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04 - algebraic geometry

# 04 - algebraic geometry - INTRODUCTION TO MANIFOLDS IV...

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INTRODUCTION TO MANIFOLDS — IV Appendix: algebraic language in Geometry 1. Algebras. Definition. A ( commutative associatetive ) algebra ( over reals ) is a lin- ear space A over R , endowed with two operations, + and · , satisfying the natural axioms of arithmetics: ( A, +) is an additive Abelian (=commutative) group with the neutral element denoted by 0, while ( A, · ) is a commutative semigroup. If there is a · -neutral element, then it is denoted by 1, though existence of such an element is not usually assumed. Example. The basic example is that of real numbers. Another elementary exam- ples: matrices Mat n ( R ). Other examples follow. The Principal Example. Let M be a smooth n -dimensional manifold, and A = C ( M ) the space of al smooth functions on it. Then if one sets ( f + g )( x ) = f ( x ) + g ( x ) , ( f · g )( x ) = f ( x ) g ( x ) , ( λf )( x ) = λf ( x ) , then al the axioms will be satisfied. Now the principal wisdom comes. The main idea of algebraic approach to geometry is to study properties of the manifold via algebraic properties of the algebra ( ring ) C ( M ) . 2. Reconstruction of points. Definition. An ideal I of an algebra A is a subalgebra with the property A · I I, which is to be understood as a A, u I au I. Problem 1. Prove that 1 I = I = A . Typeset by A M S -T E X 1

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2 APPENDIX: ALGEBRAIC LANGUAGE IN GEOMETRY Problem 2. Prove that { 0 } is always an ideal in any algebra. The principal example of an ideal is the following one. Problem 3. If Z M is a closed subset, then I Z = { f C ( M ): f | Z 0 } is an ideal in A = C ( M ).
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04 - algebraic geometry - INTRODUCTION TO MANIFOLDS IV...

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