Around Azumaya rings - An overview of ring theory in the last decades. Robert Wisbauer

Transcription

1 Around Azumaya rings - An overview of ring theory in the last decades Robert Wisbauer University of Düsseldorf, Germany Yamanashi, October 2017

2 Overview 19 th Century, Hilbert

3 Overview 19 th Century, Hilbert Wedderburn structure theorems

4 Overview 19 th Century, Hilbert Wedderburn structure theorems Category theory

5 Overview 19 th Century, Hilbert Wedderburn structure theorems Category theory Equivalence of categories (Morita, Azumaya, Hopf)

6 Overview 19 th Century, Hilbert Wedderburn structure theorems Category theory Equivalence of categories (Morita, Azumaya, Hopf) Separability revisited

7 19 th century

8 19 th century Evariste Galois, Paris, France Field extensions, group theory

9 19 th century Evariste Galois, Paris, France Field extensions, group theory George Boole, England The Mathematical Analysis of Logic 1847

10 19 th century Evariste Galois, Paris, France Field extensions, group theory George Boole, England The Mathematical Analysis of Logic 1847 Pafnuty Chebyshev, Skt. Petersburg, Russia Theorie der Kongruenzen, Berlin 1889

11 19 th century Leopold Kronecker, Berlin, Germany Grundzüge einer arithmetischen Theorie der algebraischen Grössen, 1882

12 19 th century Leopold Kronecker, Berlin, Germany Grundzüge einer arithmetischen Theorie der algebraischen Grössen, 1882 Richard Dedekind, Braunschweig, Germany On the theory of algebraic integers, 1877

13 19 th century Leopold Kronecker, Berlin, Germany Grundzüge einer arithmetischen Theorie der algebraischen Grössen, 1882 Richard Dedekind, Braunschweig, Germany On the theory of algebraic integers, 1877 Ferdinand Frobenius, Berlin, Germany Theorie der hyperkomplexen Größen, 1903

14 19 th century - Japan Samurai der Shimazu Satsuma Boshin war period,

15 19 th century - Japan Samurai der Shimazu Satsuma Boshin war period, Meiji restoration Edo period Imperial period Meiji Emperor moves from Kyoto to Tokio, 1868

16 19 th century - Japan Dairoku Kikuchi, , Tokyo sent to England 1866, Cambridge , Tokyo University, textbook Elementary Geometry

17 19 th century - Japan Dairoku Kikuchi, , Tokyo sent to England 1866, Cambridge , Tokyo University, textbook Elementary Geometry Rikitaro Fujisawa, , Tokyo England-Germany , lectures by Kronecker Ph.D. Strasbourg 1886 (Fourier series, E. Christoffel) talk at the Internat. Math. Congress in Paris, 1900

18 19 th century - Japan Dairoku Kikuchi, , Tokyo sent to England 1866, Cambridge , Tokyo University, textbook Elementary Geometry Rikitaro Fujisawa, , Tokyo England-Germany , lectures by Kronecker Ph.D. Strasbourg 1886 (Fourier series, E. Christoffel) talk at the Internat. Math. Congress in Paris, 1900 Teiji Takagi, , Tokyo Class Field Theory, Ph.D. Tokyo 1903 teacher of Goro Azumaya in class field theory

19 19-20 th century David Hilbert, , Göttingen Über den Zahlbegriff Jahresbericht Deutsch. Math. Ver., 1900

20 19-20 th century David Hilbert, , Göttingen Über den Zahlbegriff Jahresbericht Deutsch. Math. Ver., 1900 Geometry: axioms for Euclidean plane Real numbers: natural numbers, integers,...(genetic method)

21 19-20 th century David Hilbert, , Göttingen Über den Zahlbegriff Jahresbericht Deutsch. Math. Ver., 1900 Geometry: axioms for Euclidean plane Real numbers: natural numbers, integers,...(genetic method) My opinion is this: in spite of the high pedagogical and heuristic value of the genetic method, for the final presentation and full logical assurance of the content of our knowledge, the axiomatic method deserves preference.

22 19-20 th century David Hilbert, , Göttingen Über den Zahlbegriff Jahresbericht Deutsch. Math. Ver., 1900 Geometry: axioms for Euclidean plane Real numbers: natural numbers, integers,...(genetic method) My opinion is this: in spite of the high pedagogical and heuristic value of the genetic method, for the final presentation and full logical assurance of the content of our knowledge, the axiomatic method deserves preference. Axioms: (R, +, ), ring, field, order, Archimedean, completeness

23 19-20 th century Joseph Wedderburn, , Princeton On hypercomplex numbers Proc. London Math. Soc. 1908

24 19-20 th century Joseph Wedderburn, , Princeton On hypercomplex numbers Proc. London Math. Soc Finite dimensional algebras A over fields (i) A has a maximal nilpotent ideal N (radical);

