Division Theorem over the Dwork-Monsky-Washnitzer Completion of Polynomial Rings and Weyl Algebras

Division theorem over the Dwork-Monsky-Washnitzer completion of polynomial rings and Weyl algebras Luis Narv´aez Macarro∗ Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 §1 Rings of Dwork-Monsky-Washnitzer power series . . . . . . . . . 2 §2 Division over polynomial rings with coefficients in a field . . . . . 3 §3 Division over strictly convergent or DMW power series rings . . . 5 §4 Division over Weyl algebras . . . . . . . . . . . . . . . . . . . . . 10 §5 Completion of Weyl Algebras over CDV rings . . . . . . . . . . . 12 References.................................. 17 Introduction The aim of this paper is to give proofs of Weierstrass-Hironaka division theorems, in the sense of [1], over the Dwork-Monsky-Washnitzer completion of polynomial and Weyl algebras in several variables with coefficients in a complete discrete valuation ring. The case of polynomial algebras (th. 3.5) generalizes the Weierstrass division theorem stated in [14] and precises the original proof of noetherianity given in [6]. The case of Weyl algebras (th. 5.1) has been motivated by joint work with Z. Mebkhout since 1987, in order to analyze division techniques over the ring of infinite linear differential operators D† over a smooth weakly formal scheme [11, 10] (compare with [12] in the complex analytic setting). Proofs of theorems above have been obtained by the author between 1987 and 1989. A proof in the case of the full completion of Weyl algebras in one variable has been proposed independently in [7]. We thank F. Castro and Z. Mebkhout for many helpful discussions. ∗Supported by DGICYT PB94-1435. 1 §1 Rings of Dwork-Monsky-Washnitzer power series Let (W, m= (π)) be a complete discrete valuation (CDV) ring, Kits field of quotients and kits residue field. The field Kbecomes a complete ultrametric discrete valued field by defining |c|=εvm(c), for every c∈K∗and ε∈]0,1[. We say (cf. [13]) that a formal power series f(x1, . . . , xn) = Pα∈Nnfαxαwith coefficients in W(resp. in K) satisfies the Dwork-Monsky-Washnitzer condition (DMW) if there is a constant λ > 0 such that vm(fα)≥λ|α|if |α|  0. The set of power series f(x) with coefficients in W(resp. in K) satisfying the DMW condition is a sub-W-algebra (resp. a sub-K-algebra) of the W-algebra (resp. the K-Banach algebra) of strictly convergent power series Whxi=d W[x] = {X α∈Nn fαxα|lim |α|→∞ vm(fα) = +∞} (resp. Khxi=K⊗WWhxi), that will be denoted by W[x]†(resp. K[x]†). We will call the ring W[x]†the weak completion or DMW completion of W[x]. Notice that K[x]†=K⊗WW[x]†, and that the K-Banach algebra Khxiis the completion of the K-normed algebra K[x]†. If f(x) = Pα∈Nnfαxαis a power series in K⊗WW[[x]], its (m-adic) valuation is defined by v(f) := inf{vm(fα)|α∈Nn} ∈ Z. It is clear that v(fg) = v(f) + v(g), v(f+g)≥inf{v(f), v(g)}–equality holds if v(f)6=v(g)– and