A Note on Residual Variables of an Affine Fibration
aa r X i v : . [ m a t h . A C ] F e b A Note on Residual Variables of an A ffi ne Fibration Prosenjit Das
Department of Mathematics, Indian Institute of Space Science and Technology,Valiamala P.O., Trivandrum 695 547, Indiaemail: [email protected]
Amartya K. Dutta
Stat-Math Unit, Indian Statistical Institute,203, B.T. Road, Kolkata 700 108, Indiaemail: [email protected]
Abstract
In a recent paper [El 13], M.E. Kahoui has shown that if R is a polynomial ring over C , A an A -fibration over R ,and W a residual variable of A then A is stably polynomial over R [ W ]. In this article we show that the above resultholds over any Noetherian domain R provided the module of di ff erentials Ω R ( A ) of the a ffi ne fibration A (which isnecessarily a projective A -module by a theorem of Asanuma) is a stably free A -module. Keywords: Residual variable; Stably polynomial algebra; A n -fibrations, Module of di ff erentials. Mathematics Subject classifications (2010) : Primary 14R25; Secondary 13B25, 13F20
1. Introduction
Let R denote a Noetherian integral domain. A finitely generated flat R -algebra A is said to be an A n -fibration over R if A ⊗ R k ( P ) is a polynomial ring in n variables over k ( P ) for every prime ideal P of R . The concept of residualvariable was defined by S.M. Bhatwadekar and A.K. Dutta in [BD93] as an element W in the polynomial algebra R [ X , Y ] for which R [ X , Y ] ⊗ R k ( P ) is a polynomial algebra in one variable over R [ W ] ⊗ R k ( P ) for every prime ideal P of R . The following statement is a part of their main result ([BD93], Theorem 3.1) on residual variables: Theorem 1.1.
Let R be a Noetherain domain and W an element of R [ X , Y ] . Then the following are equivalent: (1) W is a residual variable in R [ X , Y ] . (2) R [ X , Y ] is a stably polynomial algebra over R [ W ] , i.e., R [ X , Y ][ Z , Z , · · · , Z r ] = R [ W ][ T , T , · · · , T r ] for someindeterminates Z i ’s, T i ’s over R. They also observed ([BD93], Remark 3.4) that an analogous result holds for a system of m algebraically indepen-dent elements W , W , · · · , W m in the polynomial ring R [ X , X , · · · , X m + ]. Recently El Kahoui [El 13] has extendedthe concept of residual variable of a polynomial ring to that of an A n -fibration: he defines an element W of an A n -fibration A over a ring R to be a residual variable of A over R if A ⊗ R k ( P ) is a polynomial algebra in n − R [ W ] ⊗ R k ( P ) for each prime ideal P of R . He shows that when n = R is a polynomial algebraover C , then the extended concept satisfies the following analogue of Theorem 1.1 ([El 13], Theorem 3.4): Theorem 1.2.
Let R be a finite-dimensional polynomial algebra over C , A an A -fibration over R and W an elementof A. Then the following are equivalent: (1) W is a residual variable of A over R. (2) A [ Z , Z , · · · , Z r ] = R [ W ][ T , T , · · · , T r , T r + ] for some for some indeterminates Z i ’s, T i ’s over R. Preprint by Prosenjit February 7, 2020 ahoui also observes in [El 13] that if the hypothesis “ R is a polynomial algebra over C ” is weakened to “ R is anydomain containing Q ”, then the conclusion (2) implies (1) still holds in Theorem 1.2. He remarks ([El 13], Pg. 39)that it is not known whether the converse also holds. Example 3.14 of our paper shows that the converse does not holdin general even when R is a regular factorial a ffi ne domain. However, using the techniques of Bhatwadekar-Dutta,we shall show that the converse (i.e., (1) = ⇒ (2) of Theorem 1.2) indeed holds over any Noetherian domain whenthe module of di ff erentials Ω R ( A ) is stably free over A . More generally, we shall prove (Corollary 3.6, Theorem 3.13,Proposition 3.16): Main Theorem.