25 19-20 th century Joseph Wedderburn, , Princeton On hypercomplex numbers Proc. London Math. Soc Finite dimensional algebras A over fields (i) A has a maximal nilpotent ideal N (radical); (ii) if N = 0, then A is a direct sum of simple matric algebras;

26 19-20 th century Joseph Wedderburn, , Princeton On hypercomplex numbers Proc. London Math. Soc Finite dimensional algebras A over fields (i) A has a maximal nilpotent ideal N (radical); (ii) if N = 0, then A is a direct sum of simple matric algebras; (iii) A simple: matric algebra over division ring;

27 19-20 th century Joseph Wedderburn, , Princeton On hypercomplex numbers Proc. London Math. Soc Finite dimensional algebras A over fields (i) A has a maximal nilpotent ideal N (radical); (ii) if N = 0, then A is a direct sum of simple matric algebras; (iii) A simple: matric algebra over division ring; (iv) A = B + N, B semisimple and N nilpotent (structure?);

28 19-20 th century Joseph Wedderburn, , Princeton On hypercomplex numbers Proc. London Math. Soc Finite dimensional algebras A over fields (i) A has a maximal nilpotent ideal N (radical); (ii) if N = 0, then A is a direct sum of simple matric algebras; (iii) A simple: matric algebra over division ring; (iv) A = B + N, B semisimple and N nilpotent (structure?); (v) not invertible elements nilpotent: A = B + N, B div. ring;

29 19-20 th century Joseph Wedderburn, , Princeton On hypercomplex numbers Proc. London Math. Soc Finite dimensional algebras A over fields (i) A has a maximal nilpotent ideal N (radical); (ii) if N = 0, then A is a direct sum of simple matric algebras; (iii) A simple: matric algebra over division ring; (iv) A = B + N, B semisimple and N nilpotent (structure?); (v) not invertible elements nilpotent: A = B + N, B div. ring; (vi) A not associative: associative subalgebras nucleus, centre.

30 19-20 th century Joseph Wedderburn, , Princeton On hypercomplex numbers Proc. London Math. Soc Finite dimensional algebras A over fields (i) A has a maximal nilpotent ideal N (radical); (ii) if N = 0, then A is a direct sum of simple matric algebras; (iii) A simple: matric algebra over division ring; (iv) A = B + N, B semisimple and N nilpotent (structure?); (v) not invertible elements nilpotent: A = B + N, B div. ring; (vi) A not associative: associative subalgebras nucleus, centre. The classification of algebras cannot be carried out much further than this till a classification of nilpotent algebras has been found...

31 20 th century Representation theory quiver representations, path algebras, Auslander-Reiten theory, tilting theory, bocses

32 20 th century Representation theory quiver representations, path algebras, Auslander-Reiten theory, tilting theory, bocses Module theory Characterisation of rings by properties of their modules: Homological classification of rings (L.A. Skornjakov, 1967),

33 20 th century Representation theory quiver representations, path algebras, Auslander-Reiten theory, tilting theory, bocses Module theory Characterisation of rings by properties of their modules: Homological classification of rings (L.A. Skornjakov, 1967), Radical theory semisimple and radical classes of rings, Jacobson radical, Brown-McCoy radical, Kurosch-Amitsur radical, torsion theory

34 20 th century Representation theory quiver representations, path algebras, Auslander-Reiten theory, tilting theory, bocses Module theory Characterisation of rings by properties of their modules: Homological classification of rings (L.A. Skornjakov, 1967), Radical theory semisimple and radical classes of rings, Jacobson radical, Brown-McCoy radical, Kurosch-Amitsur radical, torsion theory Non-associative algebras alternative algebras, Jordan algebras, Lie algebras, Malcev algebras, Leibniz algebras

35 Category theory Eilenberg - Mac Lane, General Theory of natural equivalences Trans. AMS 1945

36 Category theory Eilenberg - Mac Lane, General Theory of natural equivalences Trans. AMS 1945 Category A: objects and morphisms Mor A (A, A ),

37 Category theory Eilenberg - Mac Lane, General Theory of natural equivalences Trans. AMS 1945 Category A: objects and morphisms Mor A (A, A ), adjoint pair of functors F : A B, G : B A, bijection Mor B (F (A), B) Mor A (A, G(B)),

38 Category theory Eilenberg - Mac Lane, General Theory of natural equivalences Trans. AMS 1945 Category A: objects and morphisms Mor A (A, A ), adjoint pair of functors F : A B, G : B A, bijection Mor B (F (A), B) Mor A (A, G(B)), unit and counit (nat. transf.) η : 1 GF, ε : FG 1 F F η FGF εf F = 1 F, G ηg GFG Gε G = 1 G