that a power series fin K[x]†(resp. in Khxi) belongs to W[x]†(resp. to Whxi) if and only if v(f)≥0. The ring W[x]†can also be described in the following way (see [6]). Let Φ: W[x][[Z]] −→ W[[x]] be the homomorphism of W[x]-algebras sending Zto π. The image of Φ is the m-adic completion of the polynomial ring W[x], i.e. the ring Whxi. Let Rbe the subring of W[x][[Z]] consisting of the power series F=Pm≥0Pm(x)Zm such that there is a constant c > 0 with degx(Pm(x)) ≤c(m+ 1) for all integers m≥0. Then the ring W[x]†coincides with the image of Rby Φ. More precisely, Φ induces an isomorphism R/(Z−π)≃W[x]†. This (easy) result has the following consequence: Lemma 1.1.– Let fbe a formal power series in W[[x]] , and let f(m)∈ (W/mm)[[x]] denote its reduction modulo mmfor m≥1. For m= 1 we will write simply f:= f(1). The following properties are equivalent: a) fsatisfies the DMW condition. 2 b) f(m)is a polynomial for every m≥1, and there is a constant c > 0such that deg(f(m))≤c(m+ 1) for all m≥1. §2 Division over polynomial rings with coefficients in a field Let Abe a ring and F(x1, . . . , xn) = Pα∈NnFαxαa formal power series with coefficients in A. The support of Fis the set N(F) = {α∈Nn|Fα6= 0} ⊆ Nn. More generally, if F= (F1, . . . , Fm) is a vector of formal power series with coefficients in A, there is a unique representation F= m X j=1 X α∈Nn Fj,αxj,α where the Fj,α are in Aand xj,α denotes the vector (0, . . . , xα, . . . , 0), the monomial being at the jth place. We have Fj=Pα∈NnFj,αxαfor every j= 1, . . . , m. The support of Fis the set N(F) = {(j, α)∈ {1, . . . , m} × Nn|Fj,α 6= 0} ⊆ Nn× {1, . . . , m}. If Fis a vector of polynomials, then N(F) is a finite subset of {1, . . . , m} × Nn. As usual, if α= (α1, . . . , αn) and β= (β1, . . . , βn) are two elements in Nn, we will consider the following total order relations: Lexicographical order: α <lex βif there is an index jbetween 1 and nsuch that αi=βifor i < j and αj< βj. Inverse lexicographical order: α <invlex βif there is an index jbetween 1 and nsuch that αi=βifor i > j and αj< βj. Diagonal order: α <dβif either |α|<|β|or |α|=|β|and α <lex β. Λ-Order: Given a linear form Λ : Rn→Rwith non negative coefficients, α <Λβif either Λ(α)<Λ(β) or Λ(α) = Λ(β) and α <lex β. If Λ(α) = |α|, then <d=<Λ. If the coefficients of the linear form Λ are linearly independent over Z, then α <Λβif and only if L(α)< L(β). If (i, α) and (j, β) are two elements in {1, . . . , m} × Nn, we will also consider the following total order relations: Semi-lexicographical Λ-order for vectors: Given a linear form Λ: Rn→Rwith non negative coefficients, (i, α)<Λ(j, β) if 3 a) either Λ(α)<Λ(β) or b) Λ(α) = Λ(β) and either i < j or i=jand α <lex β. When Λ(α) = |α|we will write <dinstead of <Λ. e Λ-order for vectors: Given a linear form e Λ : R1+n→Rwith non negative coefficients linearly independent over Z, (i, α)<eΛ(j, β) if e Λ(i−1, α)< e Λ(j−1, β). Denote by ≺one of