Let R be a Noetherian domain and A an A n -fibration over R . Let B = R [ W , W , · · · , W m ] be asubring of A such that A ⊗ R k ( P ) is a polynomial algebra in n − m variables over B ⊗ R k ( P ) for every prime ideal P of R . Then A is an A n − m -fibration over B . Moreover, the following statements are equivalent:(1) Ω R ( A ) is stably free over A .(2) A is a stably polynomial algebra over B .In [BD93], Bhatwadekar-Dutta also observed the following result ([BD93], Theorem 3.2): Theorem 1.3.
Let R be a Noetherian domain such that either R contains Q or R is seminormal. Then an element Wof R [ X , Y ] is a residual variable of R [ X , Y ] over R if and only if W is a variable of R [ X , Y ] . We shall see that Theorem 1.3 holds if we replace the polynomial ring by an A -fibration whose module ofdi ff erentials is stably free (Corollary 3.19).
2. Preliminaries
Throughout the article rings will be commutative with unity. For a ring R , R [ n ] will denote the polynomial ring in n variables over R . We shall use the notation A = R [ n ] to mean that A is isomorphic, as an R -algebra, to a polynomialring in n variables over R .For a prime ideal P of R , k ( P ) will denote the residue field R P / PR P . A finitely generated flat R -algebra A is saidto be an A n -fibration over R if A ⊗ R k ( P ) = k ( P ) [ n ] for every prime ideal P of R .Note that several algebraic geometers use the term “ A n -fibration” to mean that A / m A (cid:27) ( R / m R ) [ n ] for almost allmaximal ideals m of R . We emphasise that we shall use it as defined by Sathaye in [Sat83], where the hypothesis ismade on all fibre rings.An R -algebra A is said to be stably polynomial algebra over an R -subalgebra B of A if there exist indeterminates Z , Z , · · · , Z r over A and indeterminates T , T , · · · , T s over B such that A [ Z , Z , · · · , Z r ] = B [ T , T , · · · , T s ] (as B -algebras). We state below an elementary observation on stably polynomial algebras. Lemma 2.1.
Let A be a stably polynomial algebra over R. Then Ω R ( A ) is a stably free A-module.Proof. Set D : = A [ m ] = R [ n + m ] for some m , n and M : = Ω R ( A ) ⊕ A m . Then M ⊗ A D (cid:27) ( Ω R ( A ) ⊗ A D ) ⊕ D m (cid:27) Ω R ( D ) (cid:27) D n + m (cf. [Mat80], Example 26.J, Pg. 189). Thus M ⊗ A D is a free D -module. Since A is a retract of D = A [ m ] , itfollows that M is a free A -module. Thus Ω R ( A ) is stably free over A .The following structure theorem on a ffi ne fibrations is due to T. Asanuma ([Asa87], Theorem 3.4): Theorem 2.2.
Let R be a Noetherian ring and A an A r -fibration over R. Then Ω R ( A ) is a projective A-moduleof rank r and A is an R-subalgebra (up to an isomorphism) of a polynomial ring R [ m ] for some m such that A [ m ] = Sym R [ m ] ( Ω R ( A ) ⊗ A R [ m ] ) (as R-algebras). In particular, if Ω R ( A ) is a stably free A-module, then A is a stably polynomialalgebra over R. We shall use the following result by Quillen-Suslin ([Qui76], [Sus76])
Theorem 2.3.
If R is a PID, then any finitely generated projective R [ n ] -module is free.
2e record the following result on cancellation by Hamann ([Ham75], Theorem 2.6 and Theorem 2.8).
Theorem 2.4.
Let R be a Noetherian ring such that either R contains Q or R red is seminormal. Then R [1] is R-invariant, i.e., if an R-algebra A is such that A [ m ] = R [ m + , then A = R [1] . The following result was first proved by Kambayashi-Miyanishi in ([KM78], Theorem 1). Since any rank-oneprojective module over a factorial domain is free, this result can now also be seen to follow from Theorem 2.2 andTheorem 2.4. A more general version of the result is given in ([Dut95], Theorem 3.4).
Theorem 2.5.
