A STRONGER LOCAL MONOMIALIZATION THEOREM In this note we derive extensions of the monomialization theorems for morphisms of varieties in [4] and [6]. I thank Jan Denef for conversations on this topic, suggesting that I make improvements of this type, and explaining applications of these theorems. A global “weak” monomialization theorem is established in [2] by Abramovich, Denef and Karu, generalizing an earlier theorem by Abramovich and Karu in [1]. A monomialization is “weak” if the modifications used have no further requirements; in a monomialization all modifications must be products of blow ups of nonsingular sub varieties. In this note we show that a local monomialization can be found which satisfies the extra local statements obtained in [2]. In [9] and [10] some comments are made about how the results of this paper can be used. 1. A stronger local monomomialization theorem for algebraic morphisms In this section we state and prove an extension Theorem 1.1 of the local monomialization theorem Theorem 1.4 [6]. Suppose that K is an algebraic function field over a field k. A local ring R is an algebraic local ring of K if R is a subring of K, the quotient field of R is K and R is essentially of finite type over k. Theorem 1.1. Suppose that k is a field of characteristic zero, K → K ∗ is a (possibly transcendental) extension of algebraic function fields over k, and that ν ∗ is a valuation of K ∗ which is trivial on k. Further suppose that R is an algebraic local ring of K and S is an algebraic local ring of K ∗ such that S dominates R and ν ∗ dominates S. Suppose that I is a nonzero ideal of S. Then there exist sequences of monoidal transforms R → R0 and S → S 0 along ν ∗ such that R0 and S 0 are regular local rings, S 0 dominates R0 , there exist regular parameters (y1 , . . . , yn ) in S 0 , (x1 , . . . , xm ) in R0 , units δ1 , . . . , δm ∈ S 0 and an m × n matrix (cij ) of nonnegative integers such that (cij ) has rank m, and xi = n Y c yj ij δi j=1 for 1 ≤ i ≤ m. Further, we have that 1) S 0 is a local ring of the blowup of an ideal J of S such that JS 0 = (y1α1 · · · ynαn ) for some α1 , . . . , αn ∈ N. 2) IS 0 = (y1β1 · · · ynβn ) for some β1 , . . . , βn ∈ N. The proof of Theorem 1.4 [6] is given in [4] and [6]. Also see [5] for some errata. We now explain the changes in this proof which must be made to obtain the stronger result Theorem 1.1. Theorem 1.1 is a consequence of Theorem 1.5 which is proven at the end of this section. We first indicate changes required in the proofs of [4] required to obtain Theorem 1.1 in the case of a finite extension of function fields. In the construction of Theorem 5.1 [4], we have that ν(y10 ), . . . , ν(ys0 ) are rationally independent. We first settle the case when the quotient field of S is finite over the quotient field of R and the valuation ν has rank 1. 1 Theorem 1.2. Suppose that R → S and R0 → S 0 satisfy the assumptions and conclusions of Theorem 5.1 [4] and h ∈ S 0 is nonzero. Then there exist sequences of monoidal transforms R0 → R00 and S 0 → S 00 along ν such that S 00 dominates R00 , R00 has regular parameters x001 , . . . , x00n , S 00 has regular parameters y100 , . . . , yn00 having the monomial form of the conclusions of Theorem 5.1 [4] and h = (y100 )e1 · · · (ys00 )es u where e1 , . . . , es ∈ N and u ∈ S 00 is a unit. Proof. This is an immediate consequence of Theorems 4.8 and 4.10 of [4] Corollary 1.3. Suppose that R → S and S → S 0 satisfy the assumptions and conclusions of Theorem 5.1 [4] and I ⊂ S is a nonzero ideal. Then there exist sequences of monoidal transforms R0 → R00 and S 0 → S 00 along ν such that S 00 dominates R00 , R00 has regular parameters x001 , . . . , x00n , S 00 has regular parameters y100 , . . . , yn00 having the monomial form of the conclusions of Theorem 5.1 [4] and the following holds. 