# Applications of algebraic K-theory to algebraic geometry and by Spencer J. Bloch, R. Keith Dennis, Eric M. Friedlander, By Spencer J. Bloch, R. Keith Dennis, Eric M. Friedlander, Micahel Stein (ed.)

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Additional resources for Applications of algebraic K-theory to algebraic geometry and number theory, Part 2

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B i, • • • ~r Bk = C = Cx + . . + Ct\ and suppose further that the formulas A\ = /i (Bf), • • Ak = f h (Bk)y Cx = gi (B{), . . • . , C i = gi (Bt) hold, where f x, . f 1n gly . g t are certain motions (so that A x = B l9 Cx = B X9 and so on). These formulas just mean that each of the figures A and C is equidecomposable with B . Let F tj = B t p|B j , where i =■ 1 , . . , 7c; j = 1 , . ,Z (note that some 7The father of the famous J. Bolyai, who arrived at the non-Euclidean geometry independently o f N.

This gives a rectangle aefb which is equidecomposable with the triangle abc. In fact, the triangles marked 1 in Fig. 28 are congruent, and so are those marked 2. But each of the figures abc and aefb consists of the shaded trapezoid and a pair of triangles marked 1 and 2 . Lemma 3. Any two rectangles with the same area are equi­ decomposable. In fact, we arrange the rectangles of equal area (the rectangles oabc and omnp in Fig. 30) in such a way that they have a common right angle. , FIG. 28 = -47- , which implies FIG.

We choose polygons G and H such that G CL F d II and s (H) — s (G) < e (Theorem3). H ') — s (£') = 5 (if) — 5 (G) < 8. It follows that the figure i*' = g (F) is measurable. The validity of property (7 *) for arbitrary measur­ able figures is now proved in just the same way as for rectangles (with a reference to Theorem 5 rather than to Theorem 2). The statement (7 *) is a more general property of area than axiom (7 ), and it is sometimes chosen as an axiom instead of (7 ), leading to the construction of a theory of area based on the properties (a), (P)> (Y*)> (6)* regarded as axioms.