Let R be a Noetherian factorial domain and A an A -fibration over R. Then A = R [1] .
3. Main TheoremDefinition 3.1.
Let R be a ring, A an R -algebra, n ∈ N and W : = ( W , W , · · · , W m ) an m -tuple of elements in A whichare algebraically independent over R such that A ⊗ R k ( P ) = ( R [ W ] ⊗ R k ( P )) [ n − m ] for all P ∈ Spec( R ). We shall callsuch an m -tuple W to be an m -tuple residual variable of A over R . Remark 3.2. ([BD93], Example 4.1) provides an example of a residual variable in a polynomial ring which is not avariable. For an example of a residual variable in an a ffi ne fibration which is not a polynomial ring, see Example 3.14or Remark 3.15.We first observe an elementary result. Lemma 3.3.
Let R be a ring, A an R-algebra, B , B R-subalgebras of A and B = B ⊗ R B . Suppose that A ⊗ R k ( P ) = B ⊗ R k ( P ) [ n ] for all P ∈ Spec(R). Then A ⊗ B k ( Q ) = B ⊗ B k ( Q ) [ n ] for all Q ∈ Spec(B ).Proof. Fix Q ∈ Spec( B ) and set P : = Q ∩ R ∈ Spec( R ). Set B : = B ⊗ R k ( P ). Then B ⊗ B B = B ⊗ R k ( P ) and A ⊗ B B = A ⊗ R k ( P ) so that A ⊗ B k ( Q ) = A ⊗ B B ⊗ B k ( Q ) = A ⊗ R k ( P ) ⊗ B k ( Q ) = ( B ⊗ R k ( P ) ⊗ B k ( Q )) [ n ] = ( B ⊗ B B ⊗ B k ( Q )) [ n ] = ( B ⊗ B k ( Q )) [ n ] .As a consequence, we have Remark 3.4.
If ( U , V ) : = ( U , U , · · · , U s , V , V , · · · , V t ) is an ( s + t )-tuple residual variable of A over R , then V isa t -tuple residual variable over R [ U ].Next we record a result on flatness. Lemma 3.5.
Let R ⊂ B ⊂ A be Noetherian rings such that (i)
A and B are flat over R. (ii) A ⊗ R k ( P ) is flat over B ⊗ R k ( P ) for all P ∈ Spec(R).Then A is flat over B.Proof.
We shall show that A Q is flat over B Q ∩ B for all Q ∈ Spec( A ). Fix Q ∈ Spec( A ) and set P ′ : = Q ∩ B ∈ Spec( B )and P = P ′ ∩ R ∈ Spec( R ). Then we have local homomorphisms R P −→ B P ′ −→ A Q . As B P ′ is flat over R P , to showthat A Q is flat over B P ′ , it is enough to show that A Q ⊗ R P k ( P ) is flat over B P ′ ⊗ R P k ( P ) (cf. [Mat80], 20.G, Pg. 152).Since A ⊗ R k ( P ) is flat over B ⊗ R k ( P ) and Q ∩ R = P , we see that A Q ⊗ R P k ( P ) is flat over B ⊗ R k ( P ) and hence( A Q ⊗ R P k ( P )) ⊗ B B P ′ is flat over ( B ⊗ R k ( P )) ⊗ B B P ′ . Now( A Q ⊗ R P k ( P )) ⊗ B B P ′ = ( A Q ⊗ B B P ′ ) ⊗ R P k ( P ) = ( A Q ⊗ B P ′ B P ′ ) ⊗ R P k ( P ) = A Q ⊗ R P k ( P )and( B ⊗ R k ( P )) ⊗ B B P ′ = B P ′ ⊗ R k ( P ) = B P ′ ⊗ R P k ( P ).This shows that A Q ⊗ R P k ( P ) is flat over B P ′ ⊗ R P k ( P ) and hence A Q is a flat B P ′ -algebra. Thus A is a flat B -algebra. 3rom Lemma 3.3 and Lemma 3.5 it follows that if W is an m -tuple residual variable of an A n -fibration A over aring R , then A is an A n − m -fibration over R [ W ]. More generally, we have: Corollary 3.6.