39 Category theory Eilenberg - Mac Lane, General Theory of natural equivalences Trans. AMS 1945 Category A: objects and morphisms Mor A (A, A ), adjoint pair of functors F : A B, G : B A, bijection Mor B (F (A), B) Mor A (A, G(B)), unit and counit (nat. transf.) η : 1 GF, ε : FG 1 F F η FGF εf F = 1 F, G ηg GFG Gε G = 1 G Equivalence: η and ε are natural isomorphisms

40 Application of category theory Kiiti Morita, , Tsukuba, Tokio Duality for modules and its application..., 1958 adjoint pair of functors

41 Application of category theory Kiiti Morita, , Tsukuba, Tokio Duality for modules and its application..., 1958 adjoint pair of functors counit RM S : S M R M, Hom R (M, ) : R M S M ε X : M S Hom R (M, X ) X, m f f (m), unit η Y : Y Hom R (M, M S Y ), y [m m y]

42 Application of category theory Kiiti Morita, , Tsukuba, Tokio Duality for modules and its application..., 1958 adjoint pair of functors counit RM S : S M R M, Hom R (M, ) : R M S M ε X : M S Hom R (M, X ) X, m f f (m), unit η Y : Y Hom R (M, M S Y ), y [m m y] Equivalence RM fin. gen. projective, generator, M S fin. gen. projective, Hom R (M, ) = Hom R (M, R) R, R End(M S )

43 Application of category theory Kiiti Morita, , Tsukuba, Tokio Duality for modules and its application..., 1958 adjoint pair of functors counit RM S : S M R M, Hom R (M, ) : R M S M ε X : M S Hom R (M, X ) X, m f f (m), unit η Y : Y Hom R (M, M S Y ), y [m m y] Equivalence RM fin. gen. projective, generator, M S fin. gen. projective, Hom R (M, ) = Hom R (M, R) R, R End(M S ) counit ε X : M S M R X X, unit η Y : Y M R M S Y

44 Subcategories Gen(M) = {N R M M (Λ) N 0},

45 Subcategories Gen(M) = {N R M M (Λ) N 0}, Pres(M) = {N R M M (Λ ) M (Λ) N 0},

46 Subcategories Gen(M) = {N R M M (Λ) N 0}, Pres(M) = {N R M M (Λ ) M (Λ) N 0}, σ[m] = {N R M N L, L Gen(M)} = Gen(M)

47 Subcategories Gen(M) = {N R M M (Λ) N 0}, Pres(M) = {N R M M (Λ ) M (Λ) N 0}, σ[m] = {N R M N L, L Gen(M)} = Gen(M) Q cogenerator in Gen(M), U = Hom R (M, Q) S M Cog(U) = {L S M L U-cog.}, Hom R (M, ) : σ[m] Cog(U)

48 Subcategories Gen(M) = {N R M M (Λ) N 0}, Pres(M) = {N R M M (Λ ) M (Λ) N 0}, σ[m] = {N R M N L, L Gen(M)} = Gen(M) Q cogenerator in Gen(M), U = Hom R (M, Q) S M Cog(U) = {L S M L U-cog.}, Hom R (M, ) : σ[m] Cog(U) Adjoint situations M S : S M σ[m], Hom R (M, ) : σ[m] S M,

49 Subcategories Gen(M) = {N R M M (Λ) N 0}, Pres(M) = {N R M M (Λ ) M (Λ) N 0}, σ[m] = {N R M N L, L Gen(M)} = Gen(M) Q cogenerator in Gen(M), U = Hom R (M, Q) S M Cog(U) = {L S M L U-cog.}, Hom R (M, ) : σ[m] Cog(U) Adjoint situations M S : S M σ[m], Hom R (M, ) : σ[m] S M, M S : S M Pres(M), Hom R (M, ) : Pres(M) S M,

50 Subcategories Gen(M) = {N R M M (Λ) N 0}, Pres(M) = {N R M M (Λ ) M (Λ) N 0}, σ[m] = {N R M N L, L Gen(M)} = Gen(M) Q cogenerator in Gen(M), U = Hom R (M, Q) S M Cog(U) = {L S M L U-cog.}, Hom R (M, ) : σ[m] Cog(U) Adjoint situations M S : S M σ[m], Hom R (M, ) : σ[m] S M, M S : S M Pres(M), Hom R (M, ) : Pres(M) S M, M S : Cog(U) Gen(M), Hom R (M, ) : Gen(M) Cog(U)

51 Subcategories Gen(M) = {N R M M (Λ) N 0}, Pres(M) = {N R M M (Λ ) M (Λ) N 0}, σ[m] = {N R M N L, L Gen(M)} = Gen(M) Q cogenerator in Gen(M), U = Hom R (M, Q) S M Cog(U) = {L S M L U-cog.}, Hom R (M, ) : σ[m] Cog(U) Adjoint situations M S : S M σ[m], Hom R (M, ) : σ[m] S M, M S : S M Pres(M), Hom R (M, ) : Pres(M) S M, M S : Cog(U) Gen(M), Hom R (M, ) : Gen(M) Cog(U) Equivalences Fuller s Theorem M f.gen., self-projective, self-generator

