A NOTE ON HENSEL S LEMMA IN SEVERAL VARIABLES

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 11, November 1997, Pages S (97) A NOTE ON HENSEL S LEMMA IN SEVERAL VARIABLES BENJI FISHER (Communicated by Wolmer V. Vasconcelos) Abstract. The standard hypotheses or Hensel s Lemma in several variables are slightly stronger than necessary, in the case that the Jacobian determinant is not a unit. This paper shows how to weaken the hypotheses or Hensel s Lemma and some related theorems. 0. Introduction The most amiliar version o Hensel s Lemma states that i is a polynomial with coeicients in Z p, a Z p is an approximate root o (i.e., (a) 0(modp)), and (a) is a unit (i.e., (a) 0(modp)) then there is a unique root a near a: (a )=0anda a(mod p). The irst application o this theorem is to the polynomial (X) =X p X: by Fermat s Little Theorem, every p-adic integer a is an approximate root o ; clearly, (a) 1(modp) or all a Z p. Thus there are p roots o this polynomial in Z p ; they are known as the Teichmüller representatives o the residue classes (mod p). One way to generalize ( Hensel s Lemma is to relax the requirement that (a) be a unit. Let h = v p (a) ) ( ). Letting r = v p (a), one can show that i r>2hthen there is a unique root a such that a a (mod p r h ). The irst example o this version o Hensel s Lemma is extracting square roots 2-adically: i a 2 c (mod 2 r ), with c odd and r>2, then there is a square root o c congruent to a (mod 2 r 1 ). In particular, an odd number has a square root in Z 2 i and only i it is congruent to1(mod8). A urther generalization is to consider n polynomial equations in n variables. The analogous statement, with the derivative replaced by the Jacobian determinant, is true. Let bold-ace letters denote n-tuples: a =(a 1,...,a n )andsoon. Letbe an n-tuple o polynomials in n variables and J = ( ( 1,..., n )/ (X 1,...,X n )the Jacobian determinant. Let a Z n p and h = v p J (a) ) (. I r =min i {v p i (a) ) }> 2hthen there is a unique a Z n p such that i (a )=0anda i a i (mod p r h )or all i. The point o this note is that the hypothesis r > ( 2h above is stronger than necessary. The irst improvement is to note how h = v p J (a) ) is used in the proo: by Cramer s Rule, the Jacobian matrix times its adjoint (or adjoint-transpose) has the orm p h (unit) I n ; in other words, the Jacobian matrix( M (a) divides p h I n. Any value o h or which this is true will work just as well as v p J (a) ). For example, Received by the editors May 20, Mathematics Subject Classiication. Primary 13J15; Secondary 13J05, 13B40. Key words and phrases. Hensel s lemma, power series, Henselian rings c 1997 American Mathematical Society License or copyright restrictions may apply to redistribution; see

2 3186 BENJI FISHER i M (a) is diagonal, say M (a) = diag(p,...,p,1) then one can take h = 1 instead o h = v p ( J (a) ) = n 1, a big improvement. Sometimes it is convenient to allow larger values o h. The second improvement is to note that the hypothesis that (a) be divisible by p 2h+1 can be replaced by the hypothesis that (a) be divisible by p h+1 M (a), in the sense o matrix multiplication: that is, (a) =p h M (a) bor some b (pz p ) n. For example, again suppose that M (a) is diagonal, say M (a) =diag(p h,1,...,1) (so that the irst improvement had no eect). Instead o requiring that i (a) 0 (mod p 2h+1 ) or all i, one may require this or i =1and i (a) 0(modp h+1 )or all i>1. For a number-theoretic application o this improved orm o Hensel s Lemma, see my paper with R. Dabrowski, [Da-F, Theorem 1.8]. I would be interested to see a more geometric application. In the situation described above, the proo is given in [Da-F, Lemma 1.20]. In the rest o this paper, I will veriy that the same improvements can be made in more general settings. I would like to thank Romuald Dabrowski or his insistence, while we were preparing [Da-F], that we give deinitive statements o our results. It was this insistence that orced me to ormulate this improved version o Hensel s Lemma. 1. Deinitions and notations Following Bourbaki [B2, III.4.5], I will say that a commutative ring A and an ideal m A satisy Hensel s conditions i A is complete and separated with respect to a linear topology (i.e., a topology deined by ideals) and m is closed, consisting o topologically nilpotent elements. (In the Introduction, A = Z p and m = pz p.) To avoid conusion with the n th power o m, the n-old Cartesian product o m with itsel will be denoted m (n). (In [B2], this is denoted m n.) Let A{X} denote the ring o restricted (ormal) power series in n variables, where X =(X 1,...,X n )[B2, III.4.2]. I =( 1,..., m )isanm-tuple in A{X}, let M denote the Jacobian matrix; i m = n, letj =detm be the Jacobian determinant. Let M (r) denote the matrix consisting o the irst r columns o M ; let M ( r) denote the matrix consisting o the last r columns o M. (In [B2], there is no notation or the irst r columns; the last r columns are denoted M (r).) In particular, M = [ M (r) M (r n) ]. Similarly, i a A m,leta (r) and a ( r) denote the irst and last r entries o a. Let1 n denote the n-tuple 1 n = X =(X 1,...,X n ), so that M 1n = I n (the identity matrix). I will always think o and a as column vectors. In particular, i A is a discrete ring then it is automatically complete and separated; m is automatically closed; and a restricted power series is simply a polynomial. In any case, polynomials are special cases o restricted power series. 2. Hensel s Lemma in several variables Let A be a ring and m A an ideal. I will assume that A and m satisy the simplest version o Hensel s Lemma in several variables and derive a more general version that incorporates the points discussed in the Introduction. The argument closely ollows one in Greenberg [Gr]. I will then show that i A is a Henselian local ring and m is its maximal ideal then this theorem applies. (This is surely well-known, but I do not know a reerence.) License or copyright restrictions may apply to redistribution; see

