SEMINORMAL GRADED RINGS AND WEAKLY NORMAL PROJECTIVE VARIETIES

is paper is concerned with the seminormality of reduced graded rings and the weak normality of projective varieties. One motivation for this investigation is the study of the procedure of blowing up a non-weakly normal variety along its conductor ideal.

algebraically closed field of characteristic zero) is weakly normal if and only if every globally defined continuous function which is regular outside the singular locus is in fact globally regular.In [i] it is proven that a variety is weakly normal if and only if all of its local rings are seminormal.
In section 2 of this paper the relevant commutative algebra is developed to handle questions of weak normality for projective varieties and in particular to show that the weak normalization of a projective variety is again projective.We have tried to work in as general an algebraic setting as possible, but since we wanted the normalization of a graded ring to again be graded we considered only reduced graded rings (indexed by the integers) having only a finite number of min- imal prime ideals.Given an integral extension of graded rings we give an explicit description of the relative seminormalization by adapting a construction of Swan [2].We then show that the seminormalization of a graded ring is a graded subring of its normalization and that the operations of seminormalization and homogeneous localization commute.
In section 3 we consider reduced (but not necessarily irreducible) projective varieties defined over an algebraically closed field of characteristic zero.In particular we show that such a variety is weakly normal if and only if its veronese subrings of order d are seminormal for sufficiently large d.

SEMINORMAL GRADED RINGS.
All rings are assumed to be commutative with identity.A graded ring is a ring R that admits a decomposition (as abelian groups) R @ne_ R such that R R R n m n--m+n for all m,n.We let R h denote the set of homogeneous elements of R. Notice that the identity element of R is homogeneous of degree O.In particular R is a sub-O ring of R.
If A is a ring we let A denote its normalization, i.e.A is the integral closure of A in its total ring of quotients Q(A).
(2.0) Suppose that R is a reduced graded ring having only a finite number of minimal primes 0 I,''',0.Let R i R/0 i, Ti Rh 0i and S i TIR i for i i,-..,.Notice that R is a graded subring of Rlx XR Suppose r e R 0.(RI'''R ).
n n Then there exist homogeneous elements r. of degree n and s.
p. such that r r i + s i for i i,'--,.For each i let si(n) denote the homogeneous com- ponent of s i of degree n.Replacing r i by ri + si(n) and s.1 by s i si(n) we may assume that s. (n) 0 for all i.Comparing homogeneous components of 1 Jegree n we see that r i r I for all i and consequently s i s I for all i.
Thus s I e i 0 and r r I is homogeneous of degree n.

