m a x im a l id e a ls o f T ' , a n d choose t E r
e fk + in • • • n q r \ U O i.
Then t
-1
is in te g ra l o v e r S . Say
••• +s,=0,
w ith s E S. i
S e t A = T [s i , ••• , sh],,nrE 3...,,,. Since A ' i s a finitely generated A-module, th e proof w ill be finished if w e s h o w t h a t S ' = A ' . F o r t h e n , A g S Ç S '= A ' w ill im p ly th a t S i s a finitely generated A-module and hence a sp o t over R. L e t q3 b e a m axim al ideal o f A '. Since 43nA c o n ta in s (t 1 , ••• , 4 )A , 13nT ' is m a x im a l. S in c e t l e A / , it f o llo w s th a t 43nT/= q 1 f o r some j with T h u s in A ' there a re e x a c tly k m a x im a l id e a ls q 3 ,, • • • , q 3 k a n d A ‘ , = T . so A '=S '. -
A criterion for spots
669
Since excellent local domains [M , Theorem 791 a n d 2-dimensional regular lo c a l rin g s satisfy L [L Proposition, pg. 1601, we obtain t h e following rather surprising corollaries. Corollary 1. L et (R , ni) be a 2-dim ensional re g u lar local ring. Ev ery 2dimensional quasi-unmixed local domain (S, n) birationally dom inating R is a spot over R. Corollary 2 . If (R , ni) is a d-dimensional ex cellent local dom ain, then any d-dimensional n o rm al o r quasi-unmixed local dom ain (S , n ) which birationally dom inates R is a spot over R. Corollary 3. Every d-dim enszonal norm al local dom ain (S , n ) w hich birationally dominates a d-dimensional excellent local dom ain (R , n i) is quasi-unmixed. Rem ark. Localization argum ents similar to those used in th e proof o f th e theorem can be used to show that if (R, tu) is a n analytically unramified local domain of dimension d such that every d-dimensional norm al local domain which birationally dominates R is a sp o t over R , then so is every d-dimensional quasiunmixed local domain with Noetherian integral closure which birationally dominates R. Zariski's analysis o f th e valuations birationally dominating a 2-dimensional regular local ring (R , ni) show s that there may be discrete valuation rings birationally dominating R which a re residually algebraic over R thus they a r e not spots over R . To say something about local rings (S, n) of dim ension less then d , we make th e following definition.
Definition. A local domain (R , ni) has property L k if every normal spot of dimension k birationally dominating R is analytically irreducible. Theorem 2 . L e t (R , n i) b e a d-dinz ensional analytically unranzified local domain with property L k. L e t ( S , n) be a quasi-unmixed local domain of dimension k w hich birationally dom inates R . I f S is dom inated by a quasi-local domain (W , p) w hich is of finite transcendence degree over R and satisfies the dimension equality w ith respect to R , then S is a spot over R. P ro o f . We have
dim R dim S ± tr. d. S / n R im = dim S ± tr. d. W/p : Rim — tr. d. W/p : S/tt = dim S ± dim R — dim W+ tr. d. W: R— tr. d. W/p : S/tt ^dim R
Hence, S satisfies the dim ension equality with respect to R a n d tr. d. S/rt: Rim =d — k . L e t s , — s _k be elements o f S which map onto a transcendence basis 1
d
W illiam Heinzer, Craig Huneke and Judith D. Sally
670 o f S /n over
R im
a n d se t T = R [s i , •" ,
