Mullhaupt, Ph.

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Previous Up Next ArticleCitationsFrom References: 1From Reviews: 0MR2426512 (Review) 13N10 (13N15 14R15 16S32)Bavula, V. V. (4-SHEF-PM)The inversion formulae for automorphisms of polynomial algebras and rings of differentialoperators in prime characteristic. (English summary)J. Pure Appl. Algebra 212 (2008), no. 10, 2320–2337.Inversion formulas for a class of algebras—over a field of prime characteristic—are derived.Among such algebras, the paper considers (i) the polynomial algebra Pn :=K[x1, x2, . . . , xn]; (ii)the ring of differential operators D(Pn) on Pn, i.e., D(Pn)⊗Pn; (iii) the n-th Weyl algebra An;(iv) the algebra Pn⊗Am; (v) the power series algebraK[[x1, . . . , xn]]; (vi) Tk1,...,kn ⊗Pm, whereTk1,...,kn is the subalgebra of D(Pm) generated by Pn and the higher derivations ∂[j]i , 0 ≤ j < pki,i= 1, . . . , n, where k1, . . . , kn ∈ N.The general idea for finding the inversion formula is the following. Let A be an algebra over thefieldK, σ an automorphism over A, and {xα} aK-basis of A. The identity of the algebra is firstdecomposed as idA(·) =∑λα,yα(·)xα, where λα,yα are algebraic maps. Both this decompositionand the existence of the inverse are assumed to exist for σ. Applying σ(·) to the identity mapalso has the presentation idA(·) =∑λα,σ(yα)(·)σ(xα). Then because λα,σ(yα)(A)⊆K, one simplyapplies σ−1(·) to obtain the inverse expressed as σ−1(·) =∑λα,σ(yα)(·)xα. The whole difficultyis to find suitable maps λα,σ(yα)(·).So as to be more specific, some definitions and notation are required. Let δ be aK-derivation ofan algebra A over an arbitrary field K. A finite or infinite sequence x = {x[i], 0 ≤ i ≤ l− 1} ofelements inAwhere x[0] = 1 is called an iterative sequence of length l if x[i]x[j] = (i+ji )x[i+j], 0≤i, j ≤ i− 1, i+ j ≤ l− 1. An iterative δ descent is such a sequence for which δ(x[i]) = x[i−1], 0≤i≤ l− 1, x[−1] = 0. Whenever δ is nilpotent, i.e., δl = 0 for some l ≥ 2, twoK-linear maps fromA to A can be constructed starting from an iterative δ sequence {x[i], 0≤ i < l} in the followingway:ϕ :=∑l−1i=0(−1)ix[i]δi(·) and ψ :=∑l−1i=0(−1)iδi(·)x[i]. These maps are projection maps ontothe kernelAδ of δ; that is, if c ∈A is written as c= a+ bwith a ∈Aδ and b ∈A+, which is alwayspossible since A=Aδ⊕A+ with A+ :=⊕l−1i=1 x[i]Aδ, then ψ(c) = ϕ(c) = a.In the case of an automorphism σ that preserves the ring of invariants in the sense that σ(Aδ) =Aδ, the following concepts are required. Both a non-empty well-ordered set I and a set of com-muting locally nilpotentK-derivations δ := δi, i ∈ I , are given. Suppose that for each i ∈ I thereexists an iterative δi-descent {x[j]i } of maximal length such that {x[j]i } ⊆⋂i(=k∈I Aδk . Define (i)σδ := σ|Aδ ; (ii) the twisted derivations δ′ as {δ′i := σδiσ−1, i ∈ I}; and (iii) the images of theiterative descents x′i := {x′[j]i := σ(x[j]i )}, i ∈ I . The inversion formula is shown to beσ−1(a) =∑α∈Ex[α]σ−1δ ϕσ(δ′α(a)) =∑α∈Eσ−1δ ψσ(δ′α(a))x[α].The result for σ ∈ AutK(D(K[x1, . . . , xn])) is more involved, but nevertheless still rests on suit-able locally nilpotent derivations and their nil algebras.D(K[x1, . . . , xn]) is the ring of differentialoperators on the polynomial algebraK[x1, . . . , xn]. This algebra is aK-algebra generated by theelements x1, . . . , xn and commuting