Post 9: Perverse Sheaves Part 4

07 Jan 2024

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In this post, I will discuss derived functors, the final algebraic ingredient we need to define perverse sheaves.

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Intuition

The problem

Given a certain kind of functor $F:\mc A\to \mc B$, we wish to construct a functor $D(\mc A)\to D(\mc B)$ that extends $F$ in some sense. As our construction of the derived category came in three steps, $Ch(\mc A)$, $K(\mc A)$, $D(\mc A)$, so too will our extension come in three parts.

For certain kinds of functors $F:\mc A\to \mc B$ between abelian categories, there is an “induced” functor $Ch(F):Ch(\mc A)\to Ch(\mc B)$. It is defined by sending a chain complex $A^\bullet$ to the chain complex $B^\bullet$ with $B^i = F(A^i)$ and $d_B^i = F(d_A^i)$ and sending a chain map $f$ to the chain map $g$ with $g^i = F(f^i)$. Mathematicians often denote this functor by $F$ instead of $Ch(F)$. I will walk in the middle; I will refer to the functor in the abstract with $Ch(F)$, but when I want to talk about the image of a complex or a chain map, I will use $F(A^\bullet)$ and $F(f)$ respectively.

Hopefully this definition seems “natural” or “obvious” to you. However, you might be surprised that you need “certain kinds of functors” for this to work. Recall that part of the definition of a chain complex is that each composition $d^{i+1}\circ d^i$ is a zero morphism. However, a functor does not need to send zero morphisms to zero morphisms, or even zero objects to zero objects. See this footnote¹ for a simple example. I will discuss the condition we impose on the functor later.

Recall that $K(\mc A)$ is defined by taking morphisms to be homotopy equivalence classes of chain maps. In order for $Ch(F)$ to induce a functor on $K(\mc A)$, we only need that $Ch(F)$ preserves homotopy. In other words, if $f$ and $g$ are homotopic chain maps, we need $F(f)$ and $F(g)$ to also be homotopic. It turns out that this is true with no extra work for us. Thus we define $K(F):K(\mc A)\to K(\mc B)$ by doing the same thing as $Ch(F)$ on objects and sending a morphism $[f]$ to $[F(f)]$, which we also denote by $F([f])$.

The biggest hurdle is in going to a functor between derived categories. If $K(F)$ sent quasi-isomorphisms to quasi-isomorphisms, then we would have no problem; we could define the functor $D(F)$ by sending a complex $A^\bullet$ to $F(A^\bullet)$ as usual, and sending a fraction represented by $(f,s)$ to a fraction represented by $(F(f),F(s))$. However, $K(F)$ does not send quasi-isomorphisms to quasi-isomorphisms in general.

In some special cases, even if $K(F)$ doesn’t send quasi-isomorphisms to quasi-isomorphisms, there is still a way to define a functor $D(F): D(\mc A)\to D(\mc B)$. However, it will not always agree with $Ch(F)$ and $K(F)$ on objects. You might not call this an extension, but there is a precise connection between $D(F)$ and $K(F)$ in these cases.

Another slightly technical point, but one that is nonetheless important, is that these functors will not always be defined on all of $D(\mc A)$. Instead, they will be defined on one of the subcategories $D^-(\mc A)$ or $D^+(\mc A)$, which are defined by restricting to objects whose cohomology is zero for sufficiently large integers in the positive or negative direction, respectively. For the more precise definition, see Part 3.

Special functors

Now, let me indicate the properties our functor $F$ should have, and how we will construct our functor $D(F)$.

Additive functors

Recall that in abelian categories, the set of morphisms $\Hom(A,B)$ between two objects can be given a “natural” structure of an abelian group; given two morphisms $f,g:A\to B$, there is a morphism $f+g:A\to B$. The $0$ morphism $A\to0\to B$ is the identity for this operation.

Also recall that a functor $F:\mc C\to \mc D$, by definition, induces a function $\Hom_{\mc C}(A,B)\to \Hom_{\mc D}(F(A),F(B))$ for every pair of objects $A,B\in \mc C$.

• A functor $F:\mc A\to \mc B$ between abelian categories² is called additive if for each pair of objects $A,B\in \mc A$, the function $\Hom_{\mc A}(A,B)\to \Hom_{\mc B}(F(A),F(B))$ is a group homomorphism: $F(f+g)=F(f)+F(g)$.

In particular, $F$ must send zero objects to zero objects and zero morphisms to zero morphisms.

