differential forms induces a homomorphism wâ¢/k | Ws/R -+ ~s/k of regular diffe- ... In this note we show that ~ always induces a morphism ~, which is an isomor-.
manuscripta
math.
65,
manuscripta mathematica
213 - 224 (1989)
9 Springer-Verlag1989
O N THE
TRANSITIVITY
OF REGULAR REINHOLD
DIFFERENTIAL
FORMS
Hi)BL
I f R / k resp. SIR are generically smooth algebras of finite type, equidimensional of relative dimension d resp. n, then the canonical isomorphism of meromorphic d n ,d+n differential forms induces a h o m o m o r p h i s m w•/k | Ws/R -+ ~s/k of regular differential forms, which is an isomorphism if R/k and SIR are Cohen-Macaulay or if R/k is Gorenstein.
In___troduction In [KW] E. Kunz and R. Waldi recently introduced the module w'R/~ of regular differential forms for a suitable algebra R/k. In ease R is an affine domain over a perfect ~eld k this construction was known for some time (c.f. [K1],[K2]), and it was used by a. Lipman to describe dualizing sheaves ([L1 ]). In the relative case a relative dualizing sheaf wI for a morphism f does not always exist, but for a proper m o r p h i s m f : X ~ Y of noetherian schemes which is equidimensionaI of dimension d one always has a d-dualizing pair ( f ! t f) ([Km],(4)). If in addition X and Y have no embedded components and if f is generically smooth (i.e. f is smooth at every point ~ 6 X which is a generic point of an irreducible component of X ) then it is conjectured that there is a canonical isomorphism
/ ( O y ) ~- ~ : / y and that ty(Oy) can be described in terms of traces and residues of differential forms.
Given a second morphism g: Y --r Z as above then (9 o f)! = f~ o g~ by [Km],(26)vii). Therefore, if both f and g are Cohen-Macaulay morphism, then dualizing sheaves exist, and it holds
~gof = Wf | f*~g
213
Hi)BL In view of the above conjecture this raises the following question: Let
R/k resp. SIR be generically smooth algebras of finite type, equidimen-
sional of dimension d resp. n, and assume that k, _~ and S have no embedded primes. Then when induces the canonical isomorphism d+n
of meromorphie differential forms an isomorphism d+n
In this note we show that ~ always induces a morphism ~, which is an isomorphism in either of the following two cases: i)
R/k and S/t:t are Cohen-Maeaulays algebras. (Globally this corresponds to
"full duality holds for f and g".) ii)
l:t/k is a Gorenstein-algebra. (In case n = d = 0 this is the classical "Schach-
teIungssatz" for Dedekind complementary modules; c.f. [La],III, w 1, Prop 5.) In general however ~ will not be an isomorphism as an example, due to B. Ulrich, shows. Preliminaries The theory of regular differential forms is developed in [KW],w 3,w 4. Since we only need regular differential forms of highest degree for generically smooth algebras we will include a brief description of the construction in this case. Before doing so we will fix our notation.
If R/k is an algebra we will denote by f~}~/k the universal differential algebra of lrl/k and by ft)~/k the universally finite differential algebra of R/k if it exists. If ft is a differential algebra of R we will denote by Ad(f/) the corresponding algebra of meromorphic differential forms, i . e . . M ( f t ) = fl |
K , where K = Q(R) is
the full ring of fractions of R. Furthermore if S / R is a finite generically eomptete intersection with Ass (R) = Min (_R) and Ass (S) = IVlin(S) we will write by abuse of notation
~w
~ 4 ( a S / D --* M ( a R / k )
instead of vQ(s)/Q(R) r for the trace of [KD], w 16. Finally, all rings in this paper are assumed to be noetherian and excellent. Let R / k be an algebra of finite type such that the following assumptions are met: i) Ass (k) = Min (k) and Ass (R) = Min (R).