T h e n R gT ç_S a n d , since T is a n analytically unramified local domain of dimensio n k w ith property L k , T heorem 1 m ay be applied to T a n d S. W e conclude with two examples. 1. C orollary 1 im plies that a 2-dimensional local dom ain birationally dom in a tin g a 2-dim ensional regular local ring is a sp o t precisely w hen it is quasiu n m ix e d . F o r th e r e g u la r lo c a l r in g R = k [x , A c s ,,,, w h e r e k i s a f ie ld , a N a g a ta non-quasi-unm ixed "bad exam ple" can be constructed to birationally dom inate R . L e t R = R [ y l x ] . L e t (V , N ) b e a D V R in k(x, y ) such that Nr1R =(x, y Ix ) a n d V IN is finite algebraic over R I(x , y lx ). L et (W , M )= (x, Y I x - 1 ) ) a n d s e t S = R + M r 1 N . S birationally dominates R but is n o t a sp o t over R a s it is not quasi-unmixed. -
-
-
2. T h is is a n exam ple of a 2-dim ensional analytically unram ified, normal lo c a l d o m a in th a t h a s a 2-dim ensional re g u la r lo c a l r in g which birationally dom inates it b u t is n o t a sp o t over it. T h e exam ple is E 7.1 o n p a g e 2 1 0 o f N a g a ta 's L o c al R in g s. David Shannon pointed out the existence of this example to us. T h e N a g a ta c o n stru c tio n is a s f o llo w s . K is a fie ld w ith c h a r K # 2 , x a n d y are indeterm inates, and w = a x i is a n element o f K [ [ x ] ] w h ic h is i
transcendental over I f ( x ) . If z1=z=(y+w ) and 2
zi+1=- [z — (y + ) ; a i x ))9 /x i,
th en N agata proves that R = K [x , y, z1, z2, •••],n, w h e re m is the ideal generated b y x, y, z , 2 , ••• , is a 2-dimensional regular local ring su c h th a t K [ [ x , y ] ] is t h e com pletion o f R . Moreover, z is a n irreducible elem ent o f R su c h th a t z facto rs in K [[x , y ]] a s a s q u a re , z = (y + w ) . It is sh o w n in [ N ] t h a t R [ X ] l (X — z) i s a 2-dimensional normal local domain which is analytically reducible. Note th a t R [X ]l ( X — z ).' R [y -l-w ]= R [w ]ç _ K [[x , y ]]. L e t w=w a n d w , = ( w — a x ') / x , a n d c o n sid e r V = K [x , w -1 ,w-2 , in i
2
2
2
2
1
• • •
i
w h e re n is th e ideal generated by x,w,,w ,••• . Since K[x,w1,••• is a 2-dimensional regular local ring for each j , V i s t h e u n i o n o f a strictly ascending sequence of 2-dimensional regular local rings w ith a common quotient field and hence is a valuation rin g [A , Lemma 17, p g. 3 4 6 ] . Since V is domin ated b y K [[x ]]n K ( x , w ) , V = K [[x ]]n K ( x , w ) , s o V is ra n k one discrete w ith maximal ideal generated by x . Let T V [y ]1 , 1. Then T is a 2-dimensional regular local ring w ith quotient field K (x , y , w ), m axim al id e a l (x, y)T, and coefficient field K . It follow s that I f [ [ x , y ] ] is th e completion o f T , and T = K [[x , y ]]n K ( x , y , w ) . Since R h a s c o m p le tio n K [[x , y ]], and quotient field K (x, y , z )E K (x , y , w ), w e h a v e R = K [L x , y E n K (x , y , z )= T n K (x , y , z). In particular, R g_T , so T is a birational extension of the 2-dim ensional normal 2
—
2
A criterion for spots
671
local domain R [iv ]. Since R [w ] is a n a ly tic a lly re d u c ib le a n d T is reg u lar, R [ w ] * T . It fo llo w s fro m Zariski's M ain Theorem a s form ulated by Peskine [P] and E vans [E ] th at T is n o t a sp o t over R [ w ] . For R [w ] i s normal, and x , y in R [w ] im plies that the extension of the m axim al ideal o f R [w ] to T is the m axim al ideal o f T. DEPARTM ENT O F MATHEMATICS PURDUE UNIVERSITY W E S T L A F A Y E T T E , IN D IA N A
47907
DEPARTMENT OF MATHEMATICS NORTHWESTERN UNIVERSITY EVANSTON, ILLINOIS 60201 References [ A ] S. S. A b h yan k ar, O n th e valuations centered in a local dom ain, A m er. J. Math., 78
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O . Z a r is k i, A sim ple analytical proof o f a fundamental property of birational transform ations, Proc. N a t. Acad. Sciences, 35 (1949), 62-66.