the above total order relations over {1, . . . , m} × Nnor, more generally, a monomial order in {1, . . . , m} × Nn(cf. [2]). Suppose that the ring Ais a field Land take a non-zero vector F∈L[x]m. The exponent of Fis defined by exp(F) := sup ≺ (N(F)). If Qis a non-zero polynomial in L[x] then exp(QF) = exp(Q) + exp(F), and if G∈L[x]mis another non-zero vector with exp(F)<exp(G) then exp(F+G) = exp(G). Recall the division theorem over L[x]m(cf. [2]). Theorem 2.1.– Let F1, . . . , Fp∈L[x]mbe non-zero vectors. Denote by ∆1:= Nn+ exp(F1) ∆i:= (Nn+ exp(Fi)) \ i−1 [ j=1 ∆j, i = 2, . . . , p ∆ := Nn× {1, . . . , m} \ p [ i=1 ∆i=Nn× {1, . . . , m} \ p [ i=1 (Nn+ exp(Fi)). Then, for every G∈L[x]mthere is a unique family of polynomials Q1, . . . , Qp∈ L[x]and a unique vector R∈L[x]msuch that 1. G= p X i=1 QiFi+R, 2. N(Qi) + exp(Fi)⊆∆ifor all i= 1, . . . , p, 3. N(R)⊆∆. Moreover, if ≺=<dwe also have deg(Qi)≤deg(G)−deg(Fi)and deg(R)≤ deg(G). 4 §3 Division over strictly convergent or DMW power series rings In the situation of §1, denote by Rone of the rings W[x]†or Whxiand ≺a monomial order in {1, . . . , m} × Nn. Let f(x) = Pm j=1 Pα∈Nnfj,αxj,α be a non-zero vector in (K⊗WR)m. We define the exponent of fas exp(f) := max ≺{(j, α)∈ N(f)|v(fj,α) = v(f) = min 1≤l≤mv(fl)}. It is clear that 1. If v(f) = 0, then exp(f) = exp(f), where f= (f1, . . . , fm)∈k[x]. 2. exp(f) = exp(cf), for all c∈K. Lemma 3.1.– Let q∈K⊗WRbe a non-zero power series and f, g ∈(K⊗WR)m two non-zero vectors. We have the following properties: 1. exp(qf) = exp(q) + exp(f). 2. If exp(f)6= exp(g), then exp(f+g)∈ {exp(f),exp(g)}. Proof: We may assume v(q) = v(f) = 0 and v(g) = ν≥0. Let us write q=q(0) +πq(>0), f =f(0) +πf(>0), g =πνg(0) +πν+1g(>0), where q(0) ∈W[x], f(0), g(0) ∈W[x]m, f(>0), g(>0) ∈Rm satisfying v(q(0)) = v(f(0)) = v(g(0)) = 0, v(q(>0)), v(f(>0)), v(g(>0))≥0. We have obviously exp(q) = exp(q(0)),exp(f) = exp(f(0)),exp(g) = exp(g(0)) and exp(qf) = exp(q(0)g(0)). Then, for the first part, we are reduced to the case where q∈W[x] and f∈W[x]m,v(q) = v(f) = 0, which is clear because v(qf) = 0 and exp(qf) = exp(qf) = exp(qf) = exp(q) + exp(f) = exp(q) + exp(f). Let us prove the second part. It is enough to consider the following three cases: A) exp(f)≺exp(g), v(f) = 0 < v(g) B) exp(f)≺exp(g), v(f) = 0 = v(g) 5 C) exp(g)≺exp(f), v(f) = 0 < v(g) In cases A) and C) we have v(f+g) = 0 and exp(f+g) = exp(f). In case B) we have v(f+g) = 0 and exp(f+g) = exp(g). 2 3.2 If (µ, i, α),(ν, j, β) are two elements of Z× {1, . . . , m} × Nn, consider the total order relation (µ, i, α)≺0(ν, j, β) if either ν < µ or µ=νand (i, α)≺ (j, β). If f(x) = Pm j=1 Pα∈Nnfj,αxj,α is a non-zero vector in (K⊗WR)m, denote by gexp(f) := (v(f),exp(f)) ∈Z× {1, . . . , m} × Nn, e N(f) := {(v(fj,α), j, α)|(j, α)∈ N (f)} ⊆ Z× {1, . . . , m} × Nn. The following lemma is a reformulation of lemma 3.1. Lemma 3.3.