214
HUBL
ii) R/k is equidimensional of dimension d in the sense of [KD], (B.~7). iii) R/k is generically smooth, i.e. if ~ E Min (R), then R/k is smooth at in the sense of [KD], w 8. Then for ~ E Spee (R) there exists a quasinormalization P
=
k[Xl,...,Xd] ~ Rf
in the neighborhood of ~3 by [KD], (B.20). By [KD], (C.23) R I / P is generically a complete intersection. Therefore if K = Q(P) and L = Q(RI) denotes the full rings of fractions we get a well defined trace map O'L/K: ~-~'LI k "-+ ~'~'KI k
by [KD], w 16. By Zaxiski's main theorem there exists a finite subalgebra T of
R f / P such that for ~ e Spec (R f) the canonical homomorphism T~n T -* (RI) ~ is an isomorphism. In particular it holds Q(T) = L. We set
wd/k = {w e ~tdL/k: O-L/IC(tw) e fldp/k for all t e T} and we define d
to be the Ry-submodule of f/d/k generated by ~ / k -
This construction does
not depend on the choice of T nor on the choice of the quasinormalization P . Therefore we get a global module wd/k C ~dL/k which we call module of regular d-differentials of R/k. If in the above situation R / / P is actually finite, then we have a canonical isomorphism
O-: JR,/k -~ i o m p ( n e , a ~ / k ) given by O-(~)(r) = O-L/~(r~)If ~ E S p e c ( R / ) , k~ = ~ • P , and if /~ resp. /5 denotes the completions of ( R I ) ~ resp. P a , then we can make a similar construction, using the trace k The module obtained this way will be denoted wRi ~ d k , the module of universally finite regular d~:lifferential forms of R/k. It satisfies d ~d ~nvlk - wR/k N ftdLik
More details and proofs can be found in [KW], w167 3 and 4.
215
HiJBL Transitivity of regular differential forms Let R/k be of finite type, generically smooth and equidimensional of dimension d, let S / R be of finite type, generically smooth and equidimensional of dimension n. Then S/k is equidimensional of dimension d + n. Assume that k, R and S have no embedded components. Then the various modules of regular differentials are defined. Furthermore, we have a canonical isomorphism
,~: M(a~/~) |
M(a}/R) -+ M(as/ke+")
THEOREM 1. 9 induces a well defined morphism , d-t-n
Paoos:
Clearly we get a well defined map
and it remains to show im (r
d.+n C ~"s/k
It suffices to show this after localization at a maximal ideal 9r[ of S, and then we also may replace R by Rr, and k by k~ where m = 9J[VIR and p = ffYtNk. Hence we may assume (k, p), (R, m) and
(S, ~ )
are local. Note that S / m S is
equidimensional of dimension n and that R / p R is equidimensional of dimension d. Furthermore all residue class field extensions are finite. After replacing k, R and S by their completions it suffices to show that the canonical map ~d r wR/k |
~n w~/R - , M ( f N~ d/ qk- n)
5;d+n n A//(f~d~-ff) by [KW] (4.9). Hence we has image in csq+~ because ~o~-ff = --s/k
-Slk
have reduced to the case that k, R and S are complete local rings with finite residue class field extensions. In this case we have noetherian normalizations, i.e. finite maps ~X = / ~ [ [ X l , - - - ,
Xd]]
"-~ R
and
R r = R[[Y,,..., Y,]] -~ s which induce a noetherian normalization
kx,r = t[[x~,..., x , , y~,..., G]] -* S
216
HUBL For all these extensions we have canonical traces of differential forms o n the level of meromorphic differential forms by [KD], w 16. In order to prove the theorem we have to show: If aJ~ ~ &d/a and if a;~ ~ ~5}/R, and if
denotes r
~ ) 1 " ~'02
| w~), then for all s ~ S
~'~d+n if we view
()dq-n as a submodule of ,~,~ ]tA[~'~dq-n ~kx.y/k kx.y/k)"~in the canonical way.
In this situation we also have a trace
-.
Since
34(f~R./R )
w2 E &~SiR we have that ~R/Ry(saJ2) C h~yiR, and therefore we can write aR/Ry(SW2) = fdY1.., dY,~ with f E R y
CLAIM:
wl 6~/Ry (saJ2) =col fdY1.., dye, where by abuse of notation dY/ denotes
both elements in A//(~)~y/R) and M ( ~ R y / k ).