3 A NOTE ON HENSEL S LEMMA IN SEVERAL VARIABLES 3187 Assume that A is complete and separated and that m is closed, with respect to a linear topology. Let us say that (A, m) satisies condition (H) i the ollowing version o Hensel s Lemma holds: Condition (H). Let =( 1,..., n ) be an n-tuple o restricted power series i A{X} and let J = J be the Jacobian determinant. Let a A n ; assume that (a) m (n) and that J(a) A. Then there is a unique a A n such that (a )=0and a a (mod m (n) ). Remark. I we urther assume that every element o m is topologically nilpotent then (A, m) satisy Hensel s conditions, and so condition (H) is satisied by [B2, III.4.5, Corollaire 3] (c. 3, below). Theorem 1. Assume that (A, m) satisies condition (H). Let=( 1,..., n ) be an n-tuple with i A{X}. Leta A n and let e A be such that M (a) M = ei n or some matrix M (with entries in A). Assume that (a) em (a)m (n) : i.e., (a) =em (a) b or some b m (n). Then there is some a A n such that (a )=0and a a (mod em). Ieis not a zero-divisor then a is unique. Proo. For a single ormal power series A[[X]], Taylor s theorem [B1, IV.5.8, Proposition 9] gives (X + Y) =(X)+M (X) Y+ G ij (X, Y) Y i Y j 1 i j n or some G ij A[[X, Y]]. Arguing as in [B2, III.4.5], i is restricted then so are the entries / X i o M and the G ij. Thereore, taking = i, one can replace X with a and Y with ex, obtaining (1) (a + ex) =(a)+m (a) ex+e 2 R(X)=eM (a) [b + X + M R(X)], where the remainder terms satisy R i (X) 2 A{X}. Leth(X)=b+X+M R(X) and apply condition (H) to h: since M h (0)=I n and h(0) =b m (n), there is a unique x m (n) such that h(x) =0.Leta =a+ex,sothata a(mod em (n) ) and (a )=0. Now assume that e is not a zero-divisor. Since M (a) M = ei n, J (a) = det M (a) is also not a zero-divisor. I a = a + ex is a root o, with x m (n), then multiplying both sides o Equation (1) (with X replaced by x) by the adjoint o M (a) gives0=ej(a)h(x), which implies h(x) =0. This proves uniqueness. Proposition 2. Let A be a (discrete) local ring with maximal ideal m. ThenAis Henselian i and only i (A, m) satisy condition (H). Proo. Assume that A is a Henselian local ring. The irst step is to reduce to the case n = 1, using the structure theory o étale A-algebras. Following the notation o Raynaud [R], let C = A[X], I =() C,andB=(C/I) J = A[X,T]/(,TJ 1). I R is any A-algebra then Hom A (B,R) isthesetoa R n such that (a) =0 and J(a) R. Thus we must show that the canonical map Hom A (B,A) Hom A (B,k) is an isomorphism, where k = A/m. According to the Jacobian criterion [R, V, Théorème 5] and the act that being étale is a local condition [R, II, Proposition 6], B is an étale A-algebra. The question is local on Spec B: choosing a homomorphism φ : B k determines a prime ideal q =kerφ, and liting φ toamapb A is the same as liting it to a map License or copyright restrictions may apply to redistribution; see

