Some thoughts on an example sheet. Typical examples illustrating coherent sheaves involve the geometry of varieties over a field k = kbar. == 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.) == 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. == 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. == 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. == 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.) == 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). == 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). == 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. == 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. == 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.