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Michi’s blog » Reading Merkulov: Differential geometry for an algebraist (3 in a series)

 Reading Merkulov: Differential geometry for an algebraist (3 in a series)

  • February 13th, 2006
  • 1:55 pm

Since I cannot concentrate anyway, here comes the third installment of my reading Merkulov. We now get to the funky stuff – introducing sheaves and getting to what should be at the very start of basically any modern geometry course. At least if I am to believe the geometers I know.

So, let’s launch straight to it. A presheaf \mathcal F on the topological space \mathcal M is just a contravariant functor from Top(\mathcal M) to Ab, where Top(\mathcal M) is the category of open subsets of \mathcal M with morphisms being inclusion maps.

So that’s the one-line definition. But what does it mean?
Well, a functor is a map between categories that takes objects to objects and morphisms to morphisms. So we have that \mathcal F(U) is an abelian group for any open set U\subset\mathcal M. For such a map to really be a functor, it has to be sane in a rather precisely defined sense: namely morphism composition should still be associative and the identity endomorphism on a group shouldn’t actually, ya’know, change the morphisms before or after it.
For the functor to be contravariant means precisely that for f:U\to V we get \mathcal F(f):\mathcal F(V)\to\mathcal F(U) – all arrows reverse by application of the functor.

And for our definition of presheaves? We can read out that for every inclusion of open subsets of our space U\subseteq V we get a group homomorphism \rho_U^V:\mathcal F(V)\to\mathcal F(U). Functoriality requires these homomorphisms to be sane – i.e. \rho_U^U=\mathbb 1_{\mathcal F(U)} and \rho_W^V\circ\rho_V^U=\rho_W^U whenever W\subseteq V\subseteq U.

We will most often, as soon as it is clear what sheaf we work with, stick to denoting the \rho^U_V(f) with f\mid_V and call it the restriction of f to V.

Example

Our first example will be the constant presheaf: for \mathcal M a topological space and \mathcal A an abelian group, we define \mathcal F(U)=\mathcal A if U\subseteq\mathcal M is non-empty, and \mathcal F(\emptyset)=0. The restriction homomorphisms are the identity whenever the subset is nonempty and the zero homomorphism to the empty set.

Completely analogously, presheaves of sets, graded vector spaces, algebras, modules over an algebra et.c. can be defined. Throughout, the presheaves are just contravariant functors from Top(\mathcal M) to the relevant category.

Presheaves over presheaves

Suppose \mathcal R is a presheaf of rings with restrictions \rho and \mathcal M is a presheaf of abelian groups with restrictions \hat\rho on some topological space \mathcal T such that whenever U\subseteq\mathcal T we know that \mathcal M(U) is a \mathcal R(U)-module and for any V\subseteq U\subseteq\mathcal T we also know that \hat\rho_V^U(ax)=\rho_V^U(a)\hat\rho_V^U(x) for a\in\mathcal R(U) and x\in\mathcal M(U) so that the restrictions agree with the module structure. Then we call \mathcal M a presheaf of modules over a presheaf of rings \mathcal R. By adding structure to the modules in \mathcal M we can define presheaves of algebras or Lie algebras et.c. over a fixed presheaf of (graded) commutative rings \mathcal R.

Example

Let R be a graded k-algebra for a field k. A derivation of R of degree n is an element d\in\Hom_n(R,R) satisfying the condition
d(ab)=(da)b+(-1)^{|a|}a(db)
Let Der_nR be the vector space of all derivations of degree n and let
Der R=\bigoplus_{n\in\mathbb Z}Der_n R
Now, DerR has a natural R-module structure by ad=l_a\circ d where l_a:b\mapsto ab for a,b\in R. Furthermore, there is a natural structure of graded Lie algebra to DerR with brackets given as [d_1,d_2]=d_1\circ d_2-(-1)^{|d_1||d_2|}d_2\circ d_1.

Now, if \mathcal R is a presheaf of algebras, then the associated collection Der\mathcal R is naturally a presheaf of \mathcal R-modules. It is also a presheaf of Lie algebras on the space.

Example

For an open subset U\subseteq M of a fixed topological space M let \mathcal E^0(U) be the vector space of all complex valued continuous functions on U. For every pair of open subsets V\subseteq U let \rho^U_V:\mathcal E^0(U)\to\mathcal E^0(V) be the usual restriction of a continuous function f:U\to\mathbb C to f\mid_V:V\to\mathbb C, so for v\in V f\mid_V(v)=f(v). Then \mathcal E^0 is a presheaf of continuous functions on M and is often denoted by \mathcal E^0_M.

