By Carl Faith

VI of Oregon lectures in 1962, Bass gave simplified proofs of a few "Morita Theorems", incorporating rules of Chase and Schanuel. one of many Morita theorems characterizes whilst there's an equivalence of different types mod-A R::! mod-B for 2 earrings A and B. Morita's answer organizes rules so successfully that the classical Wedderburn-Artin theorem is a straightforward outcome, and in addition, a similarity classification [AJ within the Brauer team Br(k) of Azumaya algebras over a commutative ring okay includes all algebras B such that the corresponding different types mod-A and mod-B including k-linear morphisms are similar through a k-linear functor. (For fields, Br(k) comprises similarity periods of easy critical algebras, and for arbitrary commutative ok, this is often subsumed lower than the Azumaya [51]1 and Auslander-Goldman [60J Brauer crew. ) a variety of different circumstances of a marriage of ring concept and class (albeit a shot­ gun wedding!) are inside the textual content. in addition, 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 conception. This constitutes a lot of bankruptcy four -the Morita theorem is Theorem four. 29-and the root 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 origin for a slightly whole conception of straightforward Noetherian rings-but extra approximately this within the advent.

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Example text

Gopakumar–Vafa invariants nd for O(−3) → P2 . d 1 2 3 4 5 g=0 3 −6 27 −192 1695 1 0 0 −10 231 −4452 2 0 0 0 −102 5430 3 0 0 0 15 −3672 4 0 0 0 0 1386 For open topological strings one can derive a similar expression relating open Gromov–Witten invariants to a new set of integer invariants, that we will denote by nw,g,Q . 38) g=0 1 i wi ∞ g=0 d|w Q (−1)h+g nw/d,g,Q d h−1 2 sin dgs 2 2g−2 2 sin i wi gs −dQ·t e . 2 41 Enumerative geometry and knot invariants Notice there is one such identity for each w.

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. These are in general rational numbers.

4 Integer invariants from topological strings 28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . .