ERRATA OF “LOCAL MONOMIALIZATION AND
FACTORIZATION OF MORPHISMS”
STEVEN DALE CUTKOSKY
We list here errata from our paper “Local monomialization and factorization of
morphisms” [5].
page 23, line 25 (first line of statement of Theorem 2.7): “local domain” should be
“regular local ring”.
page 33, line 5: x̃0i = x̃1 (1)Ai (h) · · · x̃s (1)Ai (h+s−1) for 1 ≤ i ≤ s.
page 34, line -1: γ1 as+1,1 + · · · + γs as+1,s + as+1,s+1 = 1c .
page 39, line 28: “fij ≥ 0 for all i, j” should be “fij ≥ 0 for 0 ≤ i, j ≤ s”.
page 40, line -1: “as1 , . . . , as+1,s+1 ” should be “as+1,1 , . . . , as+1,s+1 ”.
00
page 45, line 15: “T (1) has regular parameters” should be “T (1) has regular parameters”.
page 49, line 26: “s ≤ m ≤ l” should be “s ≤ m ≤ n”.
page 58, lines 19-22: Replace “Set σ(i) to be the largest possible . . . . . . σ(i) ≥ s”
with “Set
σ(i) = dim k(T (i))[[z 1 (i), . . . , z m (i)]]/pm (i)
where pm (i) is defined in (53). We have σ(i + 1) ≤ σ(i) for 0 ≤ i ≤ t − 1 in (45)” The
inequality σ(i + 1) ≤ σ(i) is proven in Lemma 6.3 of [6].
page 62, line 8: “By (42)” should be “By (43) and (43)”.
page 67, line 2: “u(ỹ 1 (t), . . . , ỹ l (t))” should be “Σ(ỹ 1 (t), . . . , ỹ l (t))”.
page 73, line 1: “gij ≥ 0 for all i, j” should be “gij ≥ 0 for 0 ≤ i, j ≤ s”.
page 73, line 2: “U 00 (α)[N1 , . . . , Ns , Nr ]” should be “U 00 (α)[N1 , . . . , Ns , Nr , N1r ]”.
page 73, line 6: After “in an algebraic closure of k(U (α + 1))” insert “if gs+1,s+1 > 0.
−gs+1,s+1
Mr − cα+1 =
Y
(Nr−1 − ω i d−1
α+1 )βr
i=1
where ω is a primitive −gs+1,s+1 -th root of unity if gs+1,s+1 < 0.”
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STEVEN DALE CUTKOSKY
page 73, line 7: “U 00 (α)[N1 , . . . , Ns , Nr ]” should be “U 00 (α)[N1 , . . . , Ns , Nr , N1r ]”.
page 93, line 3: “Ω ∈ m(U 00 (0))N ” should be “Ω ∈ m(U (0))N ”.
page 93, after line 7, insert:
“(5): Suppose that g = y d11 · · · y ds s Σ(y 1 , . . . , y l ) + Ω where ν(y d11 · · · y ds s ) > A
and Ω ∈ m(U (0))N with N ν(m(U (0)) > ν(y d11 · · · y ds s ). Then there exists a
CRUTS along ν as in the conclusions of Theorem 4.9 such that
0
0
g = y 1 (t0 )d1 (t ) · · · y s (t0 )ds (t ) Σ(y 1 (t0 ), . . . , y l (t0 ))
0
0
where ν(y 1 (t0 )d1 (t ) · · · y s (t0 )ds (t ) ) > A”
page 95, lines 26-31: “Let G be . . . . . . g ∈ k(c0 )[[x1 (1), . . . , xl (1)]][xl+1 ]” should be
“Since
is
y di = xf1i1 · · · xfsis φ1−fi1 · · · φ−f
s
for 1 ≤ i ≤ s, by Lemma 4.2, we can perform a MTS of type (M1) to get g 0 ∈
k(U 00 (0))[[x1 (1), . . . , xl (1)]][xl+1 ].
Let G be the Galois group of a Galois closure of k(U 00 (0)) over k(c0 ). Since xl+1 is
analytically independent of y d1 , . . . , y ds , y s+1 , . . . , y l (by Theorem 2.12) we can define
Y
g=
τ (g 0 )
τ ∈G
0
where G acts on the coefficients of g . We have g ∈ k(c0 )[[x1 (1), . . . , xl (1)]][xl+1 ]”
d so
Page 106, line 19: before ”and” on this line, insert “Σ0 divides g already in S(1),
d We have
form (126), Σ0 divides xl+1 (t) + Φ(x1 (t), . . . , xl (t)) in S(t).
(Ω + Σt )ỹ 1 (t)d1 (t) · · · ỹ s (t)ds (t) = Pt + Φ + Σt ỹ 1 (t)d1 (t) · · · ỹ s (t)ds (t) = xl+1 (t) + Φ.00
page 106, line -4: Before this line, insert: “xl+1 (t) still has a form of the type of the
equation on line -1 of page 105 or of line 2 of page 106. If the equation on line -1 of
page 105 holds, then we have the equation on line -2 of page 106. Suppose xl+1 (t) has
the form of the equation on line 2 of page 106. Then comparing with the equation on
line -5 of page 106, we see that Ω + Σt itself has an expression
Ω + Σt = P̃ (ŷ1 (t), . . . , ŷl (t)) + ŷ1 (t)e1 (t) · · · ŷs (t)es (t) Σ̂t (ŷ1 (t), . . . , ŷn (t))
where multΣ̂t (0, . . . , 0, ŷl+1 (t), 0, . . . , 0) = 1. We can assume that the series Σ̂t (ŷ1 (t), . . . , ŷn (t))
has no monomials in ŷ1 (t), . . . , ŷl (t). Since Σ0 = ŷl+1 (t) divides Ω + Σt , we have that
P̃ = 0.
page 115, line 5 of the statement of Theorem 5.3: “if mV is the maximal ideal of V
and p∗ = mV ∩ S” should be “if q is a prime ideal of V and p∗ = q ∩ S”.
page 115, line 6 of the statement of Theorem 5.3: “segments” should be “isolated
subgroups”.
page 116, line 1: After “for 1 ≤ i ≤ r”, insert the sentence: “Assume that there exist
fields ki ⊂ Rpi ∩R such that ki → (S/pi ∩ S)pi ∩S is a finitely generated field extension
for 1 ≤ i ≤ r”. This condition certainly holds whenever R and S are essentially of
finite type over a field k.
ERRATA OF “LOCAL MONOMIALIZATION AND FACTORIZATION OF MORPHISMS”
3
page 121, line 17: “t > max{aij , gij (1)}” should be “t > max{aij , gij (r)}”.
page 122, lines 4-5: “if mV is the maximal ideal of V and p∗ = mV ∩ S” should be
“if q is a prime ideal of V and p∗ = q ∩ S”.
page 122, line 14: “as in the proof of Theorem 1.10 (Chapter 7)” should be “as in the
proof of Theorem 5.1”.
page 122, lines 19-20: “if mV is the maximal ideal of V and p∗ = mV ∩ S” should be
“if q is a prime ideal of V and p∗ = q ∩ S”.
page 122, line 6 of the statement of Theorem 5.5: “segments” should be “isolated
subgroups”.
page 6, line 31: After “Theorem 1.6” insert:
“There is however uncertainty about the result of [9] (c.f. [1], [2]). There is a local
version of this result for morphisms of toric varieties which has been proven. We
may use this result instead of [9] to obtain proofs of Theorems 1.9 and 1.10 below.
Christensen [3] has proven local strong factorization of morphisms of toric 3-folds
along a toric valuation. Using the language of toric geometry, Karu [8] has extended
Christiansen’s result to prove this local result in all dimensions. A proof in the spirit
of Christensen’s original proof, using only elementary properties of determinants, is
given in [7].”
page 136, line 10: After “Theorem 7.1”, insert: “A proof of Theorem 1.9 using
Theorem 1.1 (Local Monomialization) and Local Strong Factorization of morphisms
of toric varieties along a toric valuation ([3], [8], [7]) which is thus independent of [9]
is given in Theorem 3.3 [7]. A proof of Theorem 7.1 which is independent of [9] is
now immediate from this proof of Theorem 1.9”
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STEVEN DALE CUTKOSKY
Errata of “Local monomialization of transcendental extensions”
page 1525, line 9, ”ν(fI0 ) > ν(fI )” should be ”ν(fI ) > ν(fI0 )”.
page 1525, line 10, ”ν(gJ0 ) > ν(gJ )” should be ”ν(gJ ) > ν(gJ0 )”.
page 1526, line 15, “Rational rank 1” should be “Rank 1”.
page 1536, line 3, m ≤ m0 should be m < m0 .
page 1537, line -1, r should be r.
page 1538, line -7, k(V )(t1 , . . . , tβ ).
page 1553, lines 17 and 18, “2r + m” should be “r + m”.
d
page 1556, line -3, ρ should be “ν ∗ (wd11 · · · ws s )”.
page 1562, line 11, ⇔ should be ⇒.
page 1563, line 12, w(fgI ) should be w(gI ).
References
[1] Abramovich, D., Matsuki, M, Rashid, S., A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension, Tohoku Math. J 51 (1999), 489-537.
[2] Abramovich, D., Matsuki, M, Rashid, S., Correction to A note on the factorization theorem
of toric birational maps after Morelli and its toroidal extension, Tohoku Math. J 52 (2000),
629-631.
[3] Christensen, C., Strong domination, weak factorization or three dimensional regular local local
rings, Journal of the Indian Math. Soc., 45 (1981), 21-47.
[4] Cutkosky, S.D., Local Factorization of Birational Maps, Advances in Math. 132 (1997), 167315.
[5] Cutkosky, S.D., Local Monomialization and Factorization of Morphisms, Astérisque 260, 1999.
[6] Cutkosky, S.D., Local Monomialization of Transcendental Extensions, Journal of the Fourier
Institute 55 (2005) 1517 – 1586.
[7] Cutkosky, S.D. and Srinivasan, H., Factorizations of birational extensions of local rings, Ill.
J. Math 51 (2007), 41 - 56.
[8] Karu, K., Local strong factorization of birational maps, J. Alg. Geom 14 (2005), 165 – 175.
[9] Morelli, R., The birational geometry of toric varieties, J. Algebraic Geometry 5 (1996) 751-782.
Steven Dale Cutkosky, Department of Mathematics, University of Missouri
Columbia, MO 65211, USA
[email protected]