Now R
1 is a graded subring of S i (i i,---,) so that R I..-XR is a graded subring of S S Ix---S Consequently R is a graded subring of S LEMLA 2.1.Let R be a reduced graded ring having only a finite number of minimal primes.With notation as above, the normalization R of R is a graded subring of S If R 0 for all negative integers n then R 0 for all n n negative integers m.PROOF.We know that R R I---.__R with notation as in (2.0).By the domain case ([e], vol.2, Thm.Ii, p.157) R i is a graded subring of S i for each i.
Thus is a graded subring of S S Ix'..S .Now assume that R 0 for all n n < O. Again by the domain case (Ri) 0 for all n 0 (i l,--',g) so that n R 0 for all n O. n LA 2.2.Let R be a reduced graded ring having only a finite number of minimal primes.Assume that there exists a homogeneous R-regular element of positive degree.Let T denote the set of homogeneous R-regular elements of R.
Then is a graded subring of T-IR.R i PROOF.Let Pi' Ti be as in (2.0).Let ql'''''q denote the minimal primes of R indexed so that qiR O i (i i,-..).Now suppose R. By 2.1 there exist a i Ti such that ai B R + qi (i i,--',).For each i choose a homogeneous element b i _ q#i0j 0 i.Re- placing a i by aib i we may assume that a i E j#i0 j Pi and a. B1 R. Let a be a homogeneous R-regular element of positive degree and choose m so that deg(a i) + m 0 for i 1,''- Replacing a i by ai am we may assume that deg(a i) > 0 where ai E [j#iPj Pi and a.B g R for i 1 n i 1 Let N nl...n and let m i N/n. for i i,... Replacing a i by m i l a.
we may assume that deg(a i) N > 0 for i I,.-.,4.Then c al+.--+a 1 is homogeneous and cB R. To finish the proof it suffices to see that c is R-regular.
Suppose that c is a zero divisor in R so that c 0.
for some i.Now c a. + Z a and Z a 0 implies a i Oi, a contradiction Thus c is I j#i 1 ji j i R-regular and B g T-R.Hence __T-1R.
One can easily check that T-1R is a graded subring of S TIRI'..TIR E.
Since R is a graded subring of S by 2.1 we know that R is a graded subring of T-1R.
For a graded ring R On.ZRn and a positive integer d we let R(d) denote the subring of R defined by R(d) OnRnd" Then R(d) is a graded ring where R(d) R nd" If R is the homogeneous coordinate ring of a projective variety then R(d) is called a Veronese subring of R. PROOF.Let ql'''''q% denote the minimal primes of R and let 0i q i N R for i i,.--,.By 2.1 R is a graded ring and hence each qi is a homogeneous ideal.Since R is reduced we know that [.=lqi O.In particular there exist R such that qi (O:8i) for i l,---,g Let qi(d) qi R(d) for i I,--.,E.Then ql(d) --Nq(d) O.We claim that there are no containments amongst these primes.For suppose qi(d) qj (d).Let a be a homogeneous element of qi" Then adc qi(d) qj (d) im- plies a e qj.Since qi is generated by homogeneous elements qi -qj and we must have i j.Hence ql(d),...,qE(d) are the minimal primes of R(d).Similarly, oi(d),''',o(d) are the minimal primes of R(d).
We wish to see that Q . So it suffices to establish the result when R is an integral domain and R # R. R(e) R(fm) R(m) (f).Thus replacing R and R by R(m) and R(m), respec- tively, we may assume there exists a nonzero degree i homogeneous element Let 8 e R(d)n so that B is integral over R. Say 8 m + a iBm-l+ ''-+am_lS+am 0 where ai R (i l,'-',m).For each i let c. denote the homogeneous component of a i of degree ind.Then Bm + ClBm-l+'''+cm-IB+cm 0 and c l,.--,cme R(d) so that B is integral over R(d).We will now look at the question of seminormality for a reduced graded ring R as above.We first recall the relevant definitions.
For a ring A let J(A) denote the Jacobson radical of A and let A + the normalization of A. s s e R. We note that the same argument applies to any integral extension of rings R S such that S is a graded ring and R is a graded subring of S (we say that R__ S is an integral extension of graded rings).PROPOSITION 2.4.Let R S be an integral extension of graded rings.Then R is seminormal in S if and only if R contains each homogeneous element s of 2 3 S such that s s e R.
PROOF.One half of the assertion follows immediately from Hamann's criterion.
Thus it suffices to show that if R contains each homogeneous element of S whose square and cube is in R then R is seminormal in S.

+A {b
The ring A is said to be seminormal if A +A.There is also a relative notion.
For an integral extension of rings A B, the seminormalization + B A is defined by We say that A is seminormal in B if A + B A. We refer the reader to [5] for some of the fundamental results on seminormality.
Assume that this condition is valid but R is not seminormal in S. Thus there 2 3 exists an element s of S\R such that s s R. For an element s of S let (s) denote the number of nonzero homogeneous components of s.For each nonzero element s si(1)+'''+Si(m) of S such that 0 # si(j) Si(j) and i(1)<--'<i(m) let p(s) max{k si(1),''',Si(k_l) R}.In particular if p(s) 1 then the initial nonzero homogeneous component of s is not in R. Amongst those elements of S\R whose square and cube are in R choose an element s with (s) minimal and such that if t has the same properties then p(t) _< p(s).
COROLLARY 2.5.Let R be a reduced graded ring with a finite number of minimal primes.Then the seminormalization +R of R is a graded ring and contains R as a graded subring.
PROOF.By 2.1 the normalization R of R is graded and contains R as a graded subring.Let R'

