# Abstract Algebra: A Concrete Introduction by Robert H. Redfield By Robert H. Redfield

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Intégration: Chapitres 7 et 8

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R E L A T I O N B E T W E E N T H E ANNIHILATING IDEAL OF Let L 1 , . . , Ls be general enough linear forms in ~)1. Let P = { P l , - . - , P s } C IP(~)I) ---F r-1 be the points corresponding to the forms L 1 , . . , Ls, and l e t I p be the graded ideal in R of all forms vanishing at P. Let I = A n n ( f ) be the ideal of polynomials apolar to f . Then if i satisfies s <_ dimk Rj_i, A SUM OF P O W E R S , AND A VANISHING IDEAL AT P O I N T S . we have Furthermore H ( R / I p ) d = min(dimk Rd, s) for every d >_ O.

We wish to prove t h a t H ( R / I p ) d = s for every d _> i. This e q u a l i t y is equivalent to the following restriction m a p being surjective: H~ r-x, OW-I ( d ) ) ~ H~ Op(d)). 17) Let us choose a linear form g which does not vanish on any of the points of P . 17) is epimorphic for d = i. M u l t i p l y i n g by gd-i we see it is epimorphic for every d >_ i as well. 20. COMPAalSON OF A n n ( f ) AND I p . Let 7~ = k[X,Y,Z], s = 3, f = X 4 + y 4 + z 4. T h e n Hf = ( 1 , 3 , 3 , 3 , 1) and I = A n n ( f ) = (xy, xz, yz, x 4 - y4 z4 _ z4).

We first prove P a r t (ii). Assume 2s _< j . F i x v w i t h s _< v _< j - v a n d let V~ = V s ( j - v, v; 2). First we prove t h a t Ps C Vs. Since V~ is closed it suffices to verify t h a t Ps C Vs. A n y form g E P~ C 7~j is apolar to some ~b E Rs, t h u s it is apolar to t h e vector space Rv-~q~ of dimension v - s + 1. This implies rk C a t g ( j - v , v;2) _< s, hence g E Vs. Next we prove t h a t P~ = V s ( j - s, s; 2). We proved above t h a t P~ c V ~ ( j - s , s ; 2 ) . 10) there is a nonzero form r E R~ apolar to g. 