Thoughts on Homological Algebra

So I’ve passed my homological algebra exam. Today morning, I unexpectedly realized that I’m going to have a mathematics exam (yes… with the great pain I admit that HA is mathematics :-) and I actually don’t know any proof. This is because HA is so tedious – normally, I feel very uncomfortable if I don’t check at least some details by myself, prove few things by myself, etc. This time, I didn’t check any details. I wanted to several times, but looking at them (I was using a book by Gelfand & Manin) almost always discouraged me.

It’s not that there was nothing that interested me in itself – for example, I really wanted to know why when one passes from category of chains over abelian category to category localized in quasiisomorphisms (see below) then one gets all homotopic morphisms identified. In aforementioned book there are few definitions and diagrams -“nothing is happening”, but I don’t get illuminated even after longer moment of looking at them.

Correspondingly, my knowledge of derived/triangulated categories, simplicial objects and derived functors of nonadditive functors (these were the main topics of a course) is a little bit like a knowledge of history – I know essential facts, I even see some causal connections between them, but I miss something – a spirit of mathematics?.

I’d love to see some comments on how you are/were learning homological algebra. Especially, I’d love to know how (for example) Terrence Tao learned HA (as everybody probably knows, there is non-zero chance of meeting him around wordpress.com, so who knows :-). My General Theory of Geniuses(;-) predicts that he (or maybe He?:-) spent as much time as I did (or less, of course) on the given topic, but his unconsciousness checked all the details and communicated them to his consciousness, one day or another (actually, my GTG says that my unconsiousness also checked all the details, but it saved them for itself, damn bastard.)

Below follows a survey of what I’ve learned on this HA course, part I: derived categories, with some additional thoughts on the subject. I believe you may find it worth reading if you already know some basics of HA (nothing more than classical derived functors) and you’d like to read some informal introduction, to have some view on a matter before studying it for real. Comments are appreciated.

(*)

Derived categories. Is there any motivation for them? Well, for learning them there definitely is some – gossip I heard says that physicists are using them in non-trivial way. Actually,Homological Mirror Symmmetry is formulated using derived categories. I don’t know any details. I’d love to know.

(And since this thing is “Fightin’ the resistance of matter” (and I am listening to some old-style-grungy depressing music) I’d like to use this opportunity to express my bitter sorrow that I have to learn mathematics this way. I envy everybody “on the front of research” that they have so much fun out of doing mathematics, and I learn things that I know they’re using but don’t know actually why, so they seem artificial and not funny……)

Yes… So let’s pretend I know the motivation :-) Yay! I suppose you know few things about classical derived functors. You take an additive functor F: A –> B (A and B are abelian categories) which is, say, left exact, which means that for every exact sequence 0 -> X -> Y -> Z -> 0 you get exact sequence 0 -> F(X)-> F(Y)-> F(Z). In general the last arrow isn’t onto, so you want to measure how strongly it’s not onto. And you invent derived functors RⁱF which have a property that they fit into long exact sequence:

0 -> F(X)-> F(Y)-> F(Z)->

-> R¹F(X)-> R ¹ F(Y)-> R ¹ F(X)->

-> R²F(X)-> R²F(X)-> R²F(X)->…

When we’ll have derived categories everything’ll be more elegant: All the information about derived functors will be encoded in only one functor DF: DA -> DB (DK denotes derived category of K). Greatest change in conciseness will be seen perhaps in derived functor of composition. You might remember that to compute R^*(F∘G) one needs to write down a spectral sequence(AAA!!! NOOT A SPEECTRAAL SEEEEQUEEENCE!!!!). After introducing derived categories we’ll have very elegant D(F∘G)= DF ∘ DG. This is something I still need to understand – I don’t really know where the spectral sequence has gone (but I’m happy he went away, he may stay there if he wishes to). However, this statement’s elegance is quite convincing that derived category is right thing to do.

From more philosophical point of view – what’s the most important derived functor? No, it’s not some ?%*& ext or tor. It’s cohomology. It might be that you didn’t know that, so i’ll write explicetely: Cohomology of a topological space X can be realised as a derived functor of a Global Section functor Γ:Sheaves(X)->Ab which takes a sheaf on X and gives back an abelian group of all global section of this sheaf. For example, if you want to get de Rham cohomology of a manifold you simply take sheaf which on every connected open subset takes value R (real numbers)- it’s obvious from definitions that derived functors of Γ evaluated on this sheaf are de Rham cohomology groups, because de Rham complex is (by Poincare lemma) a resolution of sheaf R(and this resolution is suitable for computing derived functor, though it’s not injective – this isn’t very hard).

