Dr. Katz DEq Homework Solutions 31

# Dr. Katz DEq Homework Solutions 31 - Algebraic Solutions of...

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Algebraic Solutions of Differential Equations 31 are the usual perfect pairings of Serre duality, and the lower row of (2.3.5.1.7) is simply the inverse image by Fab s of the lower row of (2.3.5.1.6). (2.3.5.2) Proposition. Hypotheses as in (2.3.5.1), fix integers n>O and i. The morphism (2.3.5.2.1) h(i): Fc"o-~i(R"f,(~'x/s))--~R"f,(f2"x/s)/F i+j is an isomorphism if and only if the morphism (2.3.5.2.2) h(N-i- 1): FcNo+ I +i-"(RZN-"f,(I?}/s))--, R2u-"f,(O}/s)/FN-i is an isomorphism. Proof. Just as in the proof of (2.3.4.1.7), we are immediately reduced to the case in which S is the spectrum of a field K. By the autoduality of the Hodge and conjugate filtrations (2.3.5.1.4-5), the dual of (2.3.5.2.1) is the natural mapping (2.3.5.2.3) FU-'(RZU f,(Q'x/s))-~ R2U-"f,(~l,/s)/Fi~o +1+'-". Thus (2.3.5.2.1) and (2.3.5.2.3) are isomorphisms or not together. But (2.3.5.2.2) and (2.3.5.2.3) have the same kernel, (F u-in.~'u+l-i-"~con ," ( RzN- n f, (Q*x/s)). Since the source and target of (2.3.5.2.2) have the same dimension over K, and the same is true of (2.3.5.2.2), it follows that
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