52 Subcategories Gen(M) = {N R M M (Λ) N 0}, Pres(M) = {N R M M (Λ ) M (Λ) N 0}, σ[m] = {N R M N L, L Gen(M)} = Gen(M) Q cogenerator in Gen(M), U = Hom R (M, Q) S M Cog(U) = {L S M L U-cog.}, Hom R (M, ) : σ[m] Cog(U) Adjoint situations M S : S M σ[m], Hom R (M, ) : σ[m] S M, M S : S M Pres(M), Hom R (M, ) : Pres(M) S M, M S : Cog(U) Gen(M), Hom R (M, ) : Gen(M) Cog(U) Equivalences Fuller s Theorem Sato s Theorem M f.gen., self-projective, self-generator M self-small and s-σ-quasi-projective

53 Subcategories Gen(M) = {N R M M (Λ) N 0}, Pres(M) = {N R M M (Λ ) M (Λ) N 0}, σ[m] = {N R M N L, L Gen(M)} = Gen(M) Q cogenerator in Gen(M), U = Hom R (M, Q) S M Cog(U) = {L S M L U-cog.}, Hom R (M, ) : σ[m] Cog(U) Adjoint situations M S : S M σ[m], Hom R (M, ) : σ[m] S M, M S : S M Pres(M), Hom R (M, ) : Pres(M) S M, M S : Cog(U) Gen(M), Hom R (M, ) : Gen(M) Cog(U) Equivalences Fuller s Theorem Sato s Theorem Menini-Orsatti s Th. M f.gen., self-projective, self-generator M self-small and s-σ-quasi-projective M -module (self-small tilting in σ[m])

54 Category theory Pierre Gabriel, , ETH Zürich Des categories abeliennes, thesis 1960 : Grothendieck categories: enough injectives, localisation

55 Category theory Pierre Gabriel, , ETH Zürich Des categories abeliennes, thesis 1960 : Grothendieck categories: enough injectives, localisation σ[m] is Grothendieck category M generator in σ[m] M S flat and R End(M S ) dense;

56 Category theory Pierre Gabriel, , ETH Zürich Des categories abeliennes, thesis 1960 : Grothendieck categories: enough injectives, localisation σ[m] is Grothendieck category M generator in σ[m] M S flat and R End(M S ) dense; RM faithful and M S finitely generated σ[m] = R M;

57 Category theory Pierre Gabriel, , ETH Zürich Des categories abeliennes, thesis 1960 : Grothendieck categories: enough injectives, localisation σ[m] is Grothendieck category M generator in σ[m] M S flat and R End(M S ) dense; RM faithful and M S finitely generated σ[m] = R M; M injective in σ[m] M is self-injective (Baer Lemma);

58 Category theory Pierre Gabriel, , ETH Zürich Des categories abeliennes, thesis 1960 : Grothendieck categories: enough injectives, localisation σ[m] is Grothendieck category M generator in σ[m] M S flat and R End(M S ) dense; RM faithful and M S finitely generated σ[m] = R M; M injective in σ[m] M is self-injective (Baer Lemma); every injective module in σ[m] is M-generated;

59 Category theory Pierre Gabriel, , ETH Zürich Des categories abeliennes, thesis 1960 : Grothendieck categories: enough injectives, localisation σ[m] is Grothendieck category M generator in σ[m] M S flat and R End(M S ) dense; RM faithful and M S finitely generated σ[m] = R M; M injective in σ[m] M is self-injective (Baer Lemma); every injective module in σ[m] is M-generated; M semisimple every module in σ[m] is M-injective;

60 Category theory Pierre Gabriel, , ETH Zürich Des categories abeliennes, thesis 1960 : Grothendieck categories: enough injectives, localisation σ[m] is Grothendieck category M generator in σ[m] M S flat and R End(M S ) dense; RM faithful and M S finitely generated σ[m] = R M; M injective in σ[m] M is self-injective (Baer Lemma); every injective module in σ[m] is M-generated; M semisimple every module in σ[m] is M-injective; M finite length: finite repres. type bounded repres. type

61 Category theory Multiplication algebra M(A) λ a : A A, x ax; ϱ a : A A, x xa; M(A) :=< {λ a, ϱ a a A} > End Z (A) A is M(A)-module, generated by 1 A,

62 Category theory Multiplication algebra M(A) λ a : A A, x ax; ϱ a : A A, x xa; M(A) :=< {λ a, ϱ a a A} > End Z (A) A is M(A)-module, generated by 1 A, End M(A) (A) = Z(A) (center), M(A) End(A Z(A) ),

63 Category theory Multiplication algebra M(A) λ a : A A, x ax; ϱ a : A A, x xa; M(A) :=< {λ a, ϱ a a A} > End Z (A) A is M(A)-module, generated by 1 A, End M(A) (A) = Z(A) (center), M(A) End(A Z(A) ), for every ideal I A, Hom M(A) (A, I ) = I Z(A),