4 3188 BENJI FISHER B g A, or any g B q. By the local structure theorem [R, V, Théorème 1], one can choose g so that B g is isomorphic to a standard étale A-algebra: B g = B = ( A[X]/() ) h or a single monic polynomial A[X] (in one variable), where is invertible in B. This completes the reduction step. The surjectivity o Hom A (B,A) Hom A (B,k) is just a translation o [R, VII, Proposition 3]. Injectivity ollows rom a standard argument: i (a+x) =(a)=0 with x m then Taylor s theorem gives x[ (a) +G(a, x)x] = 0. Since (a) isa unit and x m, the quantity in brackets is a unit, which implies that x =0. Conversely, i (A, m) satisy condition (H) then, taking n = 1, the converse direction o [R, VII, Proposition 3] shows that A is Henselian. 3. The Inverse and Implicit Function Theorems Theorems 3 and 4 below are improvements o Théorème 2 and Corollaire 3 o [B2, III.4.5], to which one may reer or details o the proos. Theorem 3 is an algebraic version o the Inverse Function Theorem; Theorem 4 is an algebraic version o the Implicit Function Theorem. Taking r = 0 in Theorem 4 gives Hensel s Lemma in several variables. (Instead o re-proving this theorem, one can apply Theorem 1 to produce an exact root and then use the version in [B2].) Theorem 3. Let A and m satisy Hensel s conditions. Let be an n-tuple o restricted power series i A{X} and let a A n. Let e A and let M be an n n matrix (with entries in A) such that M (a) M = ei n. There is an n-tuple g, with g i (X)A{X}, suchthat (i) M g (0) = I n. (ii) For all x A n, (a + ex)=(a)+m (a) eg(x). (iii) Let h be the n-tuple o ormal power series (not necessarily restricted) h i A[[X]] such that g h = 1 n. For all y m (n), (a + eh(y)) = (a)+m (a) ey. Proo. Taking g = 1 n +M R, (i) and (ii) ollow rom Equation (1); and (iii) ollows by replacing x with h(y). Theorem 4. Let A and m satisy Hensel s conditions. Let =( r+1,..., n ) be an (n r)-tuple o restricted power series i A{X} and let a A n. Let e A and let M be an (n r) (n r) matrix (with entries in A) such that M (r n) (a) M = ei n r. Assume that (a) =em (r n) (a) b or some b m (n r). Then there are n r power series φ i (X (r) )A[[X (r) ]] (r <i n)such that, or all t m (r), ( a (r) + e 2 t, a (r n) + eφ(t) ) =0. [ ] Proo. Apply Theorem 3 to u =(X (r) a (r) Ir 0,). Note that M u = M (r) M (r n), [ ] ei r 0 so that M u (a) M M (r) (a) M = ei n. It ollows that, in Equation (1), G ij =0 and R i =0or1 i r; and, in the proo o Theorem 3, g i = h i = X i. License or copyright restrictions may apply to redistribution; see

5 A NOTE ON HENSEL S LEMMA IN SEVERAL VARIABLES 3189 According to Theorem 3(iii), u ( a + eh(y) ) [ ] y = u(a)+m u (a) ey=m u (a) e (r) b+y (n r). Thus ( a+eh(y) ) = 0 i and only i e ( M (r) (a) y (r) +M (r n) (a) (b+y (r n) ) ) = 0. To guarantee this, it suices to set y (r) = et and y (r n) = b M M (r) (a) t, with t m (r). Thereore set φ(x (r) )=h (r n) (ex (r), b M M (r) (a) X (r) ); the theorem ollows. Reerences [B1] N. Bourbaki, Algèbre, Hermann, Paris, [B2], Algèbre Commutative, Hermann, Paris, [Da-F] R. Dabrowski and B. Fisher, A stationary-phase ormula or exponential sums over Z/p m Z and applications to GL(3)-Kloosterman sums, Acta. Arith. (to appear). [Gr] M. J. Greenberg, Rational points in Henselian discrete valuation rings, Pub. Math. IHES 31 (1966), MR 34:7515 [R] M. Raynaud, Anneaux Locaux Henseliens, Lecture Notes in Math. 169, Springer-Verlag, Berlin-Heidelberg-New York, MR 43:3252 Department o Mathematics, Columbia University, New York, New York Current address: The Bronx High School o Science, 75 West 205 th Street, Bronx, New York address: benji@math.columbia.edu License or copyright restrictions may apply to redistribution; see