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 jewelry A and B. Morita's answer organizes principles so successfully that the classical Wedderburn-Artin theorem is an easy end result, and in addition, a similarity classification [AJ within the Brauer crew Br(k) of Azumaya algebras over a commutative ring ok includes all algebras B such that the corresponding different types mod-A and mod-B such as k-linear morphisms are similar through a k-linear functor. (For fields, Br(k) involves similarity sessions of straightforward crucial algebras, and for arbitrary commutative ok, this can be subsumed below the Azumaya [51]1 and Auslander-Goldman [60J Brauer staff. ) a variety of different circumstances of a marriage of ring thought and type (albeit a shot gun wedding!) are inside the textual content. additionally, in. my try and extra simplify proofs, significantly to get rid of the necessity for tensor items in Bass's exposition, I exposed a vein of principles 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 root for it's a corre spondence theorem for projective modules (Theorem four. 7) prompt by means of the Morita context. As a spinoff, this offers origin for a slightly whole concept of straightforward Noetherian rings-but extra approximately this within the advent.

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**Algebra. Rings, modules and categories**

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1 Sect. 4] Elasticity via the Exterior Calculus 35 It has the realization x* = Q)xXX , A T) + k , t=T l + k. 22) T h e action of the group G x t thus picks the whole history of the material body u p and moves it rigidly to a new history in E x R and t h e n translates the whole history along the time axis. All is now in readiness for the variational arguments. Let Ψ d e n o t e the stored energy per unit mass of the material body on £ x i?. Noting that the Lagrangian function for a mechanical system is the kinetic energy minus the potential energy, we have 0 4 3 3 L = p { ^ V - ^ } .

44) and hence we must have d d ^ = 0. 45) i A will be such that f = άχ' only if 0 = d f = \{d F A -d F ) = 0, dF B A dX A B A dX B + {d V - d F )dX l A A 4 A A dT ; that is, e dF ABC l B c l 4 A = d V. 46) W h e n these kinematic compatability conditions are satisfied by the F ' s and the V s , the equations άχ =/' can be solved on any star-shaped region by use of the h o m o t o p y o p e r a t o r H , ι Sect. 47) w h e r e the /c's are the values of the ^ ' s at the center. 46). 32)). Because R is star-shaped with respect t o any of its points as center, the Poincare l e m m a holds in R .

So that the A"s are the same before as after G acts. This is an essential aspect of classical gauge theory. T h e group G acts only as an internal symmetry group; it does not change the underlying manifold on which the state variables are defined. It is for precisely this reason that the material description will be used in the gauge theory of defects, for in that case, the action integral is an integral over the space R of reference configuration histories, while the internal symmetry group acts on the material space Af x R.