arXiv:1802.05109v1 [] 14 Feb 2018

SIMPLE GENERAL NERON DESINGULARIZATION IN LOCAL
Q-ALGEBRAS.
DORIN POPESCU
Abstract. We give here a easier proof of the so-called General Neron Desingularization in the frame of local algebras.
Key words : Smooth morphisms, regular morphisms
2010 Mathematics Subject Classification: Primary 13B40, Secondary 14B25,13H05,13J15.

Introduction
A ring morphism u : A → A′ of Noetherian rings has regular fibers if for all prime
ideals p ∈ Spec A the ring A′ /pA′ is a regular ring. It has geometrically regular fibers
if for all prime ideals p ∈ Spec A and all finite field extensions K of the fraction field
of A/p the ring K ⊗A/p A′ /pA′ is regular. A flat morphism of Noetherian rings u is
regular if its fibers are geometrically regular. If u is regular of finite type then u is
called smooth.
The following theorem extends Néron’s desingularization (see [11], [7]) and it was
useful to solve different problems concerning the projective modules over regular
rings, or from the Artin Approximation Theory (see [2], [16], [18], [19], [20], [24],
[22], [4]).
Theorem 1. (General Néron Desingularization, Popescu [16], [17], [18], André [1],
Swan [24]) Let u : A → A′ be a regular morphism of Noetherian rings and B an
A-algebra of finite type. Then any A-morphism v : B → A′ factors through a smooth
A-algebra C, that is v is a composite A-morphism B → C → A′ .
Other authors gave similar proofs (see [23], [21], [9]) but most of them were not
constructive. In [14], [12], [13], [5], [6], [7], [8], there exist some constructive proofs
with algorithms in some special cases.
Let A be a Noetherian ring, B = A[Y ]/I, Y = (Y1 , . . . , Yn ). If f = (f1 , . . . , fr ),
r ≤ n is a system of polynomials from I then we can define the ideal ∆f generated
by all r × r-minors of the
P Jacobian matrix (∂fi /∂Yj ). After Elkik [3] let HB/A be
the radical of the ideal f ((f ) : I)∆f B, where the sum is taken over all systems of
polynomials f from I with r ≤ n. HB/A defines the non smooth locus of B over A. B
is standard smooth over A if there exists f in I as above such that B = ((f ) : I)∆f B.
In [8], it is given the following constructive theorem.
Theorem 2. Let A and A′ be Noetherian local rings, u : A → A′ be a regular
morphism and B an A-algebra of finite type. Suppose that A′ is Henselian and
the maximal ideal m of A generates the maximal ideal of A′ . Any A-morphism
1

v : B → A′ such that v(HB/A A′ ) is mA′ -primary ideal factors through a standard
smooth A-algebra B ′ .
The proof was based on the following proposition (see [8, Proposition 3], where
the conditions ”A′ is Henselian, and A, A′ have dimensions m are superfluous” and
we remove them here).
Proposition 3. Let A and A′ be Noetherian local rings and u : A → A′ be a regular
morphism. Let B = A[Y ]/I, Y = (Y1 , . . . , Yn ), f = (f1 , . . . , fr ), r ≤ n be a system
of polynomials from I as above, (Mj )j∈[l] some r × r-minors 1 of the Jacobian matrix
P
(∂fi /∂Yj ′ ), (Nj )j∈[l] ∈ ((f ) : I) and set P := lj=1 Nj Mj . Let v : B → A′ be an
A-morphism. Suppose that
(1) there exists an element d ∈ A such that d ≡ P modulo I and
(2) there exist a smooth A-algebra D and an A-morphism ω : D → A′ such that
Im v ⊂ Im ω + d2e+1 A′ and for Ā = A/(d2e+1 ) (defining e by (0 :A de ) =
(0 :A de+1 )) the map v̄ = Ā⊗A v : B̄ = B/d2e+1 B → Ā′ = A′ /d2e+1 A′ factors
through D̄ = D/d2e+1D.
Then there exists a B-algebra B ′ which is standard smooth over A such that v factors
through B ′ .
It is the purpose of our paper to provide a easier proof (see Theorem 9) of Theorem
1 in the local Q-algebras following the ideas from Theorem 2 and Proposition 3. The
proof is not constructive (see Remark 7), but Proposition 10 shows a case when this
is so.
1. A simple proof of the General Neron Desingularization
p
Let q ∈ Spec A′ be a minimal prime over ideal of hB = v(HB/A )A′ . After [24]
we say that A → B → A′ ⊃ q is resolvable if there exists an A-algebra of finite type
C such thatpv factors through C, let us say v = βt, t : B → C, β : C → A′ and
hB ⊂ hC = β(HC/A )A′ 6⊂ q.
The following lemma is in fact [18, Lemma 6.6] which unifies [24, Lemmas 12.2,
12.3]. We give here its proof in sketch for the sake of completeness.
Lemma 4. Suppose that height q = 0 and Ap → Bp = Ap ⊗A B ⊗ A′q ⊃ qA′q ,
p = u−1 (q) is resolvable. Then A → B → A′ ⊃ q is resolvable too.
Proof. By hypothesis there exists an Ap -algebra D such that vp = Ap ⊗A v factors
βp

through D, let us say vp is the composite map Bp → D −→ A′q and βp (HD/Ap ) = A′q .
Moreover we may take D to be smooth over Ap using for instance the easy lemma
[15, Lemma 2.4]
Let D = Bp [Z]/(g), Z = (Z1 , . . . , Zs ), g = (g1 , . . . , ge ) ∈ B e and βp be given
by Z → z/z0 for some z ∈ A′s , z0 ∈ A′ \ q Take the homogeneous polynomials
G(Z, Z0) = Z0c g(Z/Z0) for some c >> 1. We have G(z, z0 ) = 0 in A′q and so
aG(z, z0 ) = 0 in A′ for some a ∈ A′ \ q. Changing z, z0 by az, az0 we may assume
1

We use the notation [l] = {1, . . . , l}.
2

that G(z, z0 ) = 0 in A′ . Let C = B[Z, Z0]/(G) and β : C → A′ be the extension of
v by (Z, Z0 ) → (z, z0 ). We get a Bp -isomorphism ν : D[Z0 , Z0−1 ] → (Bp ⊗B C)Z0 by
(Z, Z0) → (Z/Z0, Z0 ) and the map Cp,Z0 → A′q induced by β, factors through ν. We
may change D p
by the smooth Ap -algebra D[Z0 , Z0−1 ], resp. g by G, C by B[Z]/(g).
We have hC = β(HC/A )A′ 6⊂ q, let us say β(t) 6∈ q for some t ∈ HC/A .
If hB ⊂ hC there exists nothing to show. Otherwise, hkB = 0 in A′q , that is whkB = 0
′
in A′ for some
P w ∈ A \ q. Let b1 , . . . , bs be a system of generators of hB and set
fj = gj − i Ui Vij , for some new variables U = (U1 , . . . , Us ), V = (Vij )i∈[s],j∈[e].
Then
E = B[Z, U, V, W ]/(f, W U1, . . . , W Us )
′
and δ : E → A extending v by U → b, V → 0, W → w resolves A → B → A′ ⊃ q.
Indeed, EUi is a polynomial algebra over B[Ui , Ui−1 ], so smooth
p over B. Thus Ebi Ui
is smooth over A and bi Ui ∈ HE/A , that is hB ⊂ hE = δ(HE/A )A′ . Note that
EW ∼
= C[V, W, W −1 ] and so EtW is smooth over A. Hence
= B[Z, V, W, W −1]/(g) ∼
β(t)w ∈ hE \ q.

