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 a subbunch of Xxk^n, the restriction of the tangent bundles to A^n to X, whose fiber over p in X is T_pX as a subspace of k^n=T_pA^n.

If we denote by u_1,..., u_n the coordinates on k^n, there is a natural set of equations defining TX in XxA^n=Spec R[u_i]:  for every f in I, we impose that \sum f_i(x)u_i=0, where f_i is the image in R of the partial derivative of f with respect to x_i. Of course it is enough to impose this condition not for every f in I, but only on a finite subset F of generators of I as an ideal.

In other words, TX is the fiber product of XxA^n and X over XxA^F, where XxA^n to XxA^F is the map given by t_f maps to \sum f_i(x)u_i, and X to XxA^F is the zero section.

We cannot describe TX as a kernel of sheaves of modules O_X^n\to O_X^F, because kernels do not commute with base change. Instead, we can describe TX as Spec Sym \Omega_X, where \Omega_X is the cokernel of O_X^F\to \Omega_{A^n}|_X.

I was taught that while in differential geometry tangent and cotangent bundle can be defined one in terms of the other, in algebraic geometry this is not true; the tangent sheaf is the dual of the cotangent, but in general not vice versa. In fact, two objects are fundamental and determine each other: one is the coherent sheaf \Omega_X, and the other is the vector bunch TX. Algebra and geometry embrace, I'll let you to decide which one is the left hand of the other (U.K. Le Guin reference).

In general, a tangent bunch on any scheme X is locally the kernel (as in, the fiber product with the zero section) of a morphism of vector bundles on X; since we work locally, the bundles can be assumed to be trivial if one so desires. Equivalently, a coherent sheaf is locally the cokernel of a morphism of locally free coherent sheaves, and again one can assume they are free if one so desires.

As someone who learned manifolds before schemes, to me it is much more natural to define the tangent sheaf TX locally and then glue (just like I do for tangent bundle of manifolds), then define the cotangent sheaf as the associated coherent sheaf, and finally show that there is a canonical isomorphism \Omega_X=\Delta^* I_{\Delta(X)} where \Delta:X \to XxX is the diagonal, but of course the main point is that all these definitions are equivalent.


Note: for simplicity I wrote the absolute version over Spec k. For a morphism f:X\to Y one can either define directly \Omega_f, or define it as the cokernel f^\Omega_Y\to \Omega_X. which after all is how we defined \Omega_X to begin with (the morphism being A^n to A^F, induced by the set F of generators of I).


 

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