Ikehata, Shuichi

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Purely Inseparable Ring Extensions and Azumaya Algebras

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Mathematical Journal of OkayamaUniversityVolume 41, Issue 1 1999 Article 6JANUARY 1999Purely Inseparable Ring Extensions andAzumaya AlgebrasShuichi Ikehata∗∗Okayama UniversityCopyright c©1999 by the authors. Mathematical Journal of Okayama University is produced byThe Berkeley Electronic Press (bepress). http://escholarship.lib.okayama-u.ac.jp/mjouMath. J. Okayama Univ. 41 (1999), 63{69PURELY INSEPARABLE RING EXTENSIONSAND AZUMAYA ALGEBRASDedicated to Professor Takasi Nagahara on his 70th birthdaySHU^ICHI IKEHATAThroughout this paper, B will mean a ring with prime characteristicp, D a derivation of B. We denote by B[X;D] the skew polynomial ringde¯ned by aX = Xa+D(a) (a 2 B). By B[X;D](0), we denote the set ofall monic polynomials g in B[X;D] such that gB[X;D] = B[X;D]g. A ringextension T=S is called a separable extension, if the T -T -homomorphismof T ­S T onto T de¯ned by a ­ b ! ab splits, and T=S is called an H-separable extension, if T ­S T is T -T -isomorphic to a direct summand ofa ¯nite direct sum of copies of T . As is well known every H-separableextension is a separable extension. A polynomial g in B[X;D](0) is calledseparable (resp. H-separable) if B[X;D]=gB[X;D] is a separable (resp.H-separable) extension of B. A ring extension B=A of commutative ringsis called a purely inseparable extension of exponent one with ±, if AB isa ¯nitely generated projective module of ¯nite rank and Hom(AB;AB) =B[±], where ± is a derivation of B and A = fa 2 Bj±(a) = 0g. (cf. [2], [10],[11])In this paper, we shall use the following conventions.Z = the center of B.VB(A) = the centralizer of A in B for a ring extension B=A.u` (resp.ur) = the left (resp. right) multiplication e®ected by u 2 B.BD = fa 2 BjD(a) = 0g, where D is a derivation of B.DjA = the restriction of D to a subring A of B.DerA(B) = the set of all A-derivations of B.Iu = the inner derivation e®ected by u, that is, Iu = u` ¡ ur.In the previous paper [7], we have studied purely inseparable exten-sions of exponent one and H-separable polynomials in the skew polynomialrings of derivation type over non commutative rings. In particular we con-sidered Azumaya algebras whose centers are purely inseparable extensions1991 Mathematics Subject Classi¯cation. 16W25, 16S36.Key words and phrases. Derivations, Skew polynomial rings, H-separable extensions,Purely inseparable extensions of exponent one, Azumaya algebras.63 1Ikehata: Purely Inseparable Ring Extensions and Azumaya AlgebrasProduced by The Berkeley Electronic Press, 199964 SHU^ICHI IKEHATAof exponent one over their constant rings. Then we constructed new Azu-maya algebras. In this paper, some results in [7] will be generalized andsharpened. For example, in [7] we have proved the following: Let B be anAzumaya Z-algebra, D a derivation of B, and ± = DjZ. Assume that Z=Z±is a purely inseparable extension of exponent one with ±, and ± satis¯es theminimal polynomial tpe+ tpe¡1®e+ ¢ ¢ ¢+ tp®2+ t®1(®i 2 Z±). If there existsan element u in BD such that Dpe+ ®eDpe¡1+ ¢ ¢ ¢ + ®2Dp + ®1D = Iu,then B[X;D] is an Azumaya Z±[f ] -algebra, where f = Xpe+Xpe¡1®e +¢ ¢ ¢ + Xp®2 + X®1 ¡ u. ([7, Theoren 2.4]). Since B is separable over Z,it is clear that Dpe+ ®eDpe¡1+ ¢ ¢ ¢ + ®2Dp + ®1D = Iu for some u in B.It is important that u is contained in BD, which is assumed in the above.However, necessarily we can take such u in BD (Theorem 2). Moreover,we have more results when we can take Iu = 0 (Theorem 5).First, we shall state the following lemma which is immediate by [4 ,Theorem 4.1].Lemma 1. Let Z be a commutative ring of prime characteristic p.Let f = Xpe+Xpe¡1®e+ ¢ ¢ ¢+Xp®2+X®1+®0 (®i 2 Z±) be in Z[X; ±](0).Then f is a separable polynomial in Z[X; ±] if and only if there exists anelement c in Z such that±pe¡1(c) + ®e±pe¡1¡1(c) + ¢ ¢ ¢+ ®2±p¡1(c) + ®1c = 1:Now, we are in a position to prove the following theorem which is asharpenning of [7, Theorem 2.4, Proposition 2.6] and a generalization of[3, Theorem4.1].Theorem 2. Let B be an Azumaya Z-algebra, D a derivation of B,and ± = DjZ. Assume that Z=Z± is a purely inseparable extension ofexponent one with ±, Z is a projective module over Z± of rank pe, and ±satis¯es the minimal polynomialtpe+ tpe¡1®e + ¢ ¢ ¢+ tp®2 + t®1(®i 2 Z±):Then there exists an element u in BD such thatDpe+ ®eDpe¡1+ ¢ ¢ ¢+ ®2Dp + ®1D = Iu:Proof. Since B is separable over Z and the derivationDpe+®eDpe¡1+¢ ¢ ¢+ ®2Dp + ®1D equals to zero on the center Z, it is an inner derivarionof B. Hence there is an element w 2 B such that Dpe + ®eDpe¡1 + ¢ ¢ ¢ +®2Dp + ®1D = Iw: Since ®i 2 Z±, we have DIw = IwD. Hence D(w) 2Z. Since Z=Z± is a purely inseparable extension of exponent one with2Mathematical Journal of Okayama University, Vol. 41 [1999], Iss. 1, Art. 6http://escholarship.lib.okayama-u.ac.jp/mjou/vol41/iss1/6PURELY INSEPARABLE RING EXTENSIONS AND AZUMAYA ALGEBRAS 65±, Xpe+ Xpe¡1®e + ¢ ¢ ¢ + Xp®2 + X®1 is an H-separable polynomial inZ[X; ±] ([5, Theorem 3.3]), so it is separable polynomial in Z[X; ±]. Thenby Lemma 1, there exists an element c in Z such that±pe¡1(c) + ®e±pe¡1¡1(c) + ¢ ¢ ¢+ ®2±p¡1(c) + ®1c = 1:By Leibniz' formula, we obtainDpj¡1(cw) =pj¡1Xº=0µpj ¡ 1º¶Dpj¡1¡º(c)Dº(w)= ±pj¡1(c)w +pj¡1Xº=1µpj ¡ 1º¶±pj¡1¡º(c)Dº(w) (j ¸ 1):SincePpj¡1º=1¡pj¡1º¢±pj¡1¡º(c)Dº(w) 2 Z, we see thatDpj¡1(cw) = ±pj¡1(c)w + (some element in Z) for all j ¸ 1:Hence we havew = (eXj=0®j+1±pj¡1(c))w=eXj=0®j+1(Dpj¡1(cw)) + (some element in Z):SinceD(eXj=0®j+1(Dpj¡1(cw))) =eXj=0®j+1Dpj (cw) = Iw(cw) = 0;we have w 2 BD + Z. Then w = u+ z, for some u 2 BD and z 2 Z, andso Iw = Iu.It is well known that if B is an Azumaya Z-algebra, then every deriva-tion on Z can be extended to a derivation of B (M. A. Knus [8]). Hencein Theorem 2, such D always exists. In the proof of Theorem 2, we usedonly the separability of f = Xpe+Xpe¡1®e+ ¢ ¢ ¢+Xp®2+X®1 2 Z[X; ±].Hence by [4, Theorem 4.1] we have the followingCorollary 3. Assume that Xpe+Xpe¡1®e+ ¢ ¢ ¢+Xp®2+X®1+®0 isa separable polynomial in Z[X; ±]. Let B be an Azumaya Z-algebra. Thenthere exists a derivation D of B which is an extention of ± and an elementu in BD such that Xpe+Xpe¡1®e+ ¢ ¢ ¢+Xp®2+X®1+®0¡u is a separablepolynomial in B[X;D].3Ikehata: Purely Inseparable Ring Extensions and Azumaya AlgebrasProduced by The Berkeley Electronic Press, 199966 SHU^ICHI IKEHATAUnder the same situation of Theorem 2, we already have the following([7, Proposition 2.3 and Theorem 2.4])(1) f = Xpe+Xpe¡1®e+¢ ¢ ¢+Xp®2+X®1¡u is an H-separable polynomialin B[X;D].