64 Category theory Multiplication algebra M(A) λ a : A A, x ax; ϱ a : A A, x xa; M(A) :=< {λ a, ϱ a a A} > End Z (A) A is M(A)-module, generated by 1 A, End M(A) (A) = Z(A) (center), M(A) End(A Z(A) ), for every ideal I A, Hom M(A) (A, I ) = I Z(A), M(A)A self-projective Z(A/I ) = Z(A)/(I Z(A)),

65 Category theory Multiplication algebra M(A) λ a : A A, x ax; ϱ a : A A, x xa; M(A) :=< {λ a, ϱ a a A} > End Z (A) A is M(A)-module, generated by 1 A, End M(A) (A) = Z(A) (center), M(A) End(A Z(A) ), for every ideal I A, Hom M(A) (A, I ) = I Z(A), M(A)A self-projective Z(A/I ) = Z(A)/(I Z(A)), σ[a] = { M(A)-modules subgenerated by A } M(A) M

66 Category theory Multiplication algebra M(A) λ a : A A, x ax; ϱ a : A A, x xa; M(A) :=< {λ a, ϱ a a A} > End Z (A) A is M(A)-module, generated by 1 A, End M(A) (A) = Z(A) (center), M(A) End(A Z(A) ), for every ideal I A, Hom M(A) (A, I ) = I Z(A), M(A)A self-projective Z(A/I ) = Z(A)/(I Z(A)), σ[a] = { M(A)-modules subgenerated by A } M(A) M Hom M(A) (A, ) : M(A)M Z(A) M, A Z(A) : Z(A)M M(A) M

67 Azumaya rings Goro Azumaya, On maximally central algebra, 1951 (Nagoya J.) Separable rings, 1980 (J. Algebra)

68 Azumaya rings Goro Azumaya, On maximally central algebra, 1951 (Nagoya J.) Separable rings, 1980 (J. Algebra) Azumaya algebras A Z(A) : Z(A) M M(A) M is equivalence of categories

69 Azumaya rings Goro Azumaya, On maximally central algebra, 1951 (Nagoya J.) Separable rings, 1980 (J. Algebra) Azumaya algebras A Z(A) : Z(A) M M(A) M Azumaya rings A Z(A) : Z(A) M σ[a] is equivalence of categories is equivalence of categories

70 Azumaya rings Goro Azumaya, On maximally central algebra, 1951 (Nagoya J.) Separable rings, 1980 (J. Algebra) Azumaya algebras A Z(A) : Z(A) M M(A) M Azumaya rings A Z(A) : Z(A) M σ[a] is equivalence of categories is equivalence of categories A Z(A) finitely generated: A Azumaya ring A Azumaya algebra

71 Azumaya rings Goro Azumaya, On maximally central algebra, 1951 (Nagoya J.) Separable rings, 1980 (J. Algebra) Azumaya algebras A Z(A) : Z(A) M M(A) M Azumaya rings A Z(A) : Z(A) M σ[a] is equivalence of categories is equivalence of categories A Z(A) finitely generated: A Azumaya ring A Azumaya algebra Artin (1969), Delale (1974), Wisbauer (1977), Burkholder (1986)

72 Semiprime algebras - σ[a] Grothendieck catgeory Central closure of semiprime algebra A (Martindale) Â injective hull of A in σ[a];

73 Semiprime algebras - σ[a] Grothendieck catgeory Central closure of semiprime algebra A (Martindale) Â injective hull of A in σ[a]; A-generated: Â = A Hom M(A)(A, Â);

74 Semiprime algebras - σ[a] Grothendieck catgeory Central closure of semiprime algebra A (Martindale) Â injective hull of A in σ[a]; A-generated: Â = A Hom M(A)(A, Â); A semiprime Hom M(A) (A, Â) = End M(A)(Â) (extended centroid, commutative, regular);

75 Semiprime algebras - σ[a] Grothendieck catgeory Central closure of semiprime algebra A (Martindale) Â injective hull of A in σ[a]; A-generated: Â = A Hom M(A)(A, Â); A semiprime Hom M(A) (A, Â) = EndM(A)(Â) (extended centroid, commutative, regular); Â = A End M(A) (Â) has ring structure (central closure);

76 Semiprime algebras - σ[a] Grothendieck catgeory Central closure of semiprime algebra A (Martindale) Â injective hull of A in σ[a]; A-generated: Â = A Hom M(A)(A, Â); A semiprime Hom M(A) (A, Â) = EndM(A)(Â) (extended centroid, commutative, regular); Â = A End M(A) (Â) has ring structure (central closure); if A is prime, then End M(A) (Â) is a field;

