By Carl Faith
VI of Oregon lectures in 1962, Bass gave simplified proofs of a couple of "Morita Theorems", incorporating principles of Chase and Schanuel. one of many Morita theorems characterizes while there's an equivalence of different types mod-A R::! mod-B for 2 earrings A and B. Morita's resolution organizes principles so successfully that the classical Wedderburn-Artin theorem is a straightforward final result, and additionally, a similarity type [AJ within the Brauer crew Br(k) of Azumaya algebras over a commutative ring okay involves all algebras B such that the corresponding different types mod-A and mod-B along with k-linear morphisms are similar by means of a k-linear functor. (For fields, Br(k) comprises similarity sessions of easy crucial algebras, and for arbitrary commutative ok, this can be subsumed below the Azumaya 1 and Auslander-Goldman [60J Brauer crew. ) a number of different situations of a marriage of ring idea and type (albeit a shot gun wedding!) are inside the textual content. moreover, in. my try to additional simplify proofs, particularly to dispose of the necessity for tensor items in Bass's exposition, I exposed a vein of rules and new theorems mendacity wholely inside of ring thought. This constitutes a lot of bankruptcy four -the Morita theorem is Theorem four. 29-and the foundation for it's a corre spondence theorem for projective modules (Theorem four. 7) steered by way of the Morita context. As a spinoff, this gives starting place for a slightly entire thought of easy Noetherian rings-but extra approximately this within the advent.
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Extra resources for Algebra I: Rings, Modules, and Categories
1 Chern–Simons theory: basic ingredients . . . . . . . . 2 Framing dependence . . . . . . . . . . . . . 3 Generating functionals for Wilson loops . . . . . . . . 1 The 1/N expansion . . . . . . . 3 The conifold transition . . . . . . 4 First test of the duality: the free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Incorporating Wilson loops .
It is very useful to fix g, h and the winding numbers, and sum over all bulk classes. This produces the following generating functional of open Gromov–Witten invariants: Fw,g (t) = Q Q −Q·t Fw,g e . 23) 36 Marcos Mariño where t = (t1 , . . , tb2 (X) ) are the complexified Kähler parameters of the Calabi–Yau manifold. In many examples relevant to knot theory, the entries Q are naturally chosen Q to be half-integers. Finally, the quantities Fw,g are the open string Gromov–Witten invariants, and they “count” in an appropriate sense the number of holomorphically embedded Riemann surfaces of genus g in X with Lagrangian boundary conditions specified by L and in the class represented by Q, w.
3 Generating functionals for Wilson loops . . . . . . . . 1 The 1/N expansion . . . . . . . 3 The conifold transition . . . . . . 4 First test of the duality: the free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Incorporating Wilson loops . . . . . . . . . . . . 3 BPS invariants for open strings from knot invariants . . . . . 70 Tests involving Wilson loops . . . . . . . . . . . 74 6 Large N transitions and toric geometry 80 7 Conclusions 86 1 Introduction Enumerative geometry and knot theory have benefitted considerably from the insights and results in string theory and topological field theory.
Algebra I: Rings, Modules, and Categories by Carl Faith