PROOF case
OF T H E C L A I M :
In case n = 0 there is nothing to show, because in this
~k/Ry(Sa~2) = ~/Ry(SW2) by trace axiom T R 2([KD], w 16). Hence we may
assume that n > 0. By the compatibility of traces with base change, the image of
6w
in
Ad(~nRy/R) will be equal to #R/Ry(sa~2) = f d Y l . . , dYn, implying
that
8w
- f d Y l . . , dY,~ E ker (A4(h'Ry/k) --* dtd(h)~v/R) ) = ( { d r : r C R})
by [KD], (11.17). Hence we have
*w
-fay1,..dYn
~
n--]
= E dr~,~ (r, e R,,~ e M(aRy/k))
implying that
,,(~w
in
, RyIk; = -Aq(SaR/k) |
- fd~q ... dry)
= ~ld~,~
= 0
M(h~y/R) since co, drl = 0 in M(h'R/a) for all i,
and the claim follows.
217
HUBL As mentioned above ~R/Rr(sw2) = f d Y 1 . . , dY,, where f =
~
p,EN n
f~Y~' with
fu 6 R. Then we get, using the trace axioms of [KD], w 16 resp. [H1]:
~kS/kx,y(s~l~2) : Cr~g/kx, Y (~r~/Ry (S~d1022))
(by TR5 (transitivity))
= ~ y / k x . r (wl&k/Rv (sT2))
(by T1~1 (linearity))
= ~ k : / k x . r ( w l f d Y 1 . . . dY,~)
(by the claim)
= ~
?rkv/kx,v(fuwl)YUdY1...dY,,
(by TR1 (linearity))
?rk/kx(f,w')Y"dY'" . . d Y ,
(by TR3 (base change))
#EN
n
= E
c h~.x/k | because ~ / k x ( f ,
h"k x , Y / k X = h d+" kx,y/k
wx) 6 ~dx/k for all /z since Wl E ~d/k
[]
In general the map ~ will not be an isomorphism, as we will show later, but under additiona~ assumptions it sometimes is: THEOREM 2. If under the assumptions of theorem i both R / k and S / R are
Cohen-Macaulay a/gebras, then ~he canonical map ~: wdlk | is
an
n
,d+n
WSlR ~ ~slk
isomorphism.
PROOF: Since both w~/R and /vf(f~d/k ) are flat as R-modules (for the first statement see [H2], lemma), the canonical map
~/~ |
~ / ~ ~ M(~/~) o~ M(~/R)
is injective, and therefore it suffices to show that r is onto. As in the proof of theorem 1 we reduce to the case that k, R and S are complete local rings with finite residue class field extensions. Let X = { X ~ , . . . , X d } be a relative system of parameters for R / k , and let Y :
{Y1,...
,Yn} be
a relative system of parameters for
S/R.
Then R / ( X )
and S / ( X , Y ) are finite flat k-modules and S / ( Y ) is a finite fiat R-module. Furthermore we get canonical maps -d ~d fl : WR/k/(X)WR/k ~ Homk(R/(X), k)
f2 : &~/R/(Y)&~/R ~ HomR(S/(Y), R)
218
H(JBL and f3 : Ws/k ~d+n / ( X , V'~ %d-.i-n --~ Homk(S/(X, Y), k) -" )"~S/ k
given by fl(W + ( X ) ) ( r Jc (X)) = t ~ S k x / k
[Xl,...,XdJ
where ]CX = /r
Xd]] --4 n
is the canonical map induced by X, and where
Reskx/k
X1,...,Xd
: ftkx/k --* k
is the canonical residue symbol, i.e.
Reskx/k [ X 1 , . . . , X a (See [L2] resp.
= fo, the constant term o f f
[H~], w 6 for the ease of universally finite differential forms.)