77 Semiprime algebras - σ[a] Grothendieck catgeory Central closure of semiprime algebra A (Martindale)  injective hull of A in σ[a]; A-generated:  = A Hom M(A)(A, Â); A semiprime Hom M(A) (A, Â) = EndM(A)(Â) (extended centroid, commutative, regular);  = A End M(A) (Â) has ring structure (central closure); if A is prime, then End M(A) (Â) is a field; if A is prime, for ideals I A, I Z(A) 0, then  is simple;

78 Semiprime algebras - σ[a] Grothendieck catgeory Central closure of semiprime algebra A (Martindale)  injective hull of A in σ[a]; A-generated:  = A Hom M(A)(A, Â); A semiprime Hom M(A) (A, Â) = EndM(A)(Â) (extended centroid, commutative, regular);  = A End M(A) (Â) has ring structure (central closure); if A is prime, then End M(A) (Â) is a field; if A is prime, for ideals I A, I Z(A) 0, then  is simple; for A = Z, Ẑ = Q.

79 Separability Ernst Steinitz, , Berlin

80 Separability Ernst Steinitz, , Berlin Algebraische Theorie der Körper, 1910

81 Separability Ernst Steinitz, , Berlin Algebraische Theorie der Körper, 1910 L : K separable field extension For a L, minimal polynomial f K[X ] has no multiple roots

82 Separability Ernst Steinitz, , Berlin Algebraische Theorie der Körper, 1910 L : K separable field extension For a L, minimal polynomial f K[X ] has no multiple roots Hassler Whitney, , Princeton Tensor products of Abelian groups, 1938

83 Separability Ernst Steinitz, , Berlin Algebraische Theorie der Körper, 1910 L : K separable field extension For a L, minimal polynomial f K[X ] has no multiple roots Hassler Whitney, , Princeton Tensor products of Abelian groups, 1938 L : K field extension: Tensor functor L K : M K M K, M L K M

84 Functor L K : M K M K Product and unit m : L K L L, ι : K L

85 Functor L K : M K M K Product and unit m : L K L L, ι : K L L-module L K M ϱ M, M ι M L M ϱ M = 1 M

86 Functor L K : M K M K Product and unit m : L K L L, ι : K L L-module L K M ϱ M, M ι M L M ϱ M = 1 M Free and forgetful functor φ L : M K M L, M (L K M, m M)

87 Functor L K : M K M K Product and unit m : L K L L, ι : K L L-module L K M ϱ M, M ι M L M ϱ M = 1 M Free and forgetful functor φ L : M K M L, M (L K M, m M) U L : M L M K, (M, ϱ) M

88 Functor L K : M K M K Product and unit m : L K L L, ι : K L L-module L K M ϱ M, M ι M L M ϱ M = 1 M Free and forgetful functor φ L : M K M L, M (L K M, m M) U L : M L M K, (M, ϱ) M Adjoint pair Hom L (L K M, N) Hom K (M, U L (N))

89 Separability L : K finite Equivalent L : K finite L : K is a separable field extension; for any field extension Q : K, Nil(L K Q) = 0; Nil(L K L) = 0;

90 Separability L : K finite Equivalent L : K finite L : K is a separable field extension; for any field extension Q : K, Nil(L K Q) = 0; Nil(L K L) = 0; L is projective as L K L-module.

91 Separability L : K finite Equivalent L : K finite L : K is a separable field extension; for any field extension Q : K, Nil(L K Q) = 0; Nil(L K L) = 0; L is projective as L K L-module. Trace tr : L K; λ a : L L, x ax tr : L λa End K (L) Tr K

92 Separability L : K finite Equivalent L : K finite L : K is a separable field extension; for any field extension Q : K, Nil(L K Q) = 0; Nil(L K L) = 0; L is projective as L K L-module. Trace tr : L K; λ a : L L, x ax tr : L λa End K (L) Tr K tr : L K is nondegenerate; ψ : L L, a tr(a ), is an isomorphism (L-linear); L K Hom K (L, ) : M K M L.

93 Separability L : K finite Dual of algebra (coalgebra) apply ( ) = Hom(, K) to m and ι L L : K separable: apply ψ : L L m L K L, L ι K, δ : L L K L, ε : L K

94 Separability L : K finite Dual of algebra (coalgebra) apply ( ) = Hom(, K) to m and ι L L : K separable: apply ψ : L L m L K L, L ι K, δ : L L K L, ε : L K Frobenius conditions L L δ L m L L L L L m L L, δ

95 Separability L : K finite Dual of algebra (coalgebra) apply ( ) = Hom(, K) to m and ι L L : K separable: apply ψ : L L m L K L, L ι K, Frobenius conditions L L δ L δ : L L K L, m L L L L L m L L, δ ε : L K L L L δ m L L L L m L L L δ

96 Separability L : K finite Dual of algebra (coalgebra) apply ( ) = Hom(, K) to m and ι L L : K separable: apply ψ : L L m L K L, L ι K, Frobenius conditions L L δ L δ : L L K L, m L L L L L m L L, δ ε : L K L L L δ m L L L L m L L L δ m left L-comodule morphism δ is left L-module morphism

97 Separability L : K finite Dual of algebra (coalgebra) apply ( ) = Hom(, K) to m and ι L L : K separable: apply ψ : L L m L K L, L ι K, Frobenius conditions L L δ L δ : L L K L, m L L L L L m L L, δ ε : L K L L L δ m L L L L m L L L δ m left L-comodule morphism δ is right L-module morphism δ is left L-module morphism m is right L-comodule morphism.