Let R ⊂ B ⊂ A be Noetherian rings such that A is an A n -fibration over R and B an A m -fibration overR with A ⊗ R k ( P ) = B ⊗ R k ( P ) [ n − m ] for all P ∈ Spec(R). Then A is an A n − m -fibration over B. Remark 3.7.
Let A be an A -fibration over R and W ∈ A . It was shown in ([El 13], Proposition 3.2) that if Q ֒ → R and A an A -fibration over R [ W ], then W is a residual variable of A over R . The converse was also proved for thecase R is a regular a ffi ne domain over C ([El 13], Theorem 3.3). Corollary 3.6 shows that the converse holds for any Noetherian domain.As a consequence of Corollary 3.6 and Theorem 2.5, we see that in an A m + -fibration over a Noetherian factorialdomain, any m -tuple residual variable is necessarily a variable. Corollary 3.8.
Let R be a Noetherian factorial domain and A an A m + -fibration over R. Then an m-tuple W of A isan m-tuple residual variable of A over R if and only if A = R [ W ] [1] = R [ m + . Remark 3.9.
Note that Corollary 3.8 need not hold for an m -tuple residual variable W of an A n -fibration A over R when n − m >
1. Example 3.14 shows that A may not be even a stably polynomial algebra over R [ W ]. The next resultshows that if R is a polynomial algebra over a PID, then A happens to be a stably polynomial algebra over R [ W ]. Corollary 3.10.
Let R be a finite-dimensional polynomial algebra over a PID, A an A n -fibration over R and W anm-tuple residual variable of A over R. Then A is a stably polynomial algebra over R [ W ] .Proof. By Corollary 3.6, A is an A n − m -fibration over R [ W ] and hence by ([Asa87], Corollary 3.5), A [ r ] = Sym R [ W ] ( M )for some r ∈ N where M is a finitely generated projective R [ W ] module of rank n − m + r . Since R is a polynomialalgebra over a PID, by Theorem 2.3, we get that A [ r ] = R [ W ] [ n − m + r ] , i.e., A is a stably polynomial algebra over R [ W ]. Remark 3.11. (1) Corollary 3.8 shows that if n − m =
1, then we have A = R [ W ] [1] in Corollary 3.10. However, if n − m >
1, thenan example of Asanuma ([Asa87], Theorem 5.1) shows that we need not have A = R [ W ] [ n − m ] even in the case R is aPID, m = n =
3. In Asanuma’s example, Q ֒ → / R . When Q ֒ → R , it is not known whether, in Corollary 3.10, onecan conclude that A = R [ W ] [ n − m ] even in the case m = n =
3. For instance in ([BD94], Example 4.13), W is aresidual variable in A = R [ X , Y , Z ], where R is a DVR containing Q , and it is not known whether A = R [ W ] [2] .(2) The proof of Corollary 3.10 shows that the hypothesis that R is “a finite-dimensional polynomial algebra over aPID” can be replaced by the condition that R is “a regular ring with trivial Grothendieck group”.The following observation on module of di ff erentials of an A n -fibration having residual variables is crucial for ourmain theorem. Lemma 3.12.
Let R be a Noetherian ring and A an A m + k -fibration over R. If W is an m-tuple residual variable of Aover R, then Ω R ( A ) = Ω R [ W ] ( A ) ⊕ A m . In particular, Ω R ( A ) is a stably free A-module if and only if Ω R [ W ] ( A ) is a stablyfree A-module.Proof. By Corollary 3.6, A is an A k -fibration over R [ W ] and hence by Theorem 2.2, A is an R [ W ]-subalgebra of apolynomial algebra B over R [ W ] and Ω R [ W ] ( A ) is a projective A -module of rank k . Since R ֒ → R [ W ] ֒ → A ֒ → B , andsince for any A -module M , every R -derivation d : R [ W ] −→ M , can be extended to an R -derivation ˜ d | A : A −→ M where ˜ d | A is the restriction of an extension ˜ d : B −→ M of d , we have the following split short exact sequence([Mat80], Theorem 57, p186):0 −→ A ⊗ R [ W ] Ω R ( R [ W ]) −→ Ω R ( A ) −→ Ω R [ W ] ( A ) −→ Ω R ( A ) = Ω R [ W ] ( A ) ⊕ A ⊗ R [ W ] Ω R ( R [ W ]) = Ω R [ W ] ( A ) ⊕ A m .4e now prove our main result. Theorem 3.13.
Let R be a Noetherian ring and A an A n -fibration over R such that Ω R ( A ) is a stably free A-module.Suppose W is an m-tuple residual variable of A over R. Then A is a stably polynomial algebra over R [ W ] ; specifically,A [ ℓ ] = R [ W ] [ n − m + ℓ ] for some ℓ ∈ N .Proof. By Corollary 3.6, A is an A n − m -fibration over R [ W ] and hence, by Lemma 3.12, Ω R [ W ] ( A ) is a stably free A -module. Therefore, we get the result by Theorem 2.2.The following example shows the necessity of the assumption “ Ω R ( A ) is a stably free A -module” in Theorem 3.13even when R is a regular factorial a ffi ne domain over the field of real numbers. Example 3.14.
Let R = R [ X , Y , Z ] / ( X + Y + Z − K ( R ), the Grothendieck group of R ,is non-trivial (in fact, it is Z ⊕ Z / (2)); in particular, there exists a finitely generated projective R -module M of rank2 which is not stably free. Let W be an indeterminate over R and A = Sym R ( M ⊕ RW ). Then A P = R P [ W ] [2] for all P ∈ Spec( R ) so that A is an A -fibration over R with W as a residual variable. If A [ ℓ ] = R [ W ] [ ℓ + , then we would haveSym R ( M ⊕ RW ⊕ R ℓ ) (cid:27) Sym R ( RW ⊕ R ℓ + ) and hence, by ([EH73], Lemma 1.3), we would have M ⊕ RW ⊕ R ℓ (cid:27) RW ⊕ R ℓ + contradicting that M is not stably free. Remark 3.15.
When R is a Noetherian factorial domain and m = n − Ω R ( A ) is stably free” may be dropped from Theorem 3.13; in fact, in this case, A = R [ W ] [1] . But even over a (non-factorial) Dedekind domain and even for n = m =
1, an A -fibration need not be stably polynomial over R [ W ]when W is a residual variable of A . For instance, choose a non-principal ideal I of the Dedekind domain R and set A = Sym R ( I ⊕ RW ). As in Example 3.14, A is an A -fibration over R , W is a residual variable of A but A is not stablypolynomial over R [ W ].The following result gives a converse of Theorem 3.13. Proposition 3.16.
Let R be a Noetherian ring and A an A n -fibration over R. Suppose that there exists an m-tupleresidual variable W of A over R such that A is a stably polynomial algebra over R [ W ] . Then Ω R ( A ) is a stably freeA-module.Proof. By Lemma 3.12, it su ffi cies to show that Ω R [ W ] ( A ) is a stably free A -module. This follows from Lemma2.1.For convenience, we state below an easy result. Lemma 3.17.
Let R ֒ → B ֒ → A be integral domains such that A is an A n -fibration over B. Then A ⊗ R k ( P ) is an A n -fibration over B ⊗ R k ( P ) for every P ∈ Spec(R) . Moreover, if B = R [ m ] , then A ⊗ R k ( P ) is a stably polynomialalgebra over B ⊗ R k ( P ) for each P ∈ Spec(R).Proof.
Let P ∈ Spec( R ), A = A ⊗ R k ( P ) and B = B ⊗ R k ( P ). Since A is a finitely generated flat B -algebra, clearly A isa finitely generated flat B -algebra. Note that A = A ⊗ R k ( P ) = A ⊗ B B ⊗ R k ( P ) = A ⊗ B B . Let Q ∈ Spec( B ) and let Q be the prime ideal of B for which QB = Q . Then Q ∩ R = P . Now k ( Q ) is the field of fractions of B / Q and hence of B P / QB P (cid:27) B / Q ; thus k ( Q ) = k ( Q ). Hence, A ⊗ B k ( Q ) = A ⊗ B B ⊗ B k ( Q ) = A ⊗ B k ( Q ) = k ( Q ) [ n ] = k ( Q ) [ n ] . Corollary 3.18.
