Fruits of procrastination

Abelian Categories

This post is about Abelian categories. This concept, like many others, has eaten away at my existence for quite some time, as there were some chinks that I never quite cleared.

Vakil says that abelian categories are “the right setting” in which one can do homological algebra. This is because one can use kernels, cokernels, etc in such a category. Before we elaborate on this, we shall write about additive categories.

An additive category \mathcal{A} is one in which for A,B\in Obj(\mathcal{A}), the set Mor(A,B) is an abelian group, there exists a zero object (an object that is both an initial and final object), and A\times B\in Obj(\mathcal{A}) too.

An abelian group structure on the objects would automatically make the morphisms an abelian group. This is seen in the category of abelian groups and the category of modules. A zero object need not always exist. For example, such an object does not exist in \textbf{Fld}. However categories like \textbf{Ab} and \textbf{Mod} do contain such objects, which is namely \{0\}. As for products, the category \textbf{Fld} again does not contain products. For example, \Bbb{Z_3}\times\Bbb{Z}_5 is not a field. However, the categories \textbf{Ab} and \textbf{Mod} do contain the zero object.

Now we shall discuss the concept of kernel and cokernel. For a map f:B\to C, a kernel is the two tuple (A,i), where i:A\to B is a morphism, such that f\circ i=0. Also, (A,i) is universal with respect to this property. What does this mean? It means that any other tuple satisfying this property factors through (A,i). This is just a fancy way of saying that A contains the whole kernel of f:B\to C.

A cokernel has an anaologous construction- that of the dual of the kernel. A cokernel of f:B\to C is a tuple of the form (D,g) such that g\circ f=0. Also, it is the universal tuple with such properties. In less fancy terms, every map from C which maps f(B) to 0 factors through D.

An abelian category is one in which every map has a kernel and a cokernel, every monomorphism is the kernel of its cokernel, and every epimorphism is the cokernel of its kernel. We shall discuss these concepts in separate paragraphs.

That every map has a kernel and a cokernel is seen in almost every category that we can think of. \textbf{Grp}, \textbf{Ring}, \textbf{Ab}, etc. Hence, this is not that much of a constraint.

What does “every monomorphism is the kernel of its cokernel” even mean? Before we can parse this statement, we need to understand what a monomorphism is. It is not just an injective map anymore. Let f,f':A\to B and g,g':B\to C. Then if f\circ g=f'\circ g\implies f=f', then g is a monomorphism. When we say every monomorphism is the kernel of its cokernel, what we mean is the following: take a monomorphism f:A\to B. As it is a map in an abelian category, it has a cokernel. Which means there exists an object Cok to which it maps via g:B\to Cok such that g\circ f=0. This map g:B\to Cok, by virtue of being a map in an abelian category, has a kernel (D,k) such that k:D\to B such that g\circ k=0. What this condition says is that (D,k)=(A,f). Why is it important that f is a monomorphism? Let us try to understand this with a concrete example in mind- that of the category \textbf{Ab}. In this setting, it all appears intuitive. Every tuple (M,l) that maps to B such that l\circ g=0 has to factor through (A,f) because (A,f) is exactly the kernel! There is no many-one mapping here. There are no added conditions. (A,f) is exactly the kernel for the map g:B\to C.

We can parse “every epimorphism is the cokernel of its kernel” in a similar way- in the category \textbf{Ab}. Take a surjective map f:A\to B. Its kernel is a tuple (C,g) such that f\circ g=0. The cokernel of g:C\to A is exactly (f,B), as B=A/ker f. This obviously is a result of the fact that f is surjective (a specific form of epimorphism). Otherwise C would contain, but not equal A/ker f, which is the cokernel of the kernel in general.

Over in all, the definition of the abelian category seems to contain some common sense requirements that most categories we encounter do fulfil. Some categories like \textbf{Fld} do not, however, as they’re not even additive categories to begin with.

Ringed Spaces

Today I’m going to be talking about the structure sheaf. Suppose \mathcal{O}_X is a sheaf of rings on a topological space X (i.e. a sheaf on X with values in the category of Rings), then (X,\mathcal{O}_X) is a ringed space. What does a sheaf of rings mean? Wikipedia says all it means is that for any open set U\subset X, F(U) is a ring. An example of a structure sheaf is the sheaf of continuous functions on an open set.

Now let us talk about \mathcal{O}_X-modules. What are these? You have another sheaf- a sheaf of abelian groups (we will call this sheaf G), over the same topological space, and then ensure that it is closed under scalar multiplication with elements of the structure sheaf. How does this make sense? We know that the union of all elements of F(U) over all open sets in X is not a ring. Like the union of all continuous functions on all open sets in \Bbb{R} is not a ring. However, each F(U) is. Hence, as long as we take elements from G(U) and multiply them only with elements of F(U), we are still dealing with elements of an abelian group being multiplied with elements from a ring, and then we’re fine. We have a module.

You also have a commutative diagram condition on ringed spaces. The diagram is attached below.

capture

What does such a diagram imply? It just means that whether you multiply and then restrict, or whether you restrict (elements of both sheaves) and then multiply, the result is the same. For example, if F is the sheaf of differentiable functions, and G is the sheaf of continuous functions, then G is an F module. Moreover, the order in which we restrict and multiply is irrelevant. Hence, the commutative diagram above is satisfied.

Why is Algebraic Geometry littered with such commutative diagrams whenever new concepts are defined? What structure are we trying to preserve? We just want to say that the “same thing is going on in both places”. For instance, when we have a mapping between sheaves, we want to say that whether we map and then restrict, or whether we restrict and then map, the result will be the same. So the operations of restrictions are “the same”, modulo mapping, in both sheaves. If we identify the elements of a sheaf with their image, then they’re being restricted in exactly the same way in both sheaves. This condition is a sanity check in some way. If any element could map to any element, regardless of the way they restrict, then we wouldn’t be able to map stalks to stalks and germs to germs, and the map between pre-sheaves would never become a map between stalks. In some sense, having such a commutative diagram makes mapping between direct limits possible.