Abelian categories and Mod R

Let record some pleasant features of $\operatorname{Mod} R$ and their abstractions.

Enriched in Ab

First, each morphism space $\operatorname{Hom}_{R}(M,N)$ is an abelian group under pointwise addition of functions \((f+g)(m) := f(m) + g(m)\)

Composition is bilinear over the abelian group struture: \(\begin{aligned} (f_1+f_2) \circ g & = f_1 \circ g + f_2 \circ g \\ f \circ (g_1 + g_2) & = f \circ g_1 + f \circ g_2 \end{aligned}\)

Definition. We say $\mathcal C$ is enriched in Ab if $\operatorname{Hom}_{\mathcal C}(X,Y)$ has the structure of abelian group such composition in $\mathcal C$ is bilinear over the group structure.

Additive categories

Next, in $\operatorname{Mod} R$, finite products and coproducts (including over $\empty$) both exist and the natural map \(\bigsqcup_{i=1}^t M_i \to \prod_{i=1}^t M_i\) is an isomorphism. Both are the direct sum $\bigoplus M_i$.

If we are in a category where this condition on finite coproducts/products is satsified we will use direct sum notation for the coproducts.

Definition. We say that $\mathcal C$ is an additive category if it is enriched in Ab and finite coproducts and products exist and coincide.

A functor $F : \mathcal C \to \mathcal D$ is additive if each \(F_{X,Y} : \operatorname{Hom}_{\mathcal C}(X,Y) \to \operatorname{Hom}_{\mathcal D}(FX,FY)\) is a group homomorphism and the natural map \(\begin{aligned} \bigoplus_{i=1}^t F X_i \to F \left(\ \bigoplus_{i=1}^t X_i \right) \end{aligned}\) is an isomorphism.

All of $- \otimes_R M$, $\operatorname{Hom}_R(M,-)$, and $\operatorname{Hom}_R(-,M)$ are examples of additive functors. Indeed, for the last two, we are just expressing the universal properties of products and coproducts.

For tensor product, the natural map \(\bigoplus_{i \in I} \left(N_i \otimes_R M\right) \to \left( \bigoplus_{i \in I} N_i \right) \otimes_R M\) is an isomorphism. So, in particular, $-\otimes_R M$ commutes with finite coproducts.

Kernels and cokernels

Each morphism $\phi : M \to N$ in $\operatorname{Mod} R$ admits a kernel and cokernel. To characterize them via universal property, let us remember the following definitions.

Definition. Let $\delta_i : A_i \to A_{i+1}$ be a collection of morphisms of abelian groups for $- N < i < M$ (with the possibility that $N = \infty$ or $M = \infty$).

We say that $(A_\bullet, \delta_\bullet)$ is a complex if \(\operatorname{im} \delta_i \subseteq \ker \delta_{i+1}\)

The collection is exact if \(\operatorname{im} \delta_i = \ker \delta_{i+1}\)

In general, we can measure the failure of cohomology using the cohomology groups of $(A_\bullet,\delta_\bullet)$: \(H^i(A_\bullet,\delta_\bullet) := \frac{\ker \delta_i}{\operatorname{im} \delta_{i-1}}\)

These definition are given for the moment in Ab but we will how to make sense of them in an arbitrary abelian category momentarily.

With this, we note that $\ker \phi$, for $\phi: M \to N$ of $R$-modules, fits into the exact sequence of abelian groups: \(0 \to \operatorname{Hom}_R(M^\prime, \ker \phi) \to \operatorname{Hom}_R(M^\prime,M) \to \operatorname{Hom}_R(M^\prime,N)\)

Similarly, for the cokernel $\operatorname{cok} \phi$, we get an exact sequence \(0 \to \operatorname{Hom}_R(\operatorname{cok} \phi, M^\prime) \to \operatorname{Hom}_R(N,M^\prime) \to \operatorname{Hom}_R(M,M^\prime)\)

Each of these is characterizations is equivalent to a universal lifting/extension property.

For the kernel, we have the diagram

which expresses that we get a unique $\tilde{\psi}$ if and only if the composition $M^\prime \to M \to N$ is $0$. In other words, we can lift map $\phi : M^\prime \to M$ uniquely over $\ker \phi \to M$ if and only if $\phi \circ \psi = 0$.

For the cokernel we have the dual lifting problem

which expresses that we get a unique $\tilde{\psi}$ if and if $M \to N \to M^\prime$ is $0$.

In a general additive category $\mathcal A$, given $\phi : X \to Y$ we ask about representability of the functors \(T \mapsto \ker\left( \operatorname{Hom}_{\mathcal A}(T,X) \overset{\phi \circ -}{\to} \operatorname{Hom}_{\mathcal A}(T,Y)\right)\) and \(T \mapsto \ker\left( \operatorname{Hom}_{\mathcal A}(Y,T) \overset{- \circ \phi}{\to} \operatorname{Hom}_{\mathcal A}(X,T)\right)\)

If the first is representable, we call it the kernel $\ker \phi$. Note that there is natural transformation \(\ker\left( \operatorname{Hom}_{\mathcal A}(-,X) \overset{\phi \circ -}{\to} \operatorname{Hom}_{\mathcal A}(-,Y)\right) \to \operatorname{Hom}_{\mathcal A}(-,X)\) given by forgeting that the map lies in the kernel. Thus we have a unique map $\alpha : \ker \phi \to X$ if the kernel exists.

For the second, we call the representing object the cokernel and write it as $\operatorname{coker} \phi$. It comes with a natural map $\beta: Y \to \operatorname{coker} \phi$.