98 Separability L : K finite (L, m, δ) satisfy Frobenius condition and m δ = 1 L

99 Separability L : K finite (L, m, δ) satisfy Frobenius condition and m δ = 1 L L-module ϱ : L K M M becomes comodule (L-linear) ω : M ι M L M δ M L L M L ϱ L M

100 Separability L : K finite (L, m, δ) satisfy Frobenius condition and m δ = 1 L L-module ϱ : L K M M becomes comodule (L-linear) ω : M ι M L M δ M L L M L ϱ L M Forgetful functor U L : M L M K, M, N M L Hom L (M, N) Hom K (U L M, U L N) splits by

101 Separability L : K finite (L, m, δ) satisfy Frobenius condition and m δ = 1 L L-module ϱ : L K M M becomes comodule (L-linear) ω : M ι M L M δ M L L M L ϱ L M Forgetful functor U L : M L M K, M, N M L Hom L (M, N) Hom K (U L M, U L N) Hom K (M, N) Hom L (M, N), splits by

102 Separability L : K finite (L, m, δ) satisfy Frobenius condition and m δ = 1 L L-module ϱ : L K M M becomes comodule (L-linear) ω : M ι M L M δ M L L M L ϱ L M Forgetful functor U L : M L M K, M, N M L Hom L (M, N) Hom K (U L M, U L N) Hom K (M, N) Hom L (M, N), splits by M f N M ω L K M L f L K N ϱ N

103 Separability L : K finite (L, m, δ) satisfy Frobenius condition and m δ = 1 L L-module ϱ : L K M M becomes comodule (L-linear) ω : M ι M L M δ M L L M L ϱ L M Forgetful functor U L : M L M K, M, N M L Hom L (M, N) Hom K (U L M, U L N) Hom K (M, N) Hom L (M, N), splits by M f N M ω L K M L f L K N ϱ N U L : M L M K is separable functor (to be defined)

104 R-algebras Maurice Auslander, Oscar Goldman, The Brauer group of a commutative ring, 1960

105 R-algebras Separable algebras Maurice Auslander, Oscar Goldman, The Brauer group of a commutative ring, 1960 (a) A is projective as an A R A o -module; (b) U L : A M M R is a separable functor; (c) A is separable over C(A) and C(A) is separable over R.

106 R-algebras Separable algebras Maurice Auslander, Oscar Goldman, The Brauer group of a commutative ring, 1960 (a) A is projective as an A R A o -module; (b) U L : A M M R is a separable functor; (c) A is separable over C(A) and C(A) is separable over R. Frobenius algebras (A, m, e) (a) A Hom R (A, R) as A-modules; (b) A R Hom R (A, ) : M R A M; (c) (A, δ, ε) is a coalgebra, Frobenius condition for (m, δ).

107 R-algebras Separable algebras Maurice Auslander, Oscar Goldman, The Brauer group of a commutative ring, 1960 (a) A is projective as an A R A o -module; (b) U L : A M M R is a separable functor; (c) A is separable over C(A) and C(A) is separable over R. Frobenius algebras (A, m, e) (a) A Hom R (A, R) as A-modules; (b) A R Hom R (A, ) : M R A M; (c) (A, δ, ε) is a coalgebra, Frobenius condition for (m, δ). Every module (M, ϱ) is comodule ω : M ι M A M δ M A A M A ϱ A M

108 Various algebras Algebra (A, m), coalgebra (A, δ), Frobenius condition A A I δ m A A A m I A A A A A δ δ I m A A A A I m A A δ

109 Various algebras Algebra (A, m), coalgebra (A, δ), Frobenius condition A A I δ m A A A m I A A A A A δ δ I m A A A A I m A A Frobenius algebra (A, m, e; δ, ε) Frobenius modules equivalence A R : A M A A M δ

110 Various algebras Algebra (A, m), coalgebra (A, δ), Frobenius condition A A I δ m A A A m I A A A A A δ δ I m A A A A I m A A Frobenius algebra (A, m, e; δ, ε) Frobenius modules equivalence A R : A M A A M δ Separable algebra (A, m, e; δ), m δ = I

