FROM FINITELY GENERATED PROJECTIVE MODULES TO A GENERALIZATION OF - PowerPoint PPT Presentation

FROM FINITELY GENERATED PROJECTIVE MODULES TO A GENERALIZATION OF SERRE-SWAN-MALLIOS THEOREM Mart Abel University of Tartu

J. P. Serre, Modules projectifs et espaces fibr´ es ` a fibre vectorielle, S` eminaire Dubreil-Pisot 1957/58, 23 , pp. 531–543: .

J. P. Serre, Modules projectifs et espaces fibr´ es ` a fibre vectorielle, S` eminaire Dubreil-Pisot 1957/58, 23 , pp. 531–543: There is an one-to-one correspondence between the finite dimensional fibers of a vector bundle over a connected affine algebraic manifold V and (some special) projective A -modules, where A is a coordinate ring of V over an algebraically closed field. .

J. P. Serre, Modules projectifs et espaces fibr´ es ` a fibre vectorielle, S` eminaire Dubreil-Pisot 1957/58, 23 , pp. 531–543: There is an one-to-one correspondence between the finite dimensional fibers of a vector bundle over a connected affine algebraic manifold V and (some special) projective A -modules, where A is a coordinate ring of V over an algebraically closed field. R. G. Swan, Vector Bundles and Projective Modules , Trans. Amer. Math. Soc. 105 (1962), pp. 264–277: .

J. P. Serre, Modules projectifs et espaces fibr´ es ` a fibre vectorielle, S` eminaire Dubreil-Pisot 1957/58, 23 , pp. 531–543: There is an one-to-one correspondence between the finite dimensional fibers of a vector bundle over a connected affine algebraic manifold V and (some special) projective A -modules, where A is a coordinate ring of V over an algebraically closed field. R. G. Swan, Vector Bundles and Projective Modules , Trans. Amer. Math. Soc. 105 (1962), pp. 264–277: There is an equivalence between the category of K -valued vector bundles over compact Hausdorff space and the category of finitely generated projective C ( X , K )-modules (where K is either R , C or H ). .

J. P. Serre, Modules projectifs et espaces fibr´ es ` a fibre vectorielle, S` eminaire Dubreil-Pisot 1957/58, 23 , pp. 531–543: There is an one-to-one correspondence between the finite dimensional fibers of a vector bundle over a connected affine algebraic manifold V and (some special) projective A -modules, where A is a coordinate ring of V over an algebraically closed field. R. G. Swan, Vector Bundles and Projective Modules , Trans. Amer. Math. Soc. 105 (1962), pp. 264–277: There is an equivalence between the category of K -valued vector bundles over compact Hausdorff space and the category of finitely generated projective C ( X , K )-modules (where K is either R , C or H ). Generally speaking, Serre-Swan (-Type) Theorem gives us an equivalence between the category of A -vector bundles over X and the category of finitely generated projective C ( X , A )-modules with different conditions on A and X .

Different versions of Serre-Swan Theorem:

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C .

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C . Kandelaki (1976): X is a compact Hausdorff space and A is a unital commutative Banach algebra over C .

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C . Kandelaki (1976): X is a compact Hausdorff space and A is a unital commutative Banach algebra over C . Fuji (1978): X is a compact Hausorff space and A is a unital commutative Banach algebra over C .

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C . Kandelaki (1976): X is a compact Hausdorff space and A is a unital commutative Banach algebra over C . Fuji (1978): X is a compact Hausorff space and A is a unital commutative Banach algebra over C . Karoubi (1978): X is a compact topological space and A is R or C .

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C . Kandelaki (1976): X is a compact Hausdorff space and A is a unital commutative Banach algebra over C . Fuji (1978): X is a compact Hausorff space and A is a unital commutative Banach algebra over C . Karoubi (1978): X is a compact topological space and A is R or C . Goodearl (1984): X is a paracompact topological space and A is R , C or H .

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C . Kandelaki (1976): X is a compact Hausdorff space and A is a unital commutative Banach algebra over C . Fuji (1978): X is a compact Hausorff space and A is a unital commutative Banach algebra over C . Karoubi (1978): X is a compact topological space and A is R or C . Goodearl (1984): X is a paracompact topological space and A is R , C or H . Mallios (1983): X is a compact Hausdorff space and A is a unital locally m -convex Q -algebra.