Similarly one defines f2 and f3. CLAIM: fl, f2 and fa are isomorphisms. PROOF OF THE CLAIM (E.KUNZ): Since R / k is Cohen-Macaulay, R is finite and
free as a k x -module, and the trace 5~/kx on the level of meromorphic differential forms induces an isomorphism
a:
-4 H max(R,~2kx/~)
as was pointed out in the preliminaries. So if r l , . . . ,rtis a k x - b a s i s of R there exists a k x - b a s i s {wl,... ,wt} of &d/k dual to r ~ , . . . , rt with respect to 5n/kx (i.e. ~n/kx (riwj) = 5ijdX1 ... dXd). Therefore we get a nondegenerate pairing ~d
~d
given by /~x(, ~
+ ( x ) , ~ + (x)) = ReSkx/k X l , . . . ,Xd3
This proves the claim for f l , and one argues similarly for f2 and f3. We have a well defined map H : Hom~(]~/(X), k) |
HomR( S / ( Y ) , R) -+ Homk(S/(X, Y), k)
219
H/~iBL given by H ( f Q g ) ( s + ( X , Y ) )
= f(g(s+(Y))+(X)).
Since R / ( X ) and S / ( X , Y )
are finite and free as k-modules and S / ( Y ) is finite and free as an R-module, the map H is easily seen to be an isomorphism. Therefore we get a diagram
aJR/ki(X)wR/k |176 ~ d
g
~ d
oJSlk I(X,Y)Wslk ~ d+n
+
~s~| H o m k ( R / ( X ) , k) |
~ d+n
s~1 H
HomR(S/(Y), R) . . . .
> Homk(S/(X, Y), k)
in which fl | f2, f3 and H are isomorphisms, and in which g is the canonical map induced by ~o. To complete the proof of theorem 2 it suffices by Nakayama's lemma to show that this diagram commutes:
H(fl | f2)((cvl + (X)) | (we + (Y)))(s + ( X , Y ) )
=ReSkx/ki gr~Ikx(wl'ResRvlRk
1,...,Rv(S~2)])ln .I /
X1,..., Xd
J
= ReSkx,v /k [ #~Y /kx.y (Wl &~/RY ( sw2 ) ) ] X1,..,,Xd, Y1,...,Y, j (this is an immediate consequence from the claim in the proof of theorem l, and from [L2] (3.8))
=ReSkx,,/k
LX1,...,Xd,Y1,...,Y~] (by the transitivity of traces)
= f3(g((w~+(X))|
II
As a first consequence we prove a generalization of the "Sehachtelungssatz" of [KW] (4.32): COROLLARY i. If under the assumptions of theorem 1 R / k is a Gorensteinalgebra, then the canonica/map
is an isomorphism. PROOF: Since wd/k is a projective R - m o d u l e of rank 1 (by [KW], (4.!7)), the map %oobviously is injeetive, and it suffices to show that it is surjective. This can
220
HUBL be done locally in S and in R. In particular we may assume that wd/k has an R - b a s i s a;0 E w~/k. Let ~3 E Spec (S), and let
nx
=
nix,,...,
ss
be a quasinormalization in a neighborhood of ~ . We may assume that S I / R x is finite. Otherwise replace Sf by a finite subalgebra T / R x for which T~n T --+ S~ is an isomorphism for each ~ 6 Spec ( S I ) . It then obviously suffices to show the surjectivity for T instead of S f (c.f. [KW], w 4). By theorem 2 the canonical map o')d/k @ R O dnR x / R -'+ ojd+n Rx/k
is an isomorphism 9 Therefore ~nx/k ,d+~ is the submodule R x w o d X l ' . . . " dX,~ of d+n ). Now let w E wd+~sf/k and let qa E 3d(12s~/n n M(f~nx/k ) be a Q(Sf)-basis of
dw/(f~}1/k ) 9 Then as an element of M(as d+~/k) = M(a
/k) |
M ( i2n st/R)
we can
write w = w0 | gr]0 for some ~ E Q(Ss). To prove the surjectivity of qo it suffices to show that g~o E w}s/n, i.e. that aR%/nx(St~o) E f ~ x / n
for all s E S f . We
can write ~r~Hnx(Sgqo ) = g d X 1 . . . . , dXn with g E Q ( R x ) , and we have to show that g E R x . Since Wo |
E o.)d+ sj/kn we have by [KW] (4.28)
k ,d+n a%/nx(wo | sg~o) E ~'nx/k = RxwodX1 . . . . . dXn
hence by trace axiom (TR1) "linearity"
Wo~rw n x ( sg~o ) E Rxc~odX1 9 ... 9dX~ Arguing as in the proof of theorem 1 (claim) we obtain k a;oCr% /nx (sgqo) = gwodX1 "..." d X ,
implying that g E R x and completing the proof. I COROLLARY 2. Under the assumptions of theorem 1 let ~3 E Spec (S), let p = q3 MR and let m = ~3 Mk. Assume that one of the following two assumptions
is satisfled. a) ~ is a Cohen-Macaulay point of S I R and p is a Cohen-Macaulay point of b) p is a Corenstein point of R / k . Then the canonical map
is an isomorphism at ~3.
