Within the Fall of 1975 we all started a joint undertaking with the final word aim of topo logically classifying genuine algebraic units. This has been a protracted satisfied collaboration (c.f., [K2)). In 1985 whereas vacationing M.S.R.1. we prepared and provided our class effects as much as that time within the M.S.R.1. preprint sequence [AK14] -[AK17]. in view that those effects are interdependent and require a few necessities in addition to familiarity with actual algebraic geometry, we made up our minds to cause them to self contained through proposing them as part of a publication in genuine algebraic geometry. even if we've not arrived to our ultimate target but we suppose that it's time to introduce them in a self contained coherent model and show their use via giving a few functions. bankruptcy I provides the review of the class application. bankruptcy II has the entire precious history for the remainder of the publication, which for that reason can be utilized as a path in actual algebraic geometry. It starts off with the simple houses of actual algebraic units and ends with the hot answer of the Nash Conjecture. bankruptcy III and bankruptcy IV improve the speculation of answer towers. answer towers are uncomplicated topologically outlined gadgets generalizing the thought of manifold.
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