Ine-r+Ri
By 2.4 R' is seminormal.But +R is the n smallest seminormal subring of that contains R and hence R' +R is a graded subring of R. Since R is a graded subring of R we know that R is graded sub- ring of +R.
In order to prove the analogue of 2.3 for the seminormalization of R we adapt a construction of Swan ([2], Thm.4.1) to describe +R.
Let R _ _ S be an integral extension of graded rings (so that R is a graded subring of S) and let T be a graded subring of S containing R. Let {s ale e be a well ordering of the set of all homogeneous elements of S T whose square and whose cube are in T. We inductively define graded subrings T of S such that Ta --c T8 whenever a _< 8. Let T_I T, Ta+I Ta[se] and for a limit ordinal let T 8 a<sTa.Set T' eATa so that T' is again a graded subring of S containing R. Now let R (0) R, R (n+l) R (n)' and set R* ">0 R(n) Note that R* is also a graded subring of S containing R.
PROPOSITION 2.6.Let R S be an integral extension of graded rings.With notation as above +S R L_v>nR(n) In particular R is a graded subring of S.
R* U nO R (n) PROOF.As observed earlier is a graded subring of S and the , seminormality of R in S follows from 2 4 Thus it suffices to see that R*c Let {sa a e A} be a well ordering of the set of all homogeneous elements of S T whose square and whose cube are in T. Define T as above.We show by transfinite induction that T c+ _ S R for all Now T_l T c * S R by assumption Suppose B > -1 and that T c R for all a < 8.If a + 1 then T8 Ta[s] where 2 3 s T. Since --., S R S we must have sa " P" and hence TB _ If is a limit ordinal then T B Ja<BTa _ +S R bY(n+lassumption.)Hence Ta cR for all a so that T' --JaCATa + But T' R so by induction R () s+R for all n and consequently sRo R* +S R. Since +S R is the smallest subring of S that contains R and is semi- , + normal in S we must have R S R. PROPOSITION 27 Let R be a reduced graded ring having only a finite number of minimal primes.For each positive integer d (+R) (d) is the seminormalization of R(d).
PROOF.Let R denote the normalization of R and let d be a positive integer.
Then R(d) is the normalization of R(d) by 2. Using the notation of 2.6 we have +R oR (n) Then 'p is a degree preserving iso- morphism of graded ring such that cp(B[X,X-I]) +R and @(A[X,X-I]) R. -i Thus C[X,X is normal and hence C is normal.Let T denote the set of homogeneous R-regular elements of R. By 2.2 we know that R is a graded subring of T-IR.So an element of C is of the form __r where r and s are homogeneous s elements of R of the same degree, s is R-regular and __r is integral over R. +R is seminormal if a e Q(B) and a a E B then a a +R Q(B) B. Thus B is seminormal by Hamann's criterion.Thus we need only show that B