If M is a smooth manifold we can take smooth functions everywhere instead, we get a presheaf \mathcal E^\infty_M of smooth functions on M.

If M is a complex manifold we can take holomorphic functions everywhere to get a presheaf \mathcal O_M of holomorphic functions on M.

Sheaves

A presheaf \mathcal F on a topological manifold is called a sheaf if, for every open set U\subseteq M and every family of open subsets U_i\subseteq U with U=\bigcup U_i the following conditions are satisfied:
If for all i, and f, g\in\mathcal F(U) we have
f\mid_{U_i}=g\mid_{U_i}
then f=g and
for every family of elements f_i\in\mathcal F(U_i) with the property that f_i\mid_{U_i\cap U_j}=f_j\mid_{U_i\cap U_j} there is some f\in\mathcal F(U) such that f\mid_{U_i}=f_i for all i.

So a sheaf is a presheaf such that if for a covering of an open set equality on all covering sets implies equality in the covered set and where if a family seems to have come from an object higher up, then that object really does exist.

Note that the constant presheaf above is only a sheaf if either the abelian group is trivial or the space contains no non-intersecting open sets. If both of these conditions fail, then for a pair of non-intersecting U,V and distinct f_1,f_2 we know that U\cap V=\emptyset and thus that f_1\mid_{U\cap V}=f_2\mid_{U\cap V}=0 but there is no f\in\mathcal f(U\cup V)=A such that f\mid_U=f_1 and f\mid_V=f_2 since all restrictions are identity homomorphisms and f_1\neq f_2. Thus it’s no sheaf.

This defect, however, is easily fixed. We define for an open subset U \mathcal F(U) to be the set of all locally constant maps f:U\to A. Clearly, if U is connected, then \mathcal F(U)=A since each map then is uniquely determined by its image. The restriction maps are then the usual restrictions of maps. The result is a sheaf of locally constant functions with values in A, and is often denoted by the same symbol A.

All the presheaves \mathcal E^0_M, \mathcal E^\infty_M and \mathcal O_M are sheaves though. Their presheaves of derivations are also sheaves.

The sheaf \mathcal E^\infty_M is called the structure sheaf of a smooth manifold M, whereas the associated sheaf \mathcal T_M=Der\mathcal E^\infty_M is called the tangent sheaf or the sheaf of smooth vector fields on M.

The sheaf \mathcal O_M is called the structure sheaf of a complex manifold M and the tangent sheaf or sheaf or holomorphic vector fields on M is defined as above as the sheaf of derivations. Note that any complex manifold also has a structure of smooth real manifold and thus also has the associated sheaves \mathcal E^\infty_{M_r} and \mathcal T_{M_r}.

Example

Let’s define a presheaf on the complex plane as a complex manifold. For open U\subseteq\mathbb C define \mathcal F(U) to be the vector space of all bounded holomorphic functions on U, with the restriction being the usual restriction of a holomorphic function. This is obviously a presheaf. However, it’s not a sheaf, since with a covering \mathbb C=\bigcup_{i\in\mathbb N}U_i for U_i=\{z\in\mathbb C\mid|z|<i\} and a family of bounded holomorphic functions f_i=z\mid_{U_i}. These functions are compatible, however, there is no bounded holomorphic function on \mathbb C such that f\mid_{U_i}=f_i.

From this we learn that non-local properties on presheaves often cause the presheaf to fail being a sheaf.

Morphisms and categorical structure

A morphism of (pre)sheaves \mathcal F\to\mathcal G is defined in the obvious way – as a family of homomorphisms of abelian groups \mathcal F(U)\to\mathcal G(U) such that the obvious diagrams commute. That is it doesn’t matter if you first restrict and then follow the morphism or first follow the morphism and then restrict – the result should be the same. These morphisms make the definition needed to have a category of sheaves, and in this category, we receive the usual definition of an isomorphism, of inclusions, et.c.

Note that the following inclusion maps are all morphisms of sheaves of rings:
\mathbb C\to\mathcal O
\mathcal O\to\mathcal E^\infty
\mathcal E^\infty\to\mathcal E^0

For the next installment, we’re going to stalks and exact sequences. And germs! Wouldya look at that? We’ve gone from maritime terminology to agricultural terminology…

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