About this blog

I am not good at writing books, also there are already so many books of algebraic geometry out there.

On the other hand, most algebraic geometry books, having to include definitions, statements and proofs while remaining within a few hundred pages, seldom discuss in detail why definitions and theorems are the way they are, and provide connections with other fields of mathematics with similar statements.

So here I am trying to collect a few musings inspired mostly by many decades of attempts to read and digest GTM52, plus other algebraic geometry content. 

There will be mistakes, from the tiny typo to the egregious fckng up of a key definition/theorem. Sorry about that. I hope the ideas will nevertheless be useful to someone out there. 


Acknowledgement

Inspiration for this post is a skeet from Zen Faulknes.




Comments

Post a Comment

Popular posts from this blog

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...
Read more

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...
Read more

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 ...
Read more