Let R be a Noetherian domain and A an A m + -fibration over R. Then an m-tuple W of A is an m-tupleresidual variable of A over R if and only if A is an A -fibration over R [ W ] .Proof. Follows from Corollary 3.6, Lemma 3.17 and Corollary 3.8.Finally, we give a condition for an m -tuple residual variable in an A m + -fibration to be a variable. Corollary 3.19.
Let R be a Noetherian domain and A an A m + -fibration over R such that Ω R ( A ) is a stably freeA-module. Suppose that either R contains Q or R is seminormal. Then for an m-tuple W of A, the following areequivalent: (I) W is an m-tuple residual variable of A over R. (II) A = R [ W ] [1] = R [ m + .Proof. Follows from Theorem 3.13 and Theorem 2.4. 5 . Appendix
Theorem 3.13 can be slightly generalised when envisaged as a statement on “tensor product decomposition”discussed in [BW77]. Let R ⊂ B ⊂ A be Noetherian rings with A an A n -fibration over R and also an A n − m -fibrationover B . Theorem 3.13 shows that if B = R [ m ] and Ω R ( A ) is stably free then A [ ℓ ] = B [ ℓ ] ⊗ R [ ℓ ] C , where C = R [ n − m + ℓ ] .One can see below (Proposition 4.2) that even when B is only an A m -fibration over R , it is stably a factor in a tensorproduct decomposition of A , even without the hypothesis that Ω R ( A ) is stably free. We first note that the proof ofLemma 3.12 can be seen to yield the following general version: Lemma 4.1.
Let R ⊂ B ⊂ A be Noetherian rings such that A is an A n -fibration over R and B an A m -fibration over Rwith A ⊗ R k ( P ) = B ⊗ R k ( P ) [ n − m ] for all P ∈ Spec(R). Then Ω R ( A ) = Ω B ( A ) ⊕ ( Ω R ( B ) ⊗ B A ) . As a consequence we have
Proposition 4.2.
Under the hypothesis of Lemma 4.1, there exists ℓ ≥ such that A [ ℓ ] = C ⊗ R [ ℓ ] B [ ℓ ] for someR [ ℓ ] -algebra C.Proof. Since A and B are a ffi ne fibrations over R , by Theorem 2.2, we can choose a su ffi ciently large positive integer ℓ such that A [ ℓ ] = Sym R [ ℓ ] ( Ω R ( A ) ⊗ A R [ ℓ ] ) and B [ ℓ ] = Sym R [ ℓ ] ( Ω R ( B ) ⊗ B R [ ℓ ] ) . Therefore, by Lemma 4.1, we have A [ ℓ ] = Sym R [ ℓ ] ( Ω R ( A ) ⊗ A R [ ℓ ] ) = Sym R [ ℓ ] (( Ω B ( A ) ⊕ ( Ω R ( B ) ⊗ B A )) ⊗ A R [ ℓ ] ) = Sym R [ ℓ ] ( Ω B ( A ) ⊗ A R [ ℓ ] ) ⊗ R [ ℓ ] Sym R [ ℓ ] (( Ω R ( B ) ⊗ B A ) ⊗ A R [ ℓ ] ) = Sym R [ ℓ ] ( Ω B ( A ) ⊗ A R [ ℓ ] ) ⊗ R [ ℓ ] Sym R [ ℓ ] ( Ω R ( B ) ⊗ B R [ ℓ ] ) = Sym R [ ℓ ] ( Ω B ( A ) ⊗ A R [ ℓ ] ) ⊗ R [ ℓ ] B [ ℓ ] = C ⊗ R [ ℓ ] B [ ℓ ] , where C = Sym R [ ℓ ] ( Ω B ( A ) ⊗ A R [ ℓ ] ). Acknowledgements:
The authors thank Neena Gupta for carefully going through the draft and the referee for sug-gestions which have been incorporated in the Appendix.
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