1) S 00 is a local ring of the blowup of an ideal J of S such that JS 00 = ((y100 )a1 · · · (ys00 )as ) for some a1 , . . . , as ∈ N. 2) IS 00 = ((y100 )b1 · · · (ys00 )bs ) for some b1 , . . . , bs ∈ N. Proof. Let K be an ideal in S such that S 0 is a local ring of the blow up of K. Let Q KS 0 = (f0 ) and IS 0 = (f1 , . . . , fl ). Let h = lj=0 fi . By Theorem 1.2, there exist sequences of monodial transforms R0 → R00 and S 0 → S 00 along ν such that R00 → S 00 and h satisfy the conclusions of Theorem 1.2. Now by Lemma 4.2 and Remark 4.1 [4], there exists a sequence of monodical transforms S 00 → S(1) such that R00 → S(1) has a monomial form as in the conclusion of Theorem 5.1 [4], KS(1) = (y1 (1)e1 · · · ys (1)es ) and IS(1) = (y1 (1)c1 · · · ys (1)cs ) for some ei and cj in N. Now S 0 → S(1) is a product of Perron transforms (as defined in Section 4.1 [4]). Thus S(1) is a local ring of the blow up of an ideal L in S 0 such that LS(1) = (y1 (1)g1 (i+1) · · · ys (1)gs (i+1) ) for some gj (i + 1) ∈ N. There exists a positive integer β such that S(1) is a local ring of the blow up of an ideal J of S such that JS(1) = K β LS(1) (this can be verified using the universal property of blowing up) and the corollary follows. We now prove the case when the quotient field of S is finite over the quotient field of R and the valuation ring has arbitrary rank. We use the notation of Theorem 5.3 [4]. Theorem 1.4. Suppose that R → S satisfies the assumptions of Theorem 5.3 [4] and I is a nonzero ideal in S. Then there exist sequences of monoidal transforms R → R0 and S → S 0 such that R0 → S 0 satisfies the conclusions of Theorem 5.3 [4] and 1) S 0 is a local ring of the blowup of an ideal J of S such that j+1 r−1 Y sY JS 0 = ( (wt1 +···+tj +l )εjl ) j=0 l=1 for some εjl ∈ N (with the convention that t1 + · · · + t0 = 0). Qr−1 Qsj+1 γjl 2) IS 0 = ( j=0 l=1 (wt1 +···+tj +l ) ) for some γjl ∈ N. 2 To obtain Theorem 1.4, we must modify the proof of Theorem 5.3 [4] as follows. On line 1 of page 117, replace the reference to Theorem 5.1 [4] with Corollary 1.3. Insert the following at the end of line 15 on page 119 (after “for 1 ≤ i ≤ λ”): “By our construction, S 00 is a local ring of the blow up of an ideal J of S such that JSq0000 r−1 j+1 r−2 Y sY =( (yt001 +···+tj +l )αjl ) j=0 l=1 for some αjl ∈ N so 00 j+1 r−2 Y sY 00 JS = H ( where H 00 is an ideal in S 00 such that ISq0000 r−1 j=0 l=1 H 00 Sq0000 r−1 j+l r−2 Y sY =( (yt001 +···+tj +l )αjl ) = Sq0000 . By our construction, r−1 (yt001 +···+tj +l )βjl ) j=0 l=1 for some βjl ∈ N so j+1 r−2 Y sY IS 00 = K 00 ( (yt001 +···+tj +l )αjl ) j=0 l=1 where K 00 is an ideal in S 00 such that K 00 Sq0000 r−1 = Sq0000 .” r−1 After line 11 of page 120 (after “for 1 ≤ i ≤ λ”), insert: “We also may obtain, using Theorem 1.2 and the argument of the proof of Corollary 1.3 (above) that γ sr 1 H 00 U 0 = (y γλ+1 ) · · · y λ+s r and δ sr 1 ).00 K 00 U 0 = (y δλ+1 · · · y λ+s r Insert the following at the end of line -2 of page 121 (at the end of the proof): “We have H 00 S(m0 + 1) = (y λ+1 (m0 + 1)γ1 · · · y λ+sr (m0 + 1)γsr + Σ) and K 00 S(m0 +1) = (y λ+1 (m0 +1)δ1 · · · y λ+sr (m0 +1)δsr +Ψ1 , . . . , y λ+1 (m0 +1)δ1 · · · y λ+sr (m0 +1)δsr +Ψe ) where Σ, Ψ1 , . . . Ψe ∈ (y 1 (m0 + 1), . . . , y λ (m0 + 1)) and γ1 , . . . , δsr ∈ N. Let εi = max{γi , δi } for 1 ≤ i ≤ sr for 1 ≤ i ≤ sr . Define a MTS S(m0 + 1) → S(m0 + 2) by Q sr 0 εj 0 for 1 ≤ i ≤ λ 0 j=1 y λ+j (m + 2) y i (m + 2) y i (m + 1) = 0 y i (m + 2) for λ < i ≤ n. Further, by our construction, S(m0 + 2) is a local ring of the blow up of an ideal B of S 00 such that j+1 r−1 Y sY BS(m0 + 2) = ( (y t1 +···+tj +l (m0 + 2))γjl ) j=0 l=1 3 for some εjl ∈ N. Thus there exists an ideal J ∗ of S and β > 0 such that S(m0 + 2) is a local ring of the blow up of J ∗ and J ∗ S(m0 + 2) = J β BS(m0 + 2). We have thus achieved the conclusions of Theorem 1.4 in R(m0 ) → S(m0 + 2). We now indicate the changes which need to be made in the statements of [6] to obtain the proof of Theorem 1.1. We first extend Theorem 1.2 to arbitrary extensions of characteristic zero algebraic function fields. We will call this “Extended Theorem 1.2”. In the statement and proof of Theorem 1.2, we replace references to Theorem 5.1 [4] with Theorem 10.1 [6], Theorems 4.8 and 4.10 [4] with Theorems 9.1 and 9.3 [6]. Also replace n with m when referring to regular parameters in birational extensions of R and s with s. We must add the following sentence to the first line of the proof: “First suppose that rank(ν) > 0, so that rank(ν) = 1 (here ν is the restriction of the given rank 1 valuation ν ∗ of the quotient field of S to the quotient field K of R)”. At the end of the proof, add: “If rank(ν) = 0, then ν is trivial so R = K and the proof is a substantial simplification.” We now extend Corollary 1.3. We will call this “Extended Corollary 1.3”. In the statement and proof of Corollary 1.3, replace references to Theorem 5.1 [4] with Theorem 10.1 [6] and Section 4.1 [4] with Section 5 [6]. Replace references to Theorem 1.2 with “Extended Theorem 1.2”. Replace n with m when referring to regular parameters in birational extensions of R. Replace s with s. We must add the following sentence to the first line of the proof: “First suppose that rank(ν) > 0, so that rank(ν) = 1 (here ν is the restriction of the given rank 1 valuation ν ∗ of the quotient field of S to the quotient field K of R)”. At the end of the proof, add: “If rank(ν) = 0, then ν is trivial so R = K and the proof is a substantial simplification.” We now adopt the notation on valuation rings introduced on page 1579 - 1581 of [6], which we will use below. We extend Theorem 1.4 to arbitrary extensions of algebraic function fields in the following Theorem 1.5. Theorem 1.5. Suppose that R → S satisfies the assumptions of Theorem 10.5 [6] and I is a nonzero ideal in S. Then there exist sequences of monoidal transforms R → R0 and S → S 0 such that R0 → S 0 satisfies the conclusions of Theorem 10.5 [6] and 1) S 0 is a local ring of the blowup of an ideal J of S such that Y JS 0 = ( (wt0 +···+ti−1 +ti,1 +···+ti,j−1 +l )εijl ) where the product is over 0 ≤ i ≤ β, 1 ≤ j ≤ σ(i), 1 ≤ l ≤ sij (with si1 = si ) and εijl ∈ N for all i, j, l. 2) We have that Y IS 0 = ( (wt0 +···+ti−1 +ti,1 +···+ti,j−1 +l )γijl ) where the product is over 0 ≤ i ≤ β, 1 ≤ j ≤ σ(i), 1 ≤ l ≤ sij 4 and γijl ∈ N for all i, j, l. Proof. In the proof of Theorem 1.4, replace references to Theorem 5.3 [4] with Theorem 10.5 [6]. Replace references to Theorem 1.4 with Theorem 1.5, references to Corollary 1.3 with Extended Corollary 1.3. and references to Theorem 1.2 with Extended Theorem 1.2. The indexing of the variables must be changed (as in the statement of Theorem 1.5). The prime ideal qr−1 in the valuation ring of the quotient field of S is replaced by the 00 prime