111 Various algebras Algebra (A, m), coalgebra (A, δ), Frobenius condition A A I δ m A A A m I A A A A A δ δ I m A A A A I m A A Frobenius algebra (A, m, e; δ, ε) Frobenius modules equivalence A R : A M A A M δ Separable algebra (A, m, e; δ), m δ = I Azumaya algebra (A, m, e; δ), m δ = I bimodules separable and central, equivalence A R : M R A M A

112 Separable functors Hom L (M, N) Hom K (U L (M), U L (N))

113 Separable functors Hom L (M, N) Hom K (U L (M), U L (N)) C. Nǎstǎsescu, M. Van den Bergh, F. Van Oystaeyen, 1989 (J. Algebra) Separable functors applied to graded rings

114 Separable functors Hom L (M, N) Hom K (U L (M), U L (N)) C. Nǎstǎsescu, M. Van den Bergh, F. Van Oystaeyen, 1989 (J. Algebra) Separable functors applied to graded rings F : A B separable functor Φ F : Hom A (A, A ) Hom B (F (A), F (A )) nat. split mono

115 Separable functors Hom L (M, N) Hom K (U L (M), U L (N)) C. Nǎstǎsescu, M. Van den Bergh, F. Van Oystaeyen, 1989 (J. Algebra) Separable functors applied to graded rings F : A B separable functor Φ F : Hom A (A, A ) Hom B (F (A), F (A )) nat. split mono Ψ : Hom B (F (A), F (A )) Hom A (A, A ), Ψ Φ F = I

116 Hopf algebras Heinz Hopf, , Zürich Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen, 1941

117 Hopf algebras Heinz Hopf, , Zürich Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen, 1941 Bialgebra: algebra (A, m, ι), coalgebra (A, δ, ε) δ : A A K A, ε : A K are algebra morphisms

118 Hopf algebras Heinz Hopf, , Zürich Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen, 1941 Bialgebra: algebra (A, m, ι), coalgebra (A, δ, ε) δ : A A K A, Bimodules (Hopf modules) ε : A K are algebra morphisms A A M A M ϱ M ω A M δ ω m ϱ A A A M A τ M A A A M

119 Hopf algebras Heinz Hopf, , Zürich Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen, 1941 Bialgebra: algebra (A, m, ι), coalgebra (A, δ, ε) δ : A A K A, Bimodules (Hopf modules) Hopf algebra ε : A K are algebra morphisms A A M A M ϱ M ω A M δ ω m ϱ A A A M A τ M A A A M A A δ A A A A A m A A = 1 A A

120 Hopf algebras Heinz Hopf, , Zürich Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen, 1941 Bialgebra: algebra (A, m, ι), coalgebra (A, δ, ε) δ : A A K A, Bimodules (Hopf modules) Hopf algebra equivalence ε : A K are algebra morphisms A A M A M ϱ M ω A M δ ω m ϱ A A A M A τ M A A A M A A δ A A A A A m A A = 1 A A A R : M R A A M

121 Hopf algebras Heinz Hopf, , Zürich Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen, 1941 Bialgebra: algebra (A, m, ι), coalgebra (A, δ, ε) δ : A A K A, Bimodules (Hopf modules) Hopf algebra ε : A K are algebra morphisms A A M A M ϱ M ω A M δ ω m ϱ A A A M A τ M A A A M A A δ A A A A A m A A = 1 A A equivalence A R : M R A AM (antipode S : A A)

122 G = Aut(L : K) Group ring K[G] algebra and coalgebra bialgebra m : K[G] K[G] K[G], g h hg,

123 G = Aut(L : K) Group ring K[G] algebra and coalgebra bialgebra m : K[G] K[G] K[G], g h hg, δ : K[G] K[G] K[G], g g g.

124 G = Aut(L : K) Group ring K[G] algebra and coalgebra bialgebra m : K[G] K[G] K[G], g h hg, δ : K[G] K[G] K[G], g g g. Dual group ring (K[G], m, δ ) L comodule algebra L is K[G] -comodule: {g i, p i } i n dual basis for K[G] ω : L L K K[G], a i g i(a) p i

125 G = Aut(L : K) Group ring K[G] algebra and coalgebra bialgebra m : K[G] K[G] K[G], g h hg, δ : K[G] K[G] K[G], g g g. Dual group ring (K[G], m, δ ) L comodule algebra L is K[G] -comodule: {g i, p i } i n dual basis for K[G] ω : L L K K[G], a i g i(a) p i Canonical map β : L K L L K K[G], b a i bg i(a) p i

126 G = Aut(L : K) Group ring K[G] algebra and coalgebra bialgebra m : K[G] K[G] K[G], g h hg, δ : K[G] K[G] K[G], g g g. Dual group ring (K[G], m, δ ) L comodule algebra L is K[G] -comodule: {g i, p i } i n dual basis for K[G] ω : L L K K[G], a i g i(a) p i Canonical map β : L K L L K K[G], b a i bg i(a) p i L : K separable and normal β is an isomorphism

127 Around Azumaya Thank you!