Some thoughts on an example sheet. Typical examples
illustrating coherent sheaves involve the geometry
of varieties over a field k = kbar.
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1. If X is a variety/k then the structure sheaf OX is
coherent. A sheaf of ideals IV in OX is coherent. (The same
applies for X a Noetherian scheme.)
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2. An ideal IV in OX is locally free if and only if it is a
locally principal ideal. This means its subscheme V(I) in X
has codimension 1 and is everywhere locally defined by one
equation (f=0) with f a nonzerodivisor.
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3. If X is nonsingular/k then it is known that its local
rings O_X,P are UFDs. Then every codimension 1 subscheme
V in X is locally defined by one equation, so
IV = OX(-V) in OX is locally free.
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4. Consider the ordinary quadratic cone
Q : (x*z=y^2) in AA^3, and the line L : (x = y = 0).
Then L is a codimension 1 subvariety of Q, but is not
locally defined by one equation near O = (0,0,0). [Hint:
x=0 and y=0 don't work. If g is any regular function then
Q intersect V(g) is singular at O.]
However the sheaf of ideals I_2L = OQ(-2L) in OQ of
functions on Q vanishing twice along L is principal. Find
its generator. Show that multiplication I_L^2 -> I_2L is not
surjective. Prove that the element
(x tensor y - y tensor x) in I_L tensor_OQ I_L
is a nonzero element in the tensor product, but its product
by x, y or z is zero. So it is a torsion element supported
at O.
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5. If F and G are coherent sheaves on a variety X then
F tensor G and the sheaf-Hom sHom_OX(F,G) are coherent
sheaves of OX modules. [I use sHom for script Hom, sim. sDer
below.] (Same over a Noetherian scheme X. We only need
Noetherian for f.g. implies finitely presented.)
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6. The tangent sheaf TX and sheaf of 1-forms Om^1_X.
Work with algebras and varieties over k. The idea is that
the tangent sheaf is the normal bundle to the diagonal DeX
in X x_k X. This is easy to visualise if X is nonsingular.
If X is singular, it leads to useful points of view
(although it gets progressively messier).
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7. For a nonsingular variety X and its nonsingular
subvariety Y, the sheaf of ideals IY in OX is locally
generated by the c = codim(Y in X) local equations of
Y in X. The sheaf IY/IY^2 is a coherent sheaf on X, killed
by the ideal IY (by construction), so is a sheaf of
OY-modules. It is the _conormal_ bundle to Y in X. You can
imagine a local section of IY/IY^2 as the first order Taylor
series of one of the defining equation of Y in the normal
direction to Y. (This may be more convincing if you work in
local analytic coordinates x1,..xn on X with Y locally
defined by x1 = .. = xc = 0.) The dual sheaf on Y,
(IY/IY^2)^dual = sHom_OY(IY/IY^2, OY)
is the normal sheaf to Y in X. A section of it is a first
order deformation of Y in X (that is, a normal vector,
trying to move Y to a neighbouring Yt).
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8. If A is an algebra over k and M an A-module, a derivation
(over k) is a map d: A -> M that satisfies
da = 0 for a in k; d(fg) = g*df - f*dg. (*)
It follows that d is k-linear, but not A-linear. One defines
the module of 1-forms of A over k in terms of a universal
derivation d: A -> Om^1_A. Think this through in the case A
= k[x1,..xn]. [Hint: the target must have elements f*dg for
all f,g in A, and be subject only to the axioms (*).] A more
algebraic treatment follows below.
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9. A vector field on a manifold acts on functions (say
C^infty) as a derivation (usually written f |-> Xf in the
differentiable context, but I do not use X for that). The
sheaf of derivations sDer_k(OX,OX) is defined as the local
k-derivations of OX to itself. Show that it is a coherent
sheaf. If X is nonsingular, it is a locally free sheaf TX,
the _tangent sheaf_. In local coordinates x1..xn, think of
TX as based by the (partial) d/dx_i. (This can be justified,
but explaining it takes longer than is interesting.)
The tangent sheaf TX can also be defined as the dual of
1-forms Om^1_X. That is, there is a canonical derivation
d: OX -> Om^1_X, and every local derivation from OX to
itself is locally the composite of d with an OX-linear
homomorphism Om^1_X -> OX.
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10. The general scheme-theoretic approach to TX is based on
the commutative algebra notion of Om^1_A/B. Namely the
tensor product algebra is
A tensor_B A
and it contains the ideal
I = ideal generated by { a tensor 1 - 1 tensor a | a in A }.
Then the module of Kaehler 1-forms is the A-module I/I^2,
and its differential d: A -> Om^1_A/B is
a |-> d(a) = a tensor 1 - 1 tensor a.
As given, I is a module over the tensor product ring. One
checks that acting on the left or on the right gives the
same action mod I^2, that d is a differential, and has the
UMP for B-differentials.
For X nonsingular over k, there are a number of things still
to check that the dual of Om^1_X/k gives the same as TX
discussed above.