The proof of Proposition 3 works for the extension given below because in the
local case s, s′, s′′ ∈ 1+(d) (see the proof of [8, Proposition 3]) are invertible elements
and so V is smooth over D, in particular hD ⊂ hB′ . In fact the idea of the following
proposition comes from [15, Proposition 7.1] and [18, Lemma 7.2].
Proposition 5. Let A and A′ be Noetherian rings and u : A → A′ be a regular
morphism. Suppose that A′ is local. Let B = A[Y ]/I, Y = (Y1 , . . . , Yn ), f =
(f1 , . . . , fr ), r ≤ n be a system of polynomials from I as above, (Mj )j∈[l] some
r × r-minors of the Jacobian matrix (∂fi /∂Yj ′ ), (Nj )j∈[l] ∈ ((f ) : I) and set P :=
Pl
′
j=1 Nj Mj . Let v : B → A be an A-morphism. Suppose that
(1) there exists an element d ∈ A such that d ≡ P modulo I and
(2) there exist an A-algebra D of finite type and an A-morphism ω : D → A′
such that Im v ⊂ Im ω + d2e+1 A′ and for Ā = A/(d2e+1 ) (defining e by (0 :A
de ) = (0 :A de+1 )) the map v̄ = Ā⊗A v : B̄ = B/d2e+1 B → Ā′ = A′ /d2e+1 A′
factors through D̄ = D/d2e+1 D.
Then there exists a B-algebra B ′ such that
v factors through B ′p
, let us say v is the
p
w
′
′
′
composite map B → B −
→ A and hD = ω(HD/A )A ⊂ hB′ = w(HB′ /A )A′ .
Next theorem is an extension of Theorem 2.
Theorem 6. Let u : A → A′ be a flat morphism of Noetherian rings and q ∈ Spec A′ .
Suppose that A′ is local, p = u−1 (q) ∈ Spec A generates the maximal ideal qA′q of A′q
and u induces a regular morphism Ap → A′q . Any A-morphism v : B → A′ such that
v(HB/A )A′q is qA′q -primary ideal factors through an A-algebra of finite type B ′ , let
p
w
us say v is the composite map B → B ′ −
→ A′ , with hB ⊂ hB′ = w(HB′ /A )A′ 6⊂ q,
that is A → B → A′ ⊃ q is resolvable.
Proof. Let m = height p = height q and choose γ1 , . . . , γm ∈ u−1 (v(HB/A A′q ), which
induces a system of parameters in A′q . The proof goes with A′q instead A′ as in [8,
Theorem 2] up to the ”Cases”, where we proceed as follows.
3

Case I If m = 0 then Ap , A′q are Artinian local rings and by [15, Corollary 3.3]
we get that Ap → Ap ⊗A B → A′q ⊃ qA′q is resolvable. Then by Lemma 4 it follows
that A → B → A′ ⊃ q is resolvable.
Case II If m > 0 we assume by induction hypothesis as in [8, Theorem 2]
γ = γm , d = dm and the existence of an Ā = A/(d2e+1)-algebra of finite type
D̄ = D/(d2e+1)D ∼
= (Ā[Z]/(ḡ))h̄M̄ , Z = (Z1 , . . . , Zr ), ḡ = (ḡ1 , . . . , ḡs ) such that
ω̄
v̄ = Ā ⊗A v factors
→ Ā′ and
q through D̄, let us say v̄ is the composite map B̄ → D̄ −
hB Ā′ ⊂ hD̄ = ω̄(HD̄/Ā )Ā′ 6⊂ q Ā′ .
Now, let g, M, h be some liftings of ḡ, M̄ , h̄ in A[Z], fi = gi − d2e+1 Ti , T =
(T1 , . . . , Ts ) and set D = A[Z, T ]/(f ). Let t ∈ A′s be such that g(z) = d2e+1 t.
The map ω : D → A′ given by (Z, Tp
) → (z, t) lifts ω̄. It follows that Im v ⊂
Im ω + d2e+1 A′ . Note that d ∈ hD = ω(HD/A )A′ . It follows that hB ⊂ hD 6⊂ q
because Ā ⊗A D ∼
= D̄[T ] is smooth over D̄ and hB Ā′ ⊂ hD̄ 6⊂ q Ā′ .
Applying Proposition 5 we see that v factors through an A-algebrapof finite type
w
B ′ , let us say v is the composite map B → B ′ −
→ A′ and hD ⊂ hB′ = w(HB′ /A )A′ .
It follows that A → B → A′ ⊃ q is resolvable.

Remark 7. The above theorem seems to be not constructive as it is Theorem 2
(more details in Proposition 10). If A′ = Q[[x1 .x2 ]] and height q = 1 then to resolve
A → B → A′ ⊃ q by a computer we might need to define precisely v modulo a power
of q, that is to give formal power series in x with infinite number of coefficients. This
is not possible. When q = (x1 , x2 ) then to define v modulo a power of q means only
to give a finite number of such coefficients (this follows from the proof of Theorem
2).
Lemma 8. Let u : A → A′ be a regular morphism of Noetherian rings, B an Aalgebra of finite type, v : B → A′ an A-algebra morphism and q ∈ Spec A′ a minimal
prime over ideal of hB . Suppose that A′ is local, height q ≥ 1 and char Ap /pAp = 0,
p = u−1 (q). Then A → B → A′ ⊃ q is resolvable, that is v factors through an
w
A-algebra of finite
→ A′ , such
p type C, let us say v is the composite map B → C −
that hB ⊂ hC = w(HC/A )A′ 6⊂ q.
Proof. Apply Theorem 6 if t := height q − height p = 0. Otherwise, A′q /pA′q is a
regular local ring of dimension t > 0 by the regularity of u. Suppose that z =
(z1 , . . . , zt ) ∈ A′t induces a regular system of parameters in A′q /pA′q . Let A[Z] be
the polynomial A-algebra in the variables Z = (Z1 , . . . , Zt ). By the local criterion
of flatness [10], the map u1 : A1 = Ap [Z](p,Z) → A′q given by Z → z is flat, even
regular since A′q /pA′q is regular and char Ap /pAp = 0. Let v1 : B1 = A1 ⊗A B → A′q
be the extension of v given by Z → z. Then v1 (HB1 /A1 )A′q is a qA′q -primary ideal
and applying Theorem 6 we see that A[Z] → B[Z] → A′ ⊃ q is resolvable and so
A → B → A′ ⊃ q is resolvable too.