ideal qβ−1,σ(β−1) of the valuation ring of the quotient field of S, so qr−1 = qr−1 ∩ S 00 00 becomes qβ−1 = qβ−1,σ(β−1) ∩ S 00 . We must add the following sentences as the first lines of the proof: “We prove the theorem by induction on rank V ∗ . If rank V ∗ = 1 then the theorem is immediate from Extended Corollary 1.3. By induction on γ = rank V ∗ , we may assume that the theorem is true whenever rank V ∗ = γ − 1. We are reduced to proving the theorem in the following two cases: Case 1. σ(β) = 1 Case 2. σ(β) > 1. Suppose that we are in Case 1, σ(β) = 1. Then V ∗ /qβ−1,σ(β−1) is a rank 1 valuation ring which dominates the rank 1 valuation ring V /pβ−1 . V ∗ /qβ−1,σ(β−1) has rational rank sβ and V /pβ−1 has rational rank rβ .” We must add the following sentences as the last lines of the proof: “Now suppose that we are in Case 2, σ(β) > 1. Then V ∗ /qβ,σ(β)−1 is a rank 1, rational rank sβ,σ(β) valuation ring which dominates the rank 0 valuation ring V /pβ , which is a field. The proof in Case 2 is thus a substantial simplification of the proof in Case 1.” 2. A geometric local monomialization theorem The main result in this section is Theorem 2.2. Uniformizing parameters on an affine k-variety V are a set of elements u1 , . . . , us ∈ A = Γ(V, OV ) such that du1 , . . . , dus is a free basis of Γ(V, ΩV /k ) as an A-module. Theorem 2.1. Suppose that k is a field of characteristic zero, ϕ : Y → X is a dominant morphism of k-varieties and ν is a zero dimensional valuation of the function field k(Y ) (the residue field of the valuation ring of ν is algebraic over k) which has a center on Y (the valuation ring of ν dominates a local ring of Y ). Further suppose that I ⊂ OY is a nonzero ideal sheaf. Then there exists a commutative diagram of morphisms of k-varieties ϕν (1) Yν → Xν βν ↓ ↓ αν ϕ Y → X where the vertical arrows are proper morphisms which are products of blow ups of nonsingular sub varieties, and if q 0 is the center of ν on Yν and p0 is the center of ν on Xν , then there exists an affine open neighborhood Vν of q 0 in Yν and an affine open neighborhood Uν of p0 in Xν,p0 , regular parameters y1 , . . . , yn in OYν,q0 which are uniformizing parameters on Vν and regular parameters x1 , . . . , xm in OXν ,p0 which are uniformizing parameters on Uν (where m = dim X, n = dim Y ) and units δ1 , . . . , δm ∈ Γ(Vν , OYν ) such that Q c 1) xi = nj=1 yj ij δi with cij ∈ N for 1 ≤ i ≤ m and rank(cij ) = m. 5 2) Vν \ Z(y1 · · · yn ) → Y is an open immersion; 3) IOVν = y1a1 · · · ynan OVν for some a1 , . . . , an ∈ N 4) ϕν : Vν → Uν is toroidal (Section 2.2 [2]) with respect to the locus of the product of the yj and the locus of the product of the xi . Proof. Let p be the center of ν on X and q be the center of ν on Y . Let ν ∗ = ν, R = OX,p , S = OY,q , I = Iq , K ∗ = k(Y ) and K = k(X). Let R0 → S 0 ↑ ↑ R → S be the diagram of the conclusions of Theorem 1.1. There exists a commutative diagram ϕν Yν → Xν βν ↓ ↓ αν ϕ Y → X where the vertical arrows are products of blow ups of nonsingular sub varieties and if p0 is the center of ν on Xν and q 0 is the center of ν on Yν then OYν ,q0 = S 0 and OXν ,p0 = R0 . Since k has characteristic zero, there exists an affine open neighborhood Uν of p0 such that x1 , . . . , xm are uniformizing parameters on Uν . Let Vν be an affine neighborhood of q 0 in Yν such that y1 , . . . , yn are uniformizing parameters on Vν and δ1 , . . . , δm ∈ Γ(Vν , OYν ) are units. Let C be an m × m sub matrix of C = (cij ) of rank m. Without loss of generality,   c11 · · · c1m   .. C =  ... . . cm1 · · · cmm Let d = Det(C). Let A = Γ(Vν , OYν ) and d B = A[z1 , . . . , zm ]/(z1d − δ1 , . . . , zm − δm ) = A[z 1 , . . . , z m ] where z i is the class of zi . Let Vν0 = Spec(B) with natural finite étale morphism π : Vν 0 → Vν . Define y j for 1 ≤ j ≤ n by m Y db yj = y j z l jl l=1 where B = (bjl ) = −C 0 −1 is an n × m matrix with coefficients in d1 Z. We have that for 1 ≤ i ≤ m,   ! n n n m Pn Y Y Y Y cij dbjl cij c c j=1 ij (2) xi = yj δ i =  yj  zl δi = y jij . j=1 j=1 l=1 6 j=1 The morphism π : Vν0 → Vν is étale so y1 , . . . , yn are uniformizing parameters on Vν0 or equivalently, dy1 , . . . , dyn are a free basis of Ω1B/k as a B-module. Let !−1 m Y dbjl zl ∈B εj = l=1 and y j = εj yj for 1 ≤ j ≤ n. Since y1 , . . . , yn are regular parameters in OVν0 ,q00 for all q 00 ∈ π −1 (q 0 ) and dy j = εj dyj + yj dεj , we have that 1 0 0 ΩVν0 /k /dy 1 OVν + · · · + dy n OVν 00 = 0 q for all q 00 ∈ π −1 (q 0 ). Let Z = Supp Ω1Vν0 /k /dy 1 OVν0 + · · · + dy n OVν0 . Z is a closed subset of Vν0 which is disjoint from π −1 (q 0 ). The morphism Vν0 → Vν is finite, so π(Z) is a closed subset of Vν which does not contain q 0 . After replacing Vν with an affine neighborhood of q 0 in Vν \ π(Z), we have that y 1 , . . . , y n are uniformizing parameters on Vν0 giving the monomial expression (2) and so Vν → Uν is toroidal with respect to the locus of the products of the yj and the products of the xi . Theorem 2.2. Suppose that k is a field of characteristic zero, ϕ : Y → X is a dominant morphism of k-varieties and I ⊂ OY is a nonzero ideal sheaf. Then there exists a finite number of commutative diagrams ϕi Yi → Xi βi ↓ ↓ αi ϕ Y → X for 1 ≤ i ≤ t such that the vertical arrows are products of blow ups of nonsingular sub varieties and there are affine open subsets Vi ⊂ Yi and Ui ⊂ Xi such that ϕi (Vi ) ⊂ Ui , ∪ti=1 βi (Vi ) = Y and the restriction ϕi : Vi → Ui is toroidal with respect to strict normal crossings divisor Ei on Vi and Di on Ui such that ϕi |(Vi \ Ei ) is an open immersion and IOVi is a divisor on Vi whose support is contained in Ei . Proof. Let Ω be the Zariski Riemann manifold of Y (Section 17 of Chapter VI [16]). The points of Ω are equivalence classes of valuations of k(Y ) which dominate a local ring of Y . There are natural continuous surjections ρZ : Ω → Z for any birational proper morphism Z → Y . Let Σ be the subset of Ω of zero dimensional valuations which dominate a local ring of Y . For each ν ∈ Σ, we construct a diagram (1) with corresponding open subset Vν of Yν . Suppose that ω is a valuation of k(Y ) which dominates a local ring of Y . If ω is not zero dimensional, then there exists ν ∈ Σ such that ν is composite with ω (Theorem 7, −1 page 16 [16]), so that ω ∈ ρ−1 Yν (Vν ). Thus {ρYν (Vν ) | ν ∈ Σ} is an open cover of Ω, and thus there is a finite sub cover since Ω is quasi compact (Theorem 40 page 113 [16]). References [1] D. Abramovich and K. Karu, Weak semistable reduction in characteristic 0, Invent. Math. 139 (2000), 241 - 273. [2] D. Abramovich, J. Denef and K. Karu, Weak toroidalization over non closed fields, Manuscripta Math. 142 (2013), 257 - 271. 7 [3] C. Banica and O. Stanasila, Algebraic methods in the global theory of complex spaces, John Wiley and sons, 1976. [4] S.D. Cutkosky, Local monomialization and factorization of morphisms, Astérisque 260, Société mathématique de France, (1999). [5] S.D. Cutkosky, Errata of Local monomialization and factorization of morphisms, www.math.missouri.edu/˜dale [6] S.D. 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