– Let q∈K⊗WRbe a non-zero power series and f, g ∈(K⊗WR)m two non-zero vectors. We have the following properties: 1. gexp(qf) = gexp(q) + gexp(f). 2. If gexp(f)≺gexp(g), then gexp(f+g) = gexp(g). The following theorem generalizes the Weierstrass division theorem of [3, 5.2.1] for strictly convergent power series with coefficients in a complete ultrametric discrete valued field. Theorem 3.4.– (Division over strictly convergent power series rings) Let f1, . . . , fp∈Khximbe non-zero vectors. Denote by ∆1:= Nn+ exp(f1) ∆i:= (Nn+ exp(fi)) \ i−1 [ j=1 ∆j, i = 2, . . . , p ∆ := Nn× {1, . . . , m} \ p [ i=1 ∆i=Nn× {1, . . . , m} \ p [ i=1 (Nn+ exp(fi)). Then, for every g∈Khximthere is a unique family of strictly convergent power series q1, . . . , qp∈Khxiand a unique vector r∈Khximsuch that 1. g= p X i=1 qifi+r, 2. N(qi) + exp(fi)⊆∆ifor all i= 1, . . . , p, 3. N(r)⊆∆. 6 Proof: The uniqueness is a standard consequence of lemma 3.1. For the existence, we may assume that v(fi) = 0 and v(g)≥0. In this case fi, g ∈ Whximand we proceed by the Hensel’s trick. First, take the canonical images F1=f1, . . . , Fp=fp, G =gin k[x] and divide them by using theorem 2.1. Second, lift the quotients and the remainder by respecting the corresponding supports. One proves easily that the iteration of this process converges and gives the desired qiand r. The details will be made precise in the proof of theorem 3.5. 2 The following theorem is the main result of this §. It generalizes the Weierstrass division theorem for DMW power series stated in [14]. Theorem 3.5.– (Division over DMW power series rings) Let us take the monomial order ≺=<d. Let f1, . . . , fp∈(K[x]†)mbe non-zero power series and keep the notations of theorem 3.4. Then, for every g∈(K[x]†)m⊂Khxim, the quotients q1, . . . , qpand the remainder rgiven by theorem 3.4 are in fact in K[x]† and (K[x]†)mrespectively. Proof: As in the proof of theorem 3.4, we may assume v(fi) = 0, v(g)≥0. After reduction modulo the maximal ideal of Wwe find vectors Fi=fi, G0=g∈k[x]m. Put di= deg(Fi), δ= max{deg(G0), d1, . . . , dm}, d = inf{d1, . . . , dm}. Since the power series fi j=Pαfi j,αxαand gj=Pαgj,αxα,j= 1, . . . , m, i= 1, . . . , p, satisfy the DMW condition, there is a constant λ > 0 and an integer n0≥0 such that vm(fi j,α), vm(gj,α)≥λ|α| for |α| ≥ n0. We can take n0≥2 λ+d. By theorem 2.1, there exist Q0 1, . . . , Q0 p∈k[x], R0∈k[x]msuch that G0=Pp i=1 Q0 iFi+R0 exp(Fi) + N(Q0 i)⊆∆i,∀i= 1, . . . , p N(R0)⊆∆ deg(Q0 i)≤deg(G0)−deg(Fi)≤δ−d, deg(R0)≤deg(G0)≤δ. Lift the polynomials Q0 i∈k[x] and the vector R0∈k[x]mto q0 i∈W[x], r0∈W[x]m with the same support. There is a vector g1∈(W[x]†)msuch that g0:= g= m X i=1 q0 ifi+r0+πg1. 