Theorem 9. Let u : A → A′ be a regular morphism of Noetherian Q-algebras, B
an A-algebra of finite type and v : B → A′ an A-algebra morphism. Suppose that A′
is a local ring. Then v factors through a standard smooth A-algebra.
4

Proof. As in [15, Corollary 3.3] we may suppose that height v(HB/A )A′ ≥ 1. This
could be done in a constructive way (see for instance the beginning of Section 2 of
[14] in the case when A′ is a domain). Let q ∈ Spec A′ be a minimal prime over
ideal of v(HB/A )A′ . Applying Lemma 8 we see that v factors through an A-algebra
w1
of finite type p
B1 , let us say v is the composite map B → B1 −→
A′ , such that
v(HB/A )A′ ⊂ w1 (HB1 /A )A′ 6⊂ q. Applying again Lemma 8 for B1 and a certain
minimal prime over ideal q1 of w1 (HB1 /A )A′ we see that w1 factors through an Aw2
algebra of finite type B2 , let
us say w1 is the composite map B1 → B2 −→
A′ ,
p
′
′
w2 (HB2 /A )A 6⊂ q1 . By Noetherianity this procesuch that w1 (HB1 /A )A ⊂
p
dure should end since the strictly increasing chain v(HB/A )A′ ⊂ w1 (HB1 /A )A′ ⊂
p
w2 (HB2 /A )A′ ⊂ . . . should stop. This could happens only when wi (HBi /A )A′ = A′
for some i. Using [15, Lemma 2.4] we may replace Bi by an A-algebra C of finite
type with HC/A = C, that is C is a smooth algebra over A. We may choose C to be
standard smooth over A using [15, Lemma 3.4] (see also [8, Lemma 4]).

Next, let u : A → A′ be a regular morphism of Noetherian local rings of dimension
two, B an A-algebra of finite type and v : B → A′ an A-morphism. If hB =
p
v(HB/A )A′ is the maximal ideal of A′ then we have a constructive General Neron
Desingularization using Theorem 2 (see also [13]). Always we may reduce to the case
height hB ≥ 1 as we noticed in the proof of Theorem 9. The difficulties to construct
a General Neron Desingularization could appear only when we show that A → B →
A′ ⊃ q is resolvable for a minimal prime over ideal q of hB with height q = 1.
Proposition 10. If there exists a minimal over ideal q of hB such that u−1 (q) has
height 1 then A → B → A′ ⊃ q is resolvable in a constructive way. Consequently,
in this case Theorem 9 holds in a constructive way.
Proof. Let B = A[Y ]/I, Y = (Y1 , . . . , Yn ) and q be a minimal over ideal of hB such
that u−1 (q) has height 1. Choose γ ∈ u−1 (v(HB/A )A′ ) such that height(γA′ ) = 1.
As in the proof of [8, Theorem 2] we may change B such that a power d of γ is
in ((f ) : I)∆f for some polynomials f = (f1 , . . . , fr ), r ≤ n from I. Choosing e
as in Proposition 5 it is enough to see as in Case 2 from Theorem 6 that Ā =
A/(d2e+1 ) → B̄ = Ā ⊗A B → Ā′ = A′ /d2e+1 A′ ⊃ q Ā′ is resolvable. But this is done
in [12] because we are now in dimension one.

