05 - algebraic geometry II

# 05 - algebraic geometry II - INTRODUCTION TO MANIFOLDS V...

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INTRODUCTION TO MANIFOLDS — V Algebraic language in Geometry ( continued ) . Everywhere below F : M N is a smooth map, and F * : C ( M ) C ( N ) the associated homomorphism of commutative algebras, F * g = g F ⇐⇒ ( F * g )( x ) = g ( F ( x )) . Let x M be a point of a smooth manifold, and m x C ( M ) the corresponding maximal ideal: m x = { f C ( M ): f ( x ) = 0 } . Deﬁnition. m 2 x := ( X α f α g α , f α ,g α m x ) . In the same way higher powers m k x of a maximal ideal are deﬁned. Problem 1. m 2 x = ' f C ( M ) : f ( y ) = O ( | y - x | 2 ) = functions without free and linear terms in the Taylor expansion centered at x . / Problem 2. If F : M N a smooth map, F ( a ) = b , then F * m k b m k a for any natural k . / Problem 3. ( F * ) - 1 m k a = m k b . / Problem 4. A tuple of functions f 1 ,...,f k m a C ( M ) has rank 1 < k at the point a ⇐⇒ ∃ c 1 ,...,c k R : k c k f k m 2 a . / Problem 5. rank a ( F * f 1 ,...,F * f k ) 6 rank F ( a ) ( f 1 ,...,f k ) . / Problem 6. Give an example of the sharp inequality in the above formula. / 1

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05 - algebraic geometry II - INTRODUCTION TO MANIFOLDS V...

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