higher derivations ∂[k]i := δkik! , i= 1, . . . , n and k ≥ 1, that sat-isfy the defining relations [xi, xj] = [∂[k]i , ∂[l]j ] = 0, ∂[k]i ∂[l]i =(k+lk)∂[k+l]i , [∂[k]i , xj] = δij∂[k−1]i forall i, j = 1, . . . , n and k, l ≥ 1 where δij is the Kronecker delta and ∂[0]i := 1, ∂[1]i = ∂∂xi . Giventwo elements xi, xj , define the inner derivation as [xi, xj] = (adxi)(xj) := xixj − xjxi. The keyprojection maps are ϕi =∑k≥0 ∂[k]i (adxi)k, ψi =∑k≥0(adxi)k∂[k]i , i= 1, . . . n, which are usedto infer the validity of the inversion formulaσ−1(a) =∑α,β∈Nn(−1)|α|ϕσ(δ′β(a)∂′[α])∂[β]xα =∑α,β∈Nnψσ(∂′[α]δ′β(a))xα∂[β],where δ′β :=∏ni=1(−adx′i)βi, the primed quantities being x′i := σ(xi) and ∂′[k]i := σ(∂[k]i ).Reviewed byPhilippe A.Mu¨llhauptReferences1. K. Adjamagbo, A. van den Essen, A differential criterion and formulas for the inversion of apolynomial map in several variables, J. Pure Appl. Algebra 65 (1990) 97–100. MR1068247(92d:14011)2. H. Bass, E.H. Connell, D. Wright, The Jacobian conjecture: reduction of degree and formalexpansion of the inverse, Bull. Amer. Math. Soc. (New Ser.) 7 (1982) 287–330. MR0663785(83k:14028)3. V.V. Bavula, The inversion formula for automorphisms of the Weyl algebras and polynomialalgebras, J. Pure Appl. Algebra 210 (2007) 147–159. ArXiv:math.RA/0512215. MR2311178(2008b:14099)4. V.V.Bavula, The JacobianConjecture2n implies theDixmier Problemn. ArXiv:math.RA/0512250.5. V.V. Bavula, Simple derivations of differentiably simpleNoetherian commutative rings in primecharacteristic, Trans. Amer. Soc., in press (doi:S0002-9947(08)04567-4), published on line on20 March 2008, ArXiv:math.RA/0602632. MR2395162 (2009c:13060)6. A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the DixmierConjecture. arXiv:math.RA/0512171. MR23378797. J. Dixmier, Sur les alge`bres deWeyl, Bull. Soc. Math. France 96 (1968) 209–242. MR0242897(39 #4224)8. N. Jacobson, Structure of Rings, Revised edition, in: American Mathematical Society Col-loquium Publications, vol. 37, American Mathematical Society, Providence, RI, 1964.MR0222106 (36 #5158)9. H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, 1994. MR0879273(88h:13001)10. J.H. McKay, S.S.-S. Wang, On the inversion formula for two polynomials in two variables, J.Pure Appl. Algebra 52 (1988) 102–119. MR0949341 (89m:14008)11. Y. Tsuchimoto, Preliminaries on Dixmier conjecture, Mem. Fac. Sci. Kochi Univ. Ser. A Math.24 (2003) 43–59. MR1967342 (2004k:16070)12. Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures, Osaka J. Math. 42 (2)(2005) 435–452. MR2147727 (2006g:14101)13. A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, in: Progress inMathematics, vol. 190, Birkhauser Verlag, Basel, 2000. MR1790619 (2001j:14082)Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.c© Copyright American Mathematical Society 2009

Review of "The inversion formulae for automorphisms of polynomial algebras and rings of differential operators in prime characteristic", by Bavula, V. V., J. Pure Appl. Algebra, 212, 2008, no. 10, pp 2320-2337

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Review of "The inversion formulae for automorphisms of polynomial algebras and rings of differential operators in prime characteristic", by Bavula, V. V., J. Pure Appl. Algebra, 212, 2008, no. 10, pp 2320-2337

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