Existence of kernels and cokernels is almost enough for the definition of an abelian category. There is one more condition that holds in $\operatorname{Mod} R$. For a morphism of modules $\phi: M \to N$, the natural map \(\operatorname{coker} \alpha \to \operatorname{ker} \beta\) is an isomorphism. The first object is often called the coimage while the second is called the image of $\phi$.

In this case, both are isomorphic to the image $\phi(M)$ submodule of $N$. In general, this is not assured.

Definition. An abelian category is an additive category $\mathcal A$ possessing all kernels and cokernels with the natural maps \(\operatorname{coker} \alpha \to \operatorname{ker} \beta\) beings isomorphisms. Since the coimage and image coincide, we will just write this as $\operatorname{im} \phi$.

Given this, we can talk about sub-objects and quotient-objects. A morphism $\phi: X \to Y$ exhibits $Y$ as a sub-object of $Y$ if $\ker \phi \cong 0$. Similarly, $Y$ is a quotient object if $\operatorname{coker} \phi \cong 0$. We will just write $X \subseteq Y$ for a sub-object and $Y = X/Z$ for a quotient object. (Here $Z = \ker \phi$).

We can now lift our definitions of complexes, exactness, and cohomology to an abelian category $\mathcal A$ without altering any notation.

Some more notation: if $0 \to A \to B$ is exact, we will write $A \hookrightarrow B$. If $A \to B \to 0$ is exact, we write $A \twoheadrightarrow B$.

Exactness

We don’t call functors respecting the abelian category structures abelian.

Definition. Let $F : \mathcal A \to \mathcal B$ be an additive functor between abelian categories. We say that $F$ is left-exact if the natural map \(F( \ker \phi ) \to \ker F(\phi)\) is an isomorphism for any $\phi : X \to Y$ in $\mathcal A$. We say that $F$ is right-exact if the natural map \(\operatorname{coker} F(\phi) \to F(\operatorname{coker} \phi)\) is an isomorphism for any $\phi$. If $F$ is both left and right exact, we simply call it exact.

Left-exactness can be expressed in terms of preservation of exact sequences of a particular shape. A functor $F$ is left-exact if and only if whenever we have an exact sequence \(0 \to A \to B \to C\) the application of $F$ yields an exact sequence \(0 \to FA \to FB \to FC\)

Similarly, $F$ is right exact if and only if for any exact sequence of the form \(A \to B \to C \to 0\) the sequence is \(FA \to FB \to FC \to 0\) is also exact.

Each of these results reduces, after to tracing out the definitions, to the fact that $0 \to A \to B \to C$ is exact is equivalent to saying that $A$ is the kernel of $B \to C$ and the analogously stated fact for cokernels.

Exactness can also be expressed in this way. $F$ is exact if and only if for any exact sequence \(0 \to A \to B \to C \to 0\) the sequence \(0 \to FA \to FB \to FC \to 0\) is also exact.

Exact sequences of the shape $0 \to A \to B \to C \to 0$ pop up frequently so they are given a name: a short exact sequence. By convention, a longer exact sequence is often termed long.

Examples

• Of course the identity functor is exact.

• The forgetful functor $\operatorname{Mod} R \to \operatorname{Ab}$ is exact.

• If $F$ is a free $R$-module, then $- \otimes_R$ is exact. Indeed, $F \cong R^{\oplus X}$ for some set $X$ and we have a natural isomorphism \(N \otimes_R R^{\oplus X} \cong N^{\oplus X}\) The result follows from the more general fact that coproducts of exact sequences remain exact.

• Similarly, if $F$ is a free module, then there is a natural isomorphism \(\operatorname{Hom}_R(R^{\oplus X}, N) \cong \prod_X N\) Products of exact sequences remain exact in $\operatorname{Mod} R$ so $\operatorname{Hom}_R(F,-)$ is also exact.

We will see next that Hom and tensor are not exact in general. They are each “half-exact”.

Adjoints and exactness

Given an adjunction $F \vdash G$ of Ab-enriched functors between abelian categories, we get a some exactness for free.

First, since $F$ commutes with coproducts and $G$ commutes with products, we automatically know that $F$ and $G$ are additive.

Next we can run the same argument that established that commutativity to get commutativity for kernels and cokernels. \(\begin{aligned} \ker \operatorname{Hom}(X,G(\phi)) & \cong \ker \operatorname{Hom}(F(X),\phi) \\ & \cong \operatorname{Hom}(F(X),\ker \phi) \\ & \cong \operatorname{Hom}(X,G(\ker \phi)) \end{aligned}\) So $G$ commutes with kernels, ie is left exact. Similarly, $F$ is right exact.

Freyd-Mitchell embedding

It seems possible that a general abelian category could be far from our touchstone of $\operatorname{Mod} R$. The Freyd-Mitchell Embedding Theorem says this not the case.

Theorem. Let $\mathcal A$ be a small abelian category. Then there exists a not-necessarily unital ring $R$ and a fully-faithful exact functor \(\mathcal A \to \operatorname{Mod} R.\)

We will not provide a proof. See this MO post for a nice dicussion of the proof.

The result becomes more immediate in examples where you already have a set of projective generators or injective co-generators in your hand. The hard work in the theorem is cook up an injective co-generator in a bigger category.

The main utility of Freyd-Mitchell is that

• we feel good that $\mathcal A$ is not so far from $\operatorname{Mod} R$ and
• each $\mathcal A$ admits a faithful functor to $\operatorname{Set}$ so we can reason using elements of each object $A$ of $\mathcal A$.

This semester our main abelian category is $\operatorname{Mod} R$. While results may be stated more generally, details will usually only be given for justifications involving this example. (Next semester is a globalization of this: $\operatorname{Qcoh} X$.)