Let n deg(r)
+A.We already have B -C _ _ Q(A) and B[X,X-I] is the seminormaliza- -i tion of A[X,X ].If PI and P2 are prime ideals in B lying over the same -i -i prime ideal of A then PIB[X,X P2B[X,X so that P1 P2" Let P be a prime ideal of B and let 0 P Q A. Then the canonical map (0) (P) must be an isomorphism since K(p) (X) K(P) (X) is an isomorphism.Hence B _ _ +A by Traverso's characterization of +A ([ ], Prop. 1.3).
We can show that normalization (respectively, seminormalization) and homogeneous localization commute.COROLLARY 2.9.Let R be a reduced graded ring having only a finite number of minimal primes and let f # 0 be homogeneous of positive degree d.Then Rf is the normalization and (+R)(f) is the seminormalization of R(f).
PROOF.R(f) is the degree 0 subring of Rf(d) and f Rf(d) is invertible and is homogeneous of degree i. (R) f(d) is the normalization of Rf(d) by 2.3   and (+R) f(d) is the seminormalization of Rf(d) by 2.7.So the assertion follows immediately from applying 2.8 to the reduced graded ring Rf(d). (Note that Rf(d) has only a finite number of minimal primes by the proof of 2.3.)RE,LARKS.D. F. Anderson [6] shows that if A is a seminormal integral domain and I an invertible fractional ideal then the Rees algebra R(1) I n is again seminormal, nO Let A be a reduced ring with a finite number of minimal primes and assume that A is seminormal.Let I be an ideal of A. It is natural to ask what conditions on I guarantee us that the Rees algebra R(1) (or the scheme Proj(R(1)) is again seminormal.We do not have any satisfactory answer to these questions at this time but we would like to make some observations.
(2.10) Since A is a reduced ring with finitely minimal primes the seminormal- ity of A implies the seminormality of the polynomial ring A[t] ( [5], Thin.In. (cf.[6] in case A is an integral domain.) (2.12) Say an ideal J of a ring A is pseudo-reduced if whenever a e A and It is well known that the normalization of a projective variety is again projec- tive.Most proofs assume that the original variety is irreducible (e.g.[8] Them.4, p. 400).The general case follows quickly from the irreducible case.We first take the normalization of each irreducible component and then construct the disjoint union of these normal projective varieties observing that this union can be embedded in some projective space.We would like to point out that if X C pn has irreducible components and > I then the homogeneous coordinate ring of the normalization of X cannot be realized as the normalization of some Veronese subring of the homogeneous coordinate ring R k[Xo, "',x n] of X.For the normalization of R (and of every Veronese subring of R) has as its degree 0 homogeneous piece I[=lk whereas the homogeneous coordinate ring of any projective variety has k as its degree 0 ILomogeneous piece.
We now show that the weak normalization of a projective variety is again pro- jective.We include a complete proof for the convenience of the reader.is the homogeneous coordinate ring of the Segre transform d(X) of X. Replacing X by d(X) we may assume that S is generated as a k-algebra by degree 1 homo- geneous elements.
Let Yo'''''YN be a k-basis for S I and let Y YN be indeterminates o Define a degree preserving map of graded k-algebras 'P:k[Yo,''',YN S by "(Yi) Yi for i O,''',N and let J ker P. Let y --]pN be the projective variety defined by J. Since R is a graded subring of S the elements x ,-.-,x are in o n S I. Let F ,---,F be linear forms whose images in S are x ,'--,x respectively o n o n and let L C]PN be the linear subspace defined by the ideal (Fo-'',Fn).Let p:N L .]pn be the projection defined by p(a) (Fo(a),.--,Fn(a)).Since R c S is an integral extension of graded rings Y L p (see [8], Proof of Thin.X.Let f R I be an R-regular element.Then Q(R(f)) Q(S(f)) are the function fields of -IX and Y, respectively, so that and V (U i) for i 0,''" n Then U and V are affine with affine i i i coordinate rings R(xi) and S(xi), respectively, and S(xi) is the seminormaliza- tion of R(xi) by 2.9.Thus (Vi, Vi is the weak normalization of U i for i 0 ,n and hence Y Li=oVi is weakly normal and (Y,n) is the weak normal- ization of X.
DEFINITION 3.3.We say that a projective variety X-pn is arithmetically weakly normal if its homogeneous coordinate ring with respect to this embedding is seminormal.
REMARK 3.4.Suppose X n is arithmetically weakly normal and let R denote its homogeneous coordinate ring with respect to this embedding.Then Rf is semi- normal for each 0 # f g P'l by 2.8.Since R(f) is the affine coordinate ring of the affine open subset Xf of X and these cover X (as f ranges through nonzero elements of R l) we see that X is weakly normal.
PROPOSITION 3.5.Let X C pn be a projective variety.Then X is weakly normal if and only if its Segre transforms Pd(X) are arithmetically weakly normal for d sufficiently large.
PROOF.Let R denote the homogeneous coordinate ring of X n.Suppose that X is weakly normal.Then Xf is weakly normal for each 0 # f a R 1 so that R(f) is seminormal for each 0 # f e R I. Thus Rf is seminormal for each 0 # f R 1 by 2.8.Hence Rf (+R)f for each 0 # f e R 1 and letting m R+ we know that Supp(+R/R) {mj.Since +R/R is a finitely generated graded R-module this implies that (+R/R) m 0 for m sufficiently large, i.e.R +R for m m m sufficiently large.Now R +R k by 3.1 so that R(d) (+R)(d) for d suf- o o ficiently large and the latter is seminormal by 2.2 and 2.4.Since R(d) is the homogeneous coordinate ring of the Segre transform d(X), the Segre transforms #d(X) are weakly normal for d sufficiently large.The converse is clear by 3.4 as d(X) is isomorphic to X. REMARK 3.6.We again remind the reader that the analogous statement for a reduc- ible normal variety is false.For if X n is a reducible normal projective var- iable with irreducible components and R is its homogeneous coordinate ring then Geometrically this is easy to see because R(d) is the affine coordinate ring of the union of cones through the origin in A n+l and hence is not normal.