221
HUBL PROOF: First suppose that assumption a) is satisfied. If B / A is a noetherian algebra of finite type, then { ~ E Spec (B) : ~ is a Cohen-Macaulay point of B/A} C Spec(B) is open by EGA IV (12.1.7). Hence there exist f E R \ p and g E S \ ~ such that R f / k and S g f / R f are Cohen-Macaulay. The claim follows from theorem
2.
If assumption b) is satisfied, then we find f E R \ p such that Rifle is Gorenstein (by EGA IV, (12.1.7) and [KW] (4.17)b)). Now the claim follows from corollary 1. COROLLARY 3 ([HK], (1.24)). In the situation of theorem 2 Iet .~ E Spec(S), p = ~ nR,
m = ~Mk,
and denote by r the type o f a Cohen-Macaulayring.
Then ~(s~/~s~)
= ~(R~/mRp) . ~(S~/pS~)
In particular S ~ / m S ~ is Gorenstein if a~d only hr Rp/rnRp and S ~ / p S ~
are
Gorenstein. PROOF: Since S~/I[p is fiat we have by [KW] (4.16) and theorem 2:
r(Sv/mSv)=
t*svk S~/k~]
= #s~,( t,ons~,/n, |
d ~n,/k.) = #S~,( c.onS~/R,) '#n,(~~ d
= r ( S ~ / p S ~ ) , r(R~/mRp) It is easy to find examples of algebras R / k and S / R as in theorem 1 for which ~o is not an isomorphism. The following example, due to B. Ulrich, shows that this can even happen if k, R and S are normal, and if R / k and S/k are CohenMacaulay. The treatment presented here is due to R.Waldi. EXAMPLE: Let k = Q[[X 2, y2, Z~]] C_ R = Q[[X 2, y 2 Z ~, XY, X Z , YZ]] c S = Q[[X, Y, Z]]. Then k, R and S are normal Cohen-Macaulay domains.
Clearly the assumptions of theorem ! are satisfied
(with d = n = 0), and both R / k and S/k are finite Cohen-Macaulay algebras. Furthermore we have (1) ~ / k = ~-r-gs, 1 as follows from [KD](G.12). (2) r176 = S : Let K = Q(R), L = Q(S). Since CrL/K(X ) = erL/K(Y)-=
CrL/K(Z ) = 0 we get S C_ co~
On the other hand let ~ E wOs/R. Then we
can write ~ = a + bX with a,b E K , and 2a that a E S and bX E W~
=
O'L/K(g ) E R C S C OdOSIR,SO
Therefore 2bX 2 = aL/K(bX" X ) E R C S and
2bXY = aL/g(bX 9Y) E R C S. Since S is factorial this implies bX E S~ hence ~ES.
222
HIJBL
(3) OdO/k= ~--~'~ ( X , 1 Y,Z)R, sinceiteanbeseeneasilythat {88 42Y,1 42z,1 u 1 is the k-basis of
w~
complementary to
{1,XY, XZ, YZ} with respect to the
canonical trace (c.f.[KW], (4.6)d).