An additive functor $F:\mc A\to \mc B$ then induces a functor $Ch(F):Ch(\mc A)\to Ch(\mc B)$, which turns out to also be additive; recall that $Ch(\mc A)$ and $Ch(\mc B)$ are abelian.

As mentioned previously, we don’t need to impose any extra condition on $F$ or $Ch(F)$ to define $K(F)$.

Exact functors

We now discuss some conditions on $F$ that are relevant for derived functors.

First, recall (or learn) that a “sequence” $A\xr f B\xr g C$ of objects and morphisms in an abelian category is exact if $\im f = \ker g$. In an abstract abelian category, this is a bit technical to define. On the other hand, in “concrete” categories, such as the category of $R$-modules, image and kernel are just how you learned in algebra: $\im f$ is the submodule of $B$ consisting of $f(a)$ for $a\in A$, and $\ker g$ is the submodule of $B$ consisting of those $b\in B$ such that $g(b)=0$. Therefore, exactness is saying that $f$ sends $A$ to exactly the elements that $g$ sends to $0$.

Any longer “sequence” of objects $\cdots\to A^{-1}\to A^0\to A^1\to \cdots$ is called exact if it is exact at each object, i.e. $A^{i-1}\to A^i\to A^{i+1}$ is exact for all $i$. The most common type of exact sequence is called a short exact sequence, which looks like $0\to A\xr f B\xr g C\to 0$. Just as before, exactness at $B$ means $\im f = \ker g$. Exactness at $A$ means $\ker f = 0$,and exactness at $C$ means $\im g = C$. In “concrete” categories, these mean $f$ is injective and $g$ is surjective. For example, if $M$ is a module and $N$ is a submodule of $M$, then there is a “canonical” short exact sequence $0\to N\to M\to M/N\to 0$, where $N\to M$ is the inclusion and $M\to M/N$ is the projection to the quotient, $m\mapsto m+N$.

For the following definitions, let $F:\mc A\to \mc B$ be an additive functor between abelian categories.

• $F$ is left exact if $0\to F(A)\to F(B)\to F(C)$ is exact for all short exact sequences $0\to A\to B\to C\to 0$ in $\mc A$.

• $F$ is right exact if $F(A)\to F(B)\to F(C)\to 0$ is exact for all short exact sequences $0\to A\to B\to C\to 0$ in $\mc A$.

• $F$ is exact if $F$ is left exact and right exact.

If $F$ is exact, then $K(F)$ takes quasi-isomorphisms to quasi-isomorphisms, so as previously discussed, we can define a functor $D(F)$ in the obvious manner. On the other hand, the more nuanced construction of derived functors is for a functor that is only left exact or only right exact.

Deriving your functors

To quote Pramod Achar’s book “Perverse Sheaves and Applications to Representation Theory”:

• To construct derived functors of nonexact functors, we need to find subcategories on which they behave “as though they were exact.”

Such a subcategory is known as a left or right adapted class. Such a category may or may not exist for a given functor. As such, a left or right exact functor might not have a derived functor. However, there are many important functors that we will encounter that do have these adapted classes. I won’t define what an adapted class is in this section, but I will nevertheless indicate their main properties and how they are used to define derived functors.

I will stick with assuming $F:\mc A\to \mc B$ is left exact; the story is only slightly different for right exact functors.

An adapted class for a left exact functor is called a right adapted class. Yes, right adapted, not left adapted. If $\mc D$ denotes the adapted class, let $Ch(\mc D)$ denote the subcategory of $Ch(\mc A)$ consisting of chain complexes whose objects are in $\mc D$. Similarly for $K(\mc D)$.

Recall that $D^+(\mc A)$ is defined by restricting to objects $X$ for which there is an integer $N$ such that $H^i(X)=0$ for $i< N$. There are also bounded variants of $Ch(\mc A)$ and $K(\mc A)$; for instance, $K^+(\mc A)$ is defined by restricting to complexes $X^\bullet$ for which there is an integer $N$ such that $X^i=0$ for $i< N$.

Here are the relevant properties of right adapted classes:

1. For any object $X\in K^+(\mc A)$, there is a quasi-isomorphism $q_X:X\to Q_X$ with $Q_X\in K^+(\mc D)$.

2. If $s:X\to Y$ is a quasi-isomorphism in $K^+(\mc D)$, then $F(s)$ is a quasi-isomorphism.

Using these three properties, we can construct a right derived functor $RF:D^+(\mc A)\to D^+(\mc B)$, as follows:

• For $X\in D^+(\mc A)$, define $RF(X)=F(Q_X)$, where $Q_X$ is given by property 1 above.