Taking other sheaves and evaluating derived functors of Γ on them leads to some different cohomology theories, which can measure some other things than ordinary cohomology theory does. So, taking derived category of Sheaves(X) and studying DΓ could be seen as “studying all cohomology theories in one time” which sounds like a cool thing to do.

Motivation for a following definition comes also from the desire to identify any object of A with its resolution. This in turn might be motivated by a following observation: if you want to compute Torⁱ(X,X), value of classical derived functors of functor _⊗X on object X, you take injective resolution of X, tensor it with X and compute homologies.(BTW, at least when resolution of X is finite you could instead take resolution of X and tensor it with resolution of X, form a Tot complex and compute homologies – this also suggests that resolution and an object are very similar from HA point of view). So, if you somehow identified object with its resolution you could say simply “to compute value of derived functor on an object just compute original functor on an object”:-). It’s a little bit tricky, but basically this is what we’ll end with.

There are also some other motivations listed in Gelfand&Manin.

Ok, let’s fix an abelian category A. Let ChA be a category of (cohomological) chain complexes over A (perhaps bounded or bounded from right or bounded from left – constructions will not change at all, outcome will be called D^b od D^– or D⁺). Motivated by all the above we want to identify two objects in ChA if there is an quasiisomorphism between them (quasiisomorphism is a morphism which induces isomorphism on homology). Accordingly, we make a following definition.

Derived category of A is a pair (DA, Θ), where DA is an additive category and Θ: ChA –> DA is a functor that carries all quasiismorphisms to isomorphisms, such that this pair is universal for this property (if there is other such pair (B,Ψ) than there exists unique functor DA–>B such that suitable diagram commutes)

There exists a very simple proof of existence. Let DA be a category whose objects are the same objects as these of ChA (that is, Ob(DA)= OB(ChA)). Set of morphisms between two objects X and Y in DA, Mor_DA(X,Y), is a set of paths between X and Y in a directed graph whose vertices are objects of A, and whose edges are

i) morphisms in A
ii) edges x_s, for every quasiisomorphism s: K->L, that go from vertex K to vertex L.

with a following relation: two paths in this graph are equivalent if you can obtain one from the other by a series of basic operations:

i) exchanging any edge f∘g with path consisting of edges g and f
ii) exchanging paths of form s∘x_s with id.

Composition of morhism is defined by conjunction of paths in the above graph.

Above construction is called localisation of ChA with respect to quasiisomorphisms.

This is it. There is obvious functor Θ from Ch(A) to DA. Pair (DA, Θ) has a suitable universal property (no, I didn‘t check all the details but it stands to reason ;-). But, DA just constructed is poor to work with – it’d very hard to decide even whether given path represents a trivial morphism. So we should work some more.

Definition: Suppose we’re given category B and class of morphisms C ⊆ MorB. We say that this class C is localizing in B iff

i) id_X∊C for ever X∊ObB

ii) if α, β ∊ C then α∘β ∊ C, if it makes sense

iii) for every α∊C and f ∊ MorB, α: X->Z, f: Y->Z there exist β∊C, g∊MorB, β: W->Y, g: W->X, such that suitable diagram commutes (draw it)

iv) for every α∊C and f ∊ MorB, α: X<-Z, f: Y<-Z there exist β∊C, g∊ MorB , β: W<-Y, g: W<-X, such that suitable diagram commutes (draw it)

v)if α∘f=α∘g for some α∊C, f,g∊MorB then there exists β∊C such that f∘β=g∘β

Proposition: If C is localizing in B then localisation of B with respect to C has a simple description. Every morphism between to objects X and Y in this localisation can be represented by a graph (denoted later (1))

X <-α- W -f-> Y

for some α∊C, f∊MorB. Moreover, two such diagrams

X <-α- W -f-> Y

X <-α’- W’-f’-> Y

represent the same morphism iff there exist β:Z->W, g:Z->W’, β∊C, g∊MorC, such that suitable diagram commutes.