C+~ 0s/k, i.e. ~ is not surjective. Furthermore .R(x--~rns) = dimQ(ms/m}) = 9, whereas #R(W~ | w~ = 12, so that ~ is not injective either. Note that in this example a dualizing sheaf for S/R From this we get ira(p)
= ~1
rn s
does not exist. Here S is reflexive as an R-module by [EG], (3.6). Hence this example also shows that in [OZ], (2.3) the additional assumption
"WATB/4 is a reflexive A-mo-
dule" is needed (e.f. [KW], (4.34) and the following remarks). REMARK: Let (R, m) be a local Gorenstein domain which is essentially of finite type over a perfect field k, let Y = Spec (R), let X be an integral k-scheme, and let
f:X-~ Y be a proper morphism, generically smooth and equidimensional of dimension d. Then there exists a canonical map
: R f*~x/y --+ Oy such that for all quasi-coherent Ox-modules ~ the f . O x - h o m o m o r p h i s m
&.:/.Hemox (~-, ~o~/y) -~ Homoy (Ref.~ -, Oy) given by 5~(a) =
f/oRdf.(c~) is an isomorphism.
Using corollary 2 this follows easily from ILl ] (5.1) (with the help of EGA IV (8.9.1) and (8.10.5), Chevalley's theorem (EGA IV (13.1.3)) and [Ha], III ex. 8.3). It should be noted as this point that P. Sastry [S] has a more funetorial approach to relative duality, at least in case R is regular. He uses techniques similar to those in [L1] and he also ties in Lipman's residue theory [L2], thus giving an explicit description of the morphism
ff.
REFERENCES: [EG] Evans, G.; Griffith,P.: Syzygies. L M S Lecture Notes 106. Cambridge University Press, Cambridge 1985 [EGA IV] Grothendieck, A.; Dieudonn6, J.: EMments de g6ometrie alg6brique. Publ. Math. ruES 20(1964), 24(1965), 28(1966), 32(1967) [Ha] Hartshorne,R.: Algebraic Geometry, Springer, Berlin, Heidelberg, New York 1977 [KH] Herzog, J.; Kunz, E. et al.: Der imaonische Modul eines CohenMacaulay-Rings. Springer Lecture Notes in Mathematics 238, 1971
223
HUBL
[H1]
Hfibl, R.: Traces of Differential Forms and Hochschild Homology. Springer Lecture Notes in Mathematics 1368~ 1989 Hiibl, R.: A Note on the Torsion of Differential Forms. To appear in: Archiv der Mathematik Kleiman, S.: Relative Duality for Quasi-coherent Sheaves. CompoIt{m] sitio Math. 41 (1980), 39-60 [KD] Kunz, E.: K~hler Differentials. Advanced Lectures in Math. Vieweg, Wiesbaden, 1986 Kunz, E.: Holomorphe Differentialformen auf algebraischen Variet~iten [KI] mit Singularit~ten I. Manuscripta Math. 15 (1975), 91-108 Kunz, E.: Holomorphe Differentialformen auf algebraischen Variet~ten [K21 mit Singularitgten IL Abh. Math. Sere. Univ. Hamburg 47 (1978), 4270 [KW] Kunz, E.; Waldi, R.: Regular Differential Forms. Contemporary Mathematics 79, Amer. Math. Soc. Providence, 1988 [La] Lang, S.: Algebraic Nmnber Theory. Addison-Wesley, Reading, 1970 Lipman, J.: Dualizing Sheaves, Differentials and Residues on AliLl] gebraic Varieties. Asterisque 117, 1984 Lipman, J.: Residues and Traces of Differential Forms via Hochschild Homology. Contemporary Mathematics 61, Amer. Math. Soc., Providence, 1986 Oneta, A.; Zatini, E,: A Note on Complementary Modules, Duality [oz] and Reflexiveness. Comm. in Alg. 12, 2631-2641 (1984) Sastry, P.: In preparation is]
Reinhold Hiibl Department of Mathematics Purdue University West Lafayette, IN 47907 USA
Present Address: Fakultgt ffir Mathematik Universit~t Regensburg Universit~tsstrai~e 31 D-8400 Regensburg
(Received A p r i l 24, 1989; in r e v i s e d form July 12, 1989)
224