(2) B[X;D] is an Azumaya Z±[f ]-algebra, VB[X;D](B) = Z[f ], andVB[X;D](Z) = B[f ].(3) B[X;D](0) = f h(f) j h(t) is a monic polynomial in Z±[t]g.(4) f g 2 B[X;D] j g is an H-separable polynomial in B[X;D]g = f f +z j z 2 Z±g.Moreover, we have the following which is a generalization of [5, The-orem 3.4]Proposition 4. Let Ã : Z±[t](0) ! B[X;D](0) be de¯ned by Ã(g0(t)) =g0(f).(1) Ã indeuces a one-to-one correspondence between Z±[t](0) and B[X;D](0).(2) For g0(t) 2 Z±[t](0), g0(t) is a separable polynomial in Z±[t] if and onlyif B[X;D]=g0(f)B[X;D] is a separable Z±-algebra. Moreover, the centerof B[X;D]=g0(f)B[X;D] is isomorphic to Z±[t]=g0(t)Z±[t].Proof. (1) is clear from the statement (3) under Corollary 3.(2) Since B[X;D] is an Azumaya Z±[f ]-algebra, the center ofB[X;D]=g0(f)B[X;D] is (Z±[f ] + g0(f)B[X;D])=g0(f)B[X;D], which isisomorphic to Z±[t]=g0(t)Z±[t]. Then the assertion is immediate by [1, The-orem 2.3.8].In Theorem 2, if VB(BD) = Z, then obviously Iu = 0. Conversely,if Iu = 0, that is, Dpe+ ®eDpe¡1+ ¢ ¢ ¢ + ®2Dp + ®1D = 0, then we haveVB(BD) = Z. This will be proved in the following theorem.Theorem 5. Let B be an Azumaya Z-algebra, D a derivation of B,and ± = DjZ. Assume that Z=Z± is a purely inseparable extension ofexponent one with ±, and ± satis¯es the minimal polynomial tpe+ tpe¡1®e+¢ ¢ ¢+ tp®2 + t®1(®i 2 Z±). If Dpe +®eDpe¡1 + ¢ ¢ ¢+®2Dp+®1D = 0, thenthere hold the following:(1) B = BDZ = BD ­Z± Z, BDB is a ¯nitely generated projective module.(2) BD is an Azumaya Z±-algebra, and VB(BD) = Z.(3) Hom(BDBBD ;BDBBD) = Z[D] = Z © ZD © ZD2 © ¢ ¢ ¢ © ZDpe¡1 .(4) DerBD(B) = ZD © ZDp © ¢ ¢ ¢ © ZDpe¡1 : In particular, DerZ±(Z) =Z± © Z±p © ¢ ¢ ¢ © Z±pe¡1.4Mathematical Journal of Okayama University, Vol. 41 [1999], Iss. 1, Art. 6http://escholarship.lib.okayama-u.ac.jp/mjou/vol41/iss1/6PURELY INSEPARABLE RING EXTENSIONS AND AZUMAYA ALGEBRAS 67(5) B[X;D] and Z[X; ±] are Azumaya Z±[f ]-algebras and B[X;D]=Z[X; ±]­Z±BD = Z[X; ±]­Z±[f ] BD[f ], where f = Xpe +Xpe¡1®e + ¢ ¢ ¢+Xp®2 +X®1.Proof. We de¯ne the map ¿ : B ! B by¿(b) =eXj=0®j+1Dpj¡1(b):Since Dpe+®eDpe¡1+ ¢ ¢ ¢+®2Dp+®1D = 0, ¿ is a BD¡BD-map, and theimage is contained in BD. Since Z=Z± is a purely inseparable extension ofexponent one with ±, it follows from [5, Theorem 3.3(d)] that there existxi; yi 2 Z such thatXi±pe¡1(xi)yi = 1 andXi±k(xi)yi = 0 (0 · k · pe ¡ 2):We de¯ne the map 'i : B ! BD by 'i = ¿(xi)r. Then we haveXi'i(b)yi =Xi¿(bxi)yi=XieXj=0®j+1Dpj¡1(bxi)yi=XieXj=0®j+1pj¡1Xº=0µpj ¡ 1º¶Dpj¡1¡º(b)±º(xi)yi=eXj=0®j+1pj¡1Xº=0µpj ¡ 1º¶Dpj¡1¡º(b)(Xi±º(xi)yi)= b: (b 2 B)This shows that B = BDZ, and BDB is a ¯nitely generated projectivemodule. Since B »= BD ­Z± Z is an Azumaya Z-algebra and Z± is a directsummand of Z, BD is an Azumaya Z±-algebra by [1, Corollary 1.1.10].B = BDZ implies VB(BD) = Z. This complets the proof of (1) and (2).