COROLLARY 2 . 3 .
Let R be a reduced graded ring having only a finite number of minimal primes.Let R denote the normalization of R. For each positive integer d, R(d) is the normalization of R(d).
) is R(d)-regular.If is a zero divisor in R then d B i 0 for some i But 0 # Bde R(d) and Bi i O, a contradiction Hence every R(d)-regular element is R-regular and Q ) and let r R h {0}.Let J {n I(T-I) # 0}.Then J is an additive subgroup of and hence J n for some positive integer m.Let e icm(d,m) and set f e/m.Then R(d) R(e) R(fm) R(m)(f) and R(d) is contained in L[t,t-I] as a graded subring.Thus R(d) (d) L[td,t-d].We claim that L _ _ Q(R(d)).For if a,b e

2 t
Thus it suffices to see that R(n)(d) _ +(R(d)) for all n.Now R(O)(d) R(d) +(R(d)).Let n > 0 and assume that R(n)(d) +(R(d)).well ordering of the set of all homogeneous elements of R T whose square and whose cube are in T. Define Ta as above.We + need to show that Ta(d) (R(d)) for all a.Since T_I T we have T_l(d _ +(R(d)).Let B -i and suppose that + Ta(d) _ _ (B(d)) for all a < 8.If B a + 1 then T B Ta[s].Let s sa and suppose t T B (d) is homogeneous of degree .Then Irks where r e Ta h and deg(r k) d k.deg(s) for all k.Now (rk sk) (rksk)3 T(d) _ (R(d))and rksk (d) R(d) (by 2.3) implies that rksk +(R(d)) for each k and hence t +(R(d)).Since t was an arbitrary homogeneous element of T 8 (d) we have TB(d) _ +(R(d)).If is a limit ordinal then T B <Ta so that TB(d) --+(R(d)) by assumption.Hence Ta(d) +(R(d)) for all a and since R (n+l) aAT we also have R(n+l) (d) --+(R(d)).Thus R (n) (d) +(r(d)) for all n and (+R)(d) jnOR(n) (d) _ +(R(d)) as desired.Consequently + (R) (d) + r(d)).LEMMA 2.8.Let R be a reduced graded ring having only a finite number of min- is the normalization imal primes and suppose that te R I is invertible Then (R) and (+R) is the seminormalization of R O O PROOF To simplify notation let A R B (+R) and C + ()o Define a O O C-algebra map :C[X,X -I] by q0(X) t.
r_ is integral over A and hence A C _ _ .Since C is normal then a J.If A and I are as above when are all powers of I ps eudo-reduced that contains R we must have R +R.m

PROPOSITION 3 . 2 .
The weak normalization of a projective variety is a projective variety.n PROOF.Let x be a projective variety with homogeneous coordinate ring R k[Xo,'-',Xn]/I k[Xo'""x hI.Let S denote the seminormalization of R. Then S is a graded ring (2.5) and S k by 3.1.Hence there exists a positive integer o d such that S(d) is generated as a k-algebra, by degree 1 homogeneous elements ([8], Lemma, p. 403).Now S(d) is the seminormalization of R(d) by 2.7 and R(d) R(d)o o i=l k normalization of R(d) by 2.2 we know that R(d) is not normal for all d 0.
Then we may identify R with a graded subring of L[t t-1] t in T-IR.Let L (T-IR) Thus to finish the proof it suffices to see that R(d) is integrally closed -i e[t,t ].