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Dr. Katz DEq Homework Solutions 28

# Dr. Katz DEq Homework Solutions 28 - (2.3.4.2.1...

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28 N.M. Katz: together with the respective degenerations (2.3.2.2-3) imply, for every 0< i< n, the formula (2.3.4.1.11) dimK H" (X, f2~./~ (log D)) = dimK F i + t + dim r Fr i. For the homomorphisms (2.3.4.1.8), source and target have the same dimension, just as for (2.3.4.1.9), and both homomorphisms have the same kernel, namely F 1 n F~,. Q.E.D. (2.3.4.2) We now discuss the sometimes defined "higher" Hasse-Witt operations (which include the usual one as a special case), still supposing the hypotheses of (2.3.2). As before, we fix an integer n>0. For each integer i, we denote by h (i) the composite mapping n--i ?1 FI ~on (R f, (F2~c/s (log D))) ~ R f, (f2x/s(log D))
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Unformatted text preview: (2.3.4.2.1) . [ R"f. (f2~,./s (log D))/F i+1 (2.3.4.2.2) Proposition. Hypotheses as in (2.3.2), and n>O fixed as above, suppose that for an integer i, the mapping h(i) (2.2.4.2.1) is an iso- morphism. Then there is a unique mapping of locally free Cs-modules, the i + 1-st Hasse- Witt operation (2.3.4.2.3) H-W(i+I): F,~sR"-i-tf,(f2ix-~sl(logD))-~ R"-i-~f,(f2~Ts~(logD)) which renders commutative the following diagram: H-W(i+I) Fa~s (R"-'- ~ f, (f~ix-~ (log D))) ~~ cohEn2 -i- 1,i+1 E i + 1,n - i-1 ~ Rn-i- If, (~r~-S1 (log V)) (2.3.4.2.4) F~'o~i-l(R"f,(f2~c/s(log D))) h(i+l), R,f,(f2~c/s(log D))/F,+2 F~:'(R"f, (f2~c/s(l 3g D))) h(0 ) f*(i x/s(l~ D))/F ~ R ~ f2" i+1 ! 0...
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