• Given $f:X\to Y$ a morphism in $D^+(\mc A)$, let $\tilde f = q_Y\circ f\circ q_X^{-1}:Q_X\to Q_Y$, where $q_X$ and $q_Y$ are understood to be the fractions represented by roof diagrams $(q_X, 1_X)$ and $(q_Y,1_Y)$. One can show that $\tilde f$ can be represented by a roof diagram $(g:Z\to Q_Y, s:Z\to Q_X)$ with $Z\in K^+(\mc D)$. We then define the morphism $RF(f):RF(X)\to RF(Y)$ to be the fraction represented by $(F(g),F(s))$. By property 2, this is a valid roof diagram.

Similarly, right exact functors need left adapted classes, and if they have one, then they have left derived functors $LF:D^-(\mc A)\to D^-(\mc B)$.

There are some well-definedness issues for sure; for instance, in property 1, there could be two quasi-isomorphisms $X\to Q$ and $X\to Q’$ with $Q,Q’\in K^+(\mc D)$. Clearly, choosing one over the other results in a different functor. However, the resulting functors will be “naturally isomorphic”, so it doesn’t really matter. In practice, one usually finds and fixes a construction of each quasi-isomorphism $q_X$.

See below to see what’s coming up next!

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Formalities

Let’s go through the previous section and fill in the details.

As I mentioned in a footnote, additive functors are defined between additive categories (see Part 2.5), with the exact same definition that I gave above. It also satisfies the property $F(A\oplus B)\cong F(A)\oplus F(B)$.

As for exactness, I suppose I never defined the image of a morphism, so let me do that here:

• Given a morphism $f:A\to B$ in an abelian category, its image is the kernel of its cokernel.

Now we define adapted classes. As before, I’m only going to do this for left exact functors $F:\mc A\to \mc B$. If you want the definitions for right exact functors, see section A.6 in Achar’s book “Perverse Sheaves and Applications to Representation Theory”.

• A full subcategory $\mc D\subset \mc A$ is a right adapted class for $F$ if it satisfies the following conditions:

1. For any object $A\in \mc A$, there is a monic map $A\to X$ with $X\in\mc D$.

2. If $0\to X’\to X\to X’‘\to 0$ is a short exact sequence with $X’,X\in \mc D$, then $X’‘\in \mc D$.

3. For any short exact sequence $0\to X’\to X\to X’‘\to 0$ with $X’\in\mc D$, the sequence $0\to F(X’)\to F(X)\to F(X’’)\to 0$ is exact.

Condition 1 is usually phrased as “$\mc D$ is large enough”, or “$\mc A$ has enough objects in $\mc D$”. For instance, you may have heard the phrases “enough injectives” or “enough projectives”.

Before I give the definition of a (right) derived functor, let me recall some notions. First, for $\mc A$ an abelian category, let $Q_{\mc A}:K(\mc A)\to D(\mc A)$ be the localization functor, defined by sending objects to themselves and morphisms $f:X\to Y$ to fractions $[(f,1_X)]$. Second, recall that there is a notion of a natural transformation, which is a kind of morphism between functors. See Part 2.5 for the precise definition.

Now, let me give the formal definition of a right derived functor for a left exact functor; it is defined via a universal property, and constructed using an adapted class. Even though this is the formalities section, I’m still skipping some details here, namely the triangulated structure on $K(\mc A)$ and $D(\mc A)$.

• Given a functor $F:\mc A\to \mc B$, a right derived functor of $F$ is a functor $RF:D^+(\mc A)\to D^+(\mc B)$ together with a natural transformation $\epsilon:Q_{\mc B}\circ F\to RF\circ Q_{\mc A}$, such that the pair $(RF,\epsilon)$ is initial with respect to this data.

It should also be understood that the localization functors have been restricted to the $+$ version of the homotopy and derived categories.

The construction of $RF$ is the same as what I gave in the previous section, but let me also give the construction of $\epsilon$.

• For $X\in K^+(\mc A)$, the component $\epsilon_X : Q_{\mc B}(F(X))\to RF(Q_{\mc A}(X))$ is the morphism $Q_{\mc B}(F(q_X))$, which makes sense because $RF(Q_{\mc A}(X))=Q_{\mc B}(F(Q_X))$.

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What’s next?

Now that we have the algebra under our belts, we can move onto the geometry and talk about sheaves. In fact, once we define sheaves, some of their related concepts, and some of their (derived) functors, we’ll be able to define perverse sheaves! I hope you’ve enjoyed reading thus far, and I hope you’ll enjoy the last bit of the ride as well!

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Footnotes: