10
N.M. Katz:
which is the
negative
of the isomorphism (1.2.1.8) explicated in (1.2.1.9),
and which we will
not
use.
In the setting of (1.2.1.9), we may choose over each V~ a morphism
~: ~1Vi~ Ker([3)[ V/~fg[ Vi which is a section of ~: fq~~ouf (i.e., such
that ~ki. ~ = id,lv, ).
(1.2.1.10.2)
In fact, we may simply choose ~biso that ~ o ~bi =
id~rlv, 
q~i ~ [3,
this being possible because, over V~, id~eCp~o[3 is a projection onto
ker([3). Then the difference ~bi ~b~ defines a morphism from ~r
V/~ V~ to
f~l V~ n ~ which vanishes on the image of ct, thus defining by passage to
quotients a morphism from ~1 V/c~ V~to fq[ V/c~ V~, still denoted ~i ~b~.
The cohomology class of {~k~~b~} in
Ha(X, Hom(~,
if)) is the element
corresponding to the extension class of (1.2.1.1) via the isomorphism
(1.2.1.10.1). To see that
this
isomorphism is the negative of (1.2.1.9), it
suffices to recall that
(1.2.1.10.3)
0~ o ~i = id~elv, 
~o~o
[3
whence
(1.2.1.10.4)
~ o (~k, ~s) = 
(~o, ~os) o [3
which gives the equality of the two cocycles {~b
i ~bs} and { (q~cp~)}.
(1.2.1.11)
The "second" isomorphism (1.2.1.10) is the
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 Fall '11
 NormanKatz
 Category theory, horn, Isomorphism, Epimorphism, canonical isomorphism Hom

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