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C . Kandelaki (1976): X is a compact Hausdorff space and A is a unital commutative Banach algebra over C . Fuji (1978): X is a compact Hausorff space and A is a unital commutative Banach algebra over C . Karoubi (1978): X is a compact topological space and A is R or C . Goodearl (1984): X is a paracompact topological space and A is R , C or H . Mallios (1983): X is a compact Hausdorff space and A is a unital locally m -convex Q -algebra. Vaserstein (1986): X is an arbitrary topological space and A is R , C or H .

Different versions of Serre-Swan Theorem: Atiyah (1967): X is a compact Hausdorff space and A is C . Kandelaki (1976): X is a compact Hausdorff space and A is a unital commutative Banach algebra over C . Fuji (1978): X is a compact Hausorff space and A is a unital commutative Banach algebra over C . Karoubi (1978): X is a compact topological space and A is R or C . Goodearl (1984): X is a paracompact topological space and A is R , C or H . Mallios (1983): X is a compact Hausdorff space and A is a unital locally m -convex Q -algebra. Vaserstein (1986): X is an arbitrary topological space and A is R , C or H . Kawamura (2003): considers Hilbert C ∗ -modules over unital non-commutative C ∗ -algebras and considers module bundles instead of vector bundles.

My goal: to get as general result as possible for topological algebras A .

My goal: to get as general result as possible for topological algebras A . Starting point: Serre-Swan-Mallios Theorem in paper A. Mallios, Vector Bundles and K-Theory over Topological Algebras , J. Math. Anal. Appl. 92 , No. 2, 1983, pp. 452–506.

My goal: to get as general result as possible for topological algebras A . Starting point: Serre-Swan-Mallios Theorem in paper A. Mallios, Vector Bundles and K-Theory over Topological Algebras , J. Math. Anal. Appl. 92 , No. 2, 1983, pp. 452–506. This talk is based on 3 papers:

My goal: to get as general result as possible for topological algebras A . Starting point: Serre-Swan-Mallios Theorem in paper A. Mallios, Vector Bundles and K-Theory over Topological Algebras , J. Math. Anal. Appl. 92 , No. 2, 1983, pp. 452–506. This talk is based on 3 papers: M. Abel, A. Mallios, On finitely generated projective modules , Rend. Circ. Mat. Palermo, 54 (2005), serie II, pp. 145–166.

My goal: to get as general result as possible for topological algebras A . Starting point: Serre-Swan-Mallios Theorem in paper A. Mallios, Vector Bundles and K-Theory over Topological Algebras , J. Math. Anal. Appl. 92 , No. 2, 1983, pp. 452–506. This talk is based on 3 papers: M. Abel, A. Mallios, On finitely generated projective modules , Rend. Circ. Mat. Palermo, 54 (2005), serie II, pp. 145–166. M. Abel, On a category of module bundles , submitted to Rend. Circ. Mat. Palermo.

My goal: to get as general result as possible for topological algebras A . Starting point: Serre-Swan-Mallios Theorem in paper A. Mallios, Vector Bundles and K-Theory over Topological Algebras , J. Math. Anal. Appl. 92 , No. 2, 1983, pp. 452–506. This talk is based on 3 papers: M. Abel, A. Mallios, On finitely generated projective modules , Rend. Circ. Mat. Palermo, 54 (2005), serie II, pp. 145–166. M. Abel, On a category of module bundles , submitted to Rend. Circ. Mat. Palermo. M. Abel, A generalization of the Serre-Swan-Mallios Theorem , submitted to Rend. Circ. Mat. Palermo.

A is an algebra with zero element θ A .

A is an algebra with zero element θ A . K is one of the fields R and C .

A is an algebra with zero element θ A . K is one of the fields R and C . A left A -module M is a linear space over K with bilinear map ( a , m ) �→ am from A × M into M , satisfying ( ab ) m = a ( bm ) for all a , b ∈ A and all m ∈ M .

A is an algebra with zero element θ A . K is one of the fields R and C . A left A -module M is a linear space over K with bilinear map ( a , m ) �→ am from A × M into M , satisfying ( ab ) m = a ( bm ) for all a , b ∈ A and all m ∈ M . An A -module M is finitely generated, if there exist k ∈ N and elements e 1 , . . . , e k ∈ M such that for every m ∈ M there are elements a 1 , . . . , a k ∈ A and numbers λ 1 , . . . , λ k ∈ K , for which k k � � m = a i e i + λ i e i . i =1 i =1