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A key commutative algebra lemma

Let A be a ring, and M an A module. What follows is the algebraic material need to define Spec A as a locally ringed space as well as quasicoherent sheaves on Spec A. For  domain R, we denote by K(R) its fraction field. Spectrum of a ring as a topological space. We denote by X:= Spec A the set of all prime ideals of A. We give it the Zariski topology, defined by requiring that a subset C of X be closed iff there exists an ideal I in A such that C= Z(I), the set of prime ideals containing I. We interpret it as the set of points p in X where f(x)=0 for every f in I. Here the value of f at p is its image in A/p and in fact of κ(p), the residue field of X at p, defined by K(A/p). A basis of the topology is given by the principal open sets D(f), loci where f ≠0, aka complement of Z((f)) where f is the principal ideal generated by f. This basis is closed under intersection, in that D(f)\cap D(g)=D(fg). Lemma D(f) is contained in D(g) iff the structural ring homomorphism A \to A_g factors...
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Properness and separatedness

Warning: this is way too long. I had to split in subsections, please consider using them. Topology background Recall (or believe me) that a topological space X is called T2, or Hausdorff, if any two points have disjoint neighborhoods, This is equivalent to the diagonal in XxX being closed. X is called compact if every open cover has a finite subcover. This is equivalent to saying that for every other topological space Y the projection XxY to Y is closed. Note that for real and complex manifolds, the product as manifolds coincides with the product as topological spaces, so no confusion is possible when mentioning XxX and XxY. T2 property says there are enough open sets on the topology; it remains true if you change topology on X to a finer one (with more open sets). Compactness says there are not too many open sets; it remains true if you change topology on X to a coarser one (with fewer open sets). Compact and Hausdorff is so to speak the perfect middle, and a theorem states that if a ...
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Sheaf of differentials and tangent bunch for a classical scheme

Classical scheme means locally of finite type over a fixed algebraically closed field k. I just made up the name tangent bunch; it should be called tangent abelian cone, but I chose this to highlight the similarity with tangent bundle. A vector bunch E over a scheme X is a scheme E affine over X with a G_m action (for which the map to X is invariant) which is isomorphic to Spec Sym F for some F\in Coh(X). A morphism is a G_m-equivariant morphism of X schemes.  Theorem The functor Spec Sym defines an equivalence of categories Coh(X)^{op} to Vector Bunches on X, which commutes with pullbacks. We know that for a smooth scheme X over k, the individual tangent spaces T_pX neatly organize themselves into a vector bundle TX over X, just as they do for manifolds. But in general, the dimension of T_pX depends on p.  We start with an affine scheme X=Spec R, and write R=k[x,_1,...,x_n]/I, i.e. we choose a closed embedding of X in A^n. We want to define the tangent vector bunch TX to X as...
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On quasicoherent sheaves

On an arbitrary ringed space, one can define sheaves of modules (easily) and coherent sheaves (in general, very painfully).  On a scheme X, there is a third option, which is defining quasicoherent sheaves. If X=Spec R is affine, the functor (sheaves of modules on X) to (R modules) sending F to F(X) has a left adjoint, denoted by M maps to tilde M.  If M=R, then \tilde M is canonically isomorphic to the structure sheaf.  The left adjoint is of course unique up to canonical isomorphism, or one can choose an explicit construction and get a functor; we then define a sheaf of modules F as being quasicoherent if the natural map from tilde (F(X)) to F is an isomorphism. I much prefer to say that a sheaf of modules on X is quasicoherent if for every principal open D(f), the natural map F(X)_f\to F(D(f)), which exists for every sheaf of modules, is an isomorphism. Exercise Let X = Spec R be an affine scheme. Show that global sections define an equivalence of categories from Qcoh(X...
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