7 We have obviously exp(fi) + N(q0 i)⊆∆i,∀i= 1, . . . , p N(r0)⊆∆ deg(q0 i)≤δ−d, deg(r0)≤δ. Let us write q0 i=X |β|≤δ−d q0 i,βxβ, r0= m X j=1 X |α|≤δ r0 j,αxj,α, g1= m X j=1 X α∈Nn g1 j,αxj,α. For α∈Nn, we have πg1 j,α =gj,α −X 1≤i≤p,|β|≤δ−d q0 i,βfi j,α−β−r0 j,α. If |α| ≥ n0+ (δ−d)> δ, then r0 j,α = 0 and 1 + vm(g1 j,α)≥ inf{vm(gj,α), vm(qi,β)0+vm(fj,γ )i|1≤i≤p, |β| ≤ δ−d, β +γ=α} ≥ inf{λ|α|, λ|γ|s.t. |β| ≤ δ−d, β +γ=α}=· · · =λ(|α| − (δ−d))j for every j= 1, . . . , m. Therefore, vm(g1 j,α)≥1 if |α| ≥ n0+ (δ−d), and deg(g1 j)< n0+ (δ−d). By induction, assume that there are polynomials ql i∈W[x],1≤i≤p, 0≤ l≤Nand vectors rl∈W[x]m,0≤l≤N,gl∈(W[x]†)m,0≤l≤N+ 1 such that g0=g gl= p X i=1 ql ifi+rl+πgl+1,0≤l≤N exp(fi) + N(ql i)⊆∆i,1≤i≤p, 0≤l≤N(1) N(rl)⊆∆,0≤l≤N(2) deg(ql i)≤(δ−d) + l(n0−d),1≤i≤p, 0≤l≤N(3) deg(rl)≤δ+l(n0−d),0≤l≤N(4) 1 + vm(gl j,α)≥λ(|α| − [(δ−d)+(l−1)(n0−d)]),(5) for |α| ≥ δ+l(n0−d), j= 1, . . . , m and l= 1, . . . , N +1. In particular, we have deg(gl)< δ +l(n0−d) for every l= 1, . . . , N + 1. 8 By proceeding as in the first step of the induction we find polynomials qN+1 1, . . . , qN+1 m∈W[x] and vectors rN+1 ∈W[x]m, gN+2 = m X j=1 X α∈Nn gN+1 j,α xj,α ∈(W[x]†)m such that gN+1 =Pm i=1 qN+1 ifi+rN+1 +πgN+2 exp(fi) + N(qN+1)⊆∆i,∀i= 1, . . . , p N(rN+1)⊆∆ deg(qN+1 i)≤δ−d+ (N+ 1)(n0−d) deg(rN+1)≤δ+ (N+ 1)(n0−d), hence 1 + vm(gN+2 j,α )≥λ(|α| − [(δ−d)+(N+ 1)(n0−d)]) for |α| ≥ δ+ (N+ 2)(n0−d). In particular, deg(gN+2)< δ + (N+ 2)(n0−d). Therefore, we can find sequences {ql i}1≤i≤p,l≥0⊂W[x], {rl}l≥0⊂W[x]m, {gl}l≥0⊂(W[x]†)msatisfying the above properties. For every N≥0, we have g= m X i=1 ( N X l=0 πlql i)fi+ N X l=0 πlrl+πN+1gN+1. To end the proof of the theorem we define qi:= ∞ X l=0 πlql i, i = 1, . . . , p r:= ∞ X l=0 πlrl. The inequalities (3) and (4) imply that the quotients qiand the remainder rsatisfy the DMW condition and, by (1) and (2), exp(fi) + N(qi)⊆∆i,∀i= 1, . . . , p, N(r)⊆∆. 2 Remark 3.6.– In the proof of theorem 3.5 it is essential to control the degree of the quotients and the remainder in each induction step. For this, we have considered the exponent of power series and vectors with respect to the monomial order relation ≺=<d(see the last part of the theorem 2.1). However, theorem 3.4 remains true for all monomial order relations. 9 {rl}0≤l<∞⊂Wn(W)m, {al}0≤l<∞⊂(Wn(W)†)m satisfying that a0=a, al=rl+Pp i=1 ql ibi+πgl+1,∀l≥0, exp(bi) + N(ql i)⊆∆ifor all i= 1, . . . , m,l≥0, N(rl)⊆∆, for all l≥0, 1+vm(al j,α,β)≥λ((|α|+|β|−[(δ−d)+(l−1)(n0−d)]) for |α|+|β| ≥ δ+l(n0−d) and for all l≥0, deg(gl)< n0+l(δ−d) for all l≥0. To end the proof of the existence, define qi:= ∞ X l=0 πlql i, i = 1, . . . , p, r:= ∞ X l=0 πlrl. By the above properties, it is clear that qi∈Wn(W)†,r∈(Wn(W)†)mand N(qi) + exp(bi)⊆∆ifor all i= 1, . . . , p, N(r)⊆∆. 2 Remark 5.2.– Theorems 3.4 and 3.5 give obviously another proof (see [6]) of noetherianity for the rings Khxiand K[x]†. In fact they can be extended to the rings Whxiand W[x]†. For that, we must use the extended exponent gexp of vectors in Whximor (W[x]†)mas it was defined in 3.2. Remark 5.3.– As in the preceeding remark, theorem 5.1 can be extended to the rings d Wn(W)and Wn(W)†. 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