References
[1] M. André, Cinq exposés sur la désingularisation, Handwritten manuscript Ecole
Polytechnique Fédérale de Lausanne, (1991).
[2] M. Artin, Algebraic approximation of structures over complete local rings, Publ.
Math. IHES, 36, (1969), 23-58.
[3] R. Elkik, Solutions d’équations à coéfficients dans un anneaux hensélien, Ann.
Sci. Ecole Normale Sup., 6 (1973), 553-604.
[4] H. Hauser, The Classical Artin Approximation Theorems, Bull. Amer. Math.
Soc. 54, (2017), 595-633.
5

[5] A. Khalid, Z. Kosar, An Easy proof of the General Neron Desingularization in
dimension 1, Bull. Math. Soc. Sci. Math. Roumanie, 59 (107), 2016, 349-353,
arXiv:AC/1605.04677.
[6] A. Khalid, G. Pfister, D. Popescu, A uniform General Neron Desingularization in dimension one, J. of Algebra and Its Applications, 16, (2018),
arXiv:AC/1612.03416.
[7] A. Khalid, A. Popescu, D. Popescu, Algorithms in the classical Néron Desingularization, Bull. Math. Soc. Sci. Math. Roumanie, 61(109), (2018), 73-83,
arXiv:AC/1702.01445.
[8] Z. Kosar, G. Pfister, D. Popescu, Constructive Néron Desingularization of algebras with big smooth locus, Communications in Algebra,
http://dx.doi.org/10.1080/00927872.2017.1360333, arXiv:1702.01867.
[9] J. Majadas, A. Rodicio, Smoothness, Regularity and Complete Intersection, London Math. Soc. Lect. Note 373, 2010.
[10] H. Matsumura, Commutative Algebra, Benjamin, New-York, 1980.
[11] A. Néron, Modèles minimaux des varietés abéliennes sur les corps locaux et
globaux, Publ. Math. IHES, 21, 1964, 5-128.
[12] G. Pfister, D. Popescu, Constructive General Neron Desingularization for one
dimensional local rings, Journal of Symbolic Computation, 80, (2017), 570-580.
[13] G. Pfister, D. Popescu, Construction of Neron Desingularization for Two Dimensional Rings, to appear in ”Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory”, G. Böckle et al. (eds.), Springer AG
2017, https://doi.org/10.1007/978-3-319-70566-8-24, arXiv:AC/1612.01827.
[14] A. Popescu, D. Popescu, A method to compute the General Neron Desingularization in the frame of one dimensional local domains, to appear in ”Singularities
and Computer Algebra - Festschrift for Gert-Martin Greuel, On the Occasion of
his 70th Birthday”, Editors Wolfram Decker, Gerhard Pfister, Mathias Schulze,
Springer Monograph., DOI 10.1007/978-3-319-28829-1, arXiv:AC/1508.05511.
[15] D. Popescu, General Neron Desingularization, Nagoya Math. J., 100 (1985),
97-126.
[16] D. Popescu, General Neron Desingularization and approximation, Nagoya
Math. J., 104, (1986), 85-115.
[17] D. Popescu, Letter to the Editor, General Neron Desingularization and approximation, Nagoya Math. J., 118, (1990), 45-53.
[18] D. Popescu, Artin Approximation, in ”Handbook of Algebra”, vol. 2, Ed. M.
Hazewinkel, Elsevier, 2000, 321-355.
[19] D. Popescu, Around General Neron Desingularization, Journal of Algebra
and Its Applications, 16, No. 2 (2017), doi: 10.1142/S0219498817500724,
arXiv:1504.06938.
[20] D. Popescu, Artin approximation property and the General Neron Desingularization, Revue Roum. Math. Pures et Appl., 62, (2017), 171-189
arXiv:AC/1511.06967.
[21] R. Quarez, The Artin Conjecture for Q-algebras, Rev. Mat. Univ. Complut.
Madrid 10, (1997), 229-263.
6

[22] G. Rond, Artin Approximation, arXiv:1506.04717.
[23] M. Spivakovsky, A new proof of D. Popescu’s theorem on smoothing of ring
homomorphisms, J. Amer. Math. Soc., 294, (1999), 381-444.
[24] R. Swan, Neron-Popescu desingularization, in ”Algebra and Geometry”, Ed. M.
Kang, International Press, Cambridge, (1998), 135-192.
Dorin Popescu, Simion Stoilow Institute of Mathematics of the Romanian Academy, Research unit 5, University of Bucharest, P.O.Box 1-764, Bucharest 014700,
Romania
E-mail address: dorin.popescu@imar.ro

7