(3) Let ' be in Hom(BDBBD ;BDBBD). Then we have'(b) =Xi¿(bxi)'(yi): (b 2 B)5Ikehata: Purely Inseparable Ring Extensions and Azumaya AlgebrasProduced by The Berkeley Electronic Press, 199968 SHU^ICHI IKEHATASince VB(BD) = Z, it is easy to see '(Z) ½ Z. Hence we obtain'(b) =Xi'(yi)¿(bxi)=Xi'(yi)eXj=0®j+1pj¡1Xº=0µpj ¡ 1º¶±pj¡1¡º(xi)Dº(b):This implies that ' 2Ppj¡1º=0 ZDº . Since f = Xpe +Xpe¡1®e + ¢ ¢ ¢+Xp®2+X®1¡u is anH-separable polynomial in B[X;D],1;D;D2;¢ ¢ ¢;Dpe¡1are linearly independent over Z ([7, Lemma 2.1]). Thus we haveHom(BDBBD ;BDBBD) = Z[D] = Z © ZD © ZD2 © ¢ ¢ ¢ © ZDpe¡1.(4) Let ¢ be any derivation in DerBD(B). By (3), we have¢ =Ppe¡1k=1 zkDk (zk 2 Z). (Note that ¢ has no constant term). Forany a; b 2 B, we obtain¢(ab) =pe¡1Xk=1zk(kXº=0µkº¶Dk¡º(a)Dº(b))=pe¡1Xº=0(pe¡1Xk=ºµkº¶zkDk¡º(a))Dº(b):On the other hand¢(a)b+ a¢(b) = (pe¡1Xk=1zkDk(a))b+pe¡1Xº=1azºDº(b):Since 1; D;D2; ¢ ¢ ¢ ; Dpe¡1 are linearly independent overB ([7, Lemma 2.1]),we obtainazº =pe¡1Xk=ºµkº¶zkDk¡º(a) (a 2 B; 1 · º · pe ¡ 1);and hence,pe¡1Xk=º+1µkº¶zkDk¡º = 0:Since 1; D;D2; ¢ ¢ ¢ ; Dpe¡1 are linearly independent over B again, we haveµkº¶zk = 0 (1 · º < k · pe ¡ 1):Then by the same arguments in the proof of [5, Theorem 3.1], we see that¢ is in ZD © ZDp © ¢ ¢ ¢ © ZDpe¡1 . This completes the proof of (4).6Mathematical Journal of Okayama University, Vol. 41 [1999], Iss. 1, Art. 6http://escholarship.lib.okayama-u.ac.jp/mjou/vol41/iss1/6PURELY INSEPARABLE RING EXTENSIONS AND AZUMAYA ALGEBRAS 69(5) is immediate by [1, Theorem 2.4.3].Under the same situation of Theorem 5 we have the followingCorollary 6. Let ¢ be an another derivation of B which is an ex-tension of ±. If B¢ = BD, then ¢ = D.Proof. By Theorem 5(4), ¢ =Pe¡1j=0 zjDpj (zj 2 Z). Since ¢jZ =DjZ = ±, we have ± = Pe¡1j=0 zj±pj , and so z0 = 1 and zj = 0 (1 · j ·e¡ 1).References[1] F. DeMeyer and F. Ingraham: Separable algebras over a commutative ring, LectureNotes in Math. 181, Springer, Berlin, 1971.[2] G. Georgantas: Inseparable Galois cohomology, J. Algebra 38 (1976), 368{379.[3] K. Hoechsmann: Simple algebras and derivations, Trans. Amer. Math. Soc. 180(1963),1{12.[4] S. Ikehata: On separable polynomials and Frobenius polynomials in skew polynomialrings, Math. J. Okayama Univ. 22 (1980), 115{129.[5] S. Ikehata: Azumaya algebras and skew polynomial rings. Math, J. Okayama Univ.23 (1981), 19{32.[6] S. Ikehata: Azumaya algebras and skew polynomial rings. II, Math. J. OkayamaUniv. 26 (1984), 49{57.[7] S. Ikehata: Purely inseparable ring extensions and H-separable polynomials, Math.J. Okayama Univ. 40 (1998), 55{63.[8] K. A. Knus: Sur le theoreme de Skolem-Noether et sur les derivation des algebresd'Azumaya, C. R. Acad. Sci. Paris Ser. A-B 270, (1970) A637{A639.[9] K. A. Knus, M. Ojanguren and D. J. Saltman: On Brauer groups in characteristicp, Lecture Notes in Mathematics 549, Springer, Berlin 1976, 25{49.[10] S. Yuan: On logarithmic derivatives, Bull. Soc. Math. France, 96 (1968), 41{52.[11] S. Yuan: Inseparable Galois theory of exponent one, Trans. Amer. Math. Soc. 149(1970), 163{170.Shu^ichi IkehataDepartment of Mathematical ScienceFaculty of Environmental Science and TechnologyOkayama UniversityTsushima, Okayama 700-8530, Japane-mail address: ikehata@ems.okayama-u.ac.up(Received May 30, 2000)7Ikehata: Purely Inseparable Ring Extensions and Azumaya AlgebrasProduced by The Berkeley Electronic Press, 1999

Purely Inseparable Ring Extensions and Azumaya Algebras

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