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By Lapidus M., van Frankenhuijsen M. (eds.)

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Additional info for Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Part 1

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F / X where f stands for a tuple of functions f1 ; : : : ; fr 2 A. 1/21. Then consider Ahf i D Ahf1 ; : : : ; fr i D Ah 1 ; : : : ; r i=. i fi I i D 1; : : : ; r/: as an affinoid K-algebra. There is a canonical morphism of affinoid K-algebras ✲ Ahf i and, associated to it, a morphism between affinoid K-spaces à WA ÃW Sp Ahf i ✲ X . f / admit a unique factorization of affinoid K-spaces 'W Y through Ã. ✲ X and let To check this consider a morphism of affinoid K-spaces 'W Y ✲ it correspond to a morphism of affinoid K-algebras ' W A B.

The latter is unique as the image of Adbf0 1 ec is dense in Ah ff0 i. t u Proposition 12 (Transitivity of Affinoid Subdomains). For an affinoid K-space X , consider an affinoid subdomain V X , and an affinoid subdomain U V . Then U is an affinoid subdomain in X as well. Proof. Consider a morphism of affinoid K-spaces 'W Y ✲ X having image in U . Then, as U V and V is an affinoid subdomain of X , there is a unique factorization ✲ V of ' through V ✲ X . Furthermore, ' 0 admits a unique '0W Y 00 ✲ ✲ V , as U is an affinoid subdomain U through U factorization ' W Y 00 ✲ V ✲ X that, of V .

Proposition 14. Let X be an affinoid K-space and let U ,V X be affinoid subdomains. Then U \V is an affinoid subdomain of X . If U and V are Weierstraß, resp. Laurent, resp. rational domains, the same is true for U \ V . ✲ X be the morphism defining U as an affinoid subdomain Proof. Let 'W U of X . V / and we see that U \ V is an affinoid subdomain of U by Proposition 13. Hence, by Proposition 12, U \ V is an affinoid subdomain of X . 1/. The product of both ideals is the unit ideal again and we see that the functions fi gj , i D 0; : : : ; r, j D 0; : : : ; s have no common zero on X .

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