Word about a proof: from a property iv)it follows that indeed every morphism in a localisation has suitable representation. Proof of the latter part is carried out in Gelfand&Manin in a following way: they define category B’ to be a category with objects same as in B and morphisms given by diagrams of form (1). They define suitably composition of such diagrams ((1) can be regarded as a fraction of form f/α and composition is realized by adding such fractions) and use all the properties i)–v)to prove that everything is well defined and that B’ has a suitable universal property. So indeed this B’ must be itself a localisation.

However, class of quasiismorphisms is not localizing in ChA. Cha!:-) We have to build an intermediate category out of ChA – so called homotopy category of A, denoted KA. Its objects are same as these of ChA, and

Mor_KA(X,Y):= Mor_ChA(X,Y)/{morphisms which are chain homotopic to 0}

One painfully checks that quasiisomorphisms are localizing in KA, so KA localized in quasiisomorphisms has nice description guaranteed by above proposition.

Now follows actually something interesting. We don’t know whether localized KA is equivalent do DA. We have a functor from DA which is identity on objects and which is obviously surjective on morphisms (it’s really obvious). We don’t know whether it’s injective on morphisms. That is, we don’t know if the following is true:

Fact: If f,g: X->Y are homotopic in ChA then their images are equal in DA.

It turns out that it is true (G&M, III.4.3), but for me it’s kind of a miracle – DA is a localisation in quasiisomorphisms. The construction doesn’t say a word about homotopic morphisms. Proof occupies one page in G&M (so it’s about 0.9 pages to long for me to really understand :-).If anybody has any philosophical remarks on this I’ll greatly appreciate if he’ll share them!

Now, when we have our shiny derived category, we move slowly to derived functors. Before, let’s observe that we have a functor from our original category A to DA (realize object of A as a chain complex concentrated in dimension 0, send it to DA). I claim that this functor is an equivalence of categories A and full subcategory of DA consisting of chain complexes with nonzero homology only in dimension 0 (called “category of H⁰ complexes”).(This is one of very few things I actually checked so here you are :-). We have to check two things, that our functor gives a bijection Mor_A(X,Y)–> Mor_DA(X,Y) for every pair of objects X,Y ∊ A, and that every H^{0 ­} complex of DA is isomorphic to one concentrated in dimension 0.

Both are easy: for the latter one we need to take H⁰ complex Z and build a diagram

Z <-α- V -β-> H⁰(Z)

with both α and β quasiisomorphisms. For Vⁱ one can take Zⁱ for i<0, ker(d⁰) for i=0 and 0 for i>0. For the first one let’s check surjectivity. So we have a diagram

X <-α- Z -f-> Y

and we need to check that it’s equivalent to a diagram

X <-id- X -g-> Y

(these diagrams are images of morphisms from A). For g one takes what f∘α^-1 generates on H⁰, and for suitable β:V->Z, h:V->X one takes V as above, with obvious morphisms. Injectivity is pretty obvious,’cause no matter what representation for morphism you take, you always get the same thing on homologies.

For chain complex X^*(or rather cocomplex), or its image in KA or DA, we define shifted complex X[i]ⁿ:= X[i+n]. For ~2 years I couldn’t remember this definition – there is always a problem “in which direction is it moving?”. But 2 days ago I finally managed to invent a suitable visualisation: I see integer points on a line which never move.Under this line I see an original complex and then I imagine that this complex moves violently in a direction opposite to the order of integer numbers by i places.

We turn to derived functors, for some simple case for now: suppose we’d like to compute classical Extⁱ_A(X,Y) for some X,Y∊A. You can do it in a standard way, but there is also following:

Fact: Extⁱ_A(X,Y) ≅ Hom_DA(X[k], Y[k+i])

(As I said in a motivation part, we have a functor Hom(_, Y): A -> A and we’d like to define DHom(_,Y): DA -> DA that would encode all the information about classical Extⁱ(X,Y). Additionally it should be defined by pointwise use of normal Hom. The fact shows that we should have a chance to do something in this spirit)

Notice that left side doesn’t depend on k. However, it’s easy to see that right side doesn’t as well (up to natural isomorphism induced by “shifting functor”). This is very elegant: for example, one can now define product in Ext in a very simple manner:

Extⁱ_A(X,Y) x Ext^j_A(X,Y)-> Ext^i+j_A(X,Y)

is realized by a composition of morphisms in DA. Things like long exact sequences associated to short exact sequence 0 -> X -> Y -> Z ->0 also follow from general abstract nonsense.(Namely, in more general setting of triangulated category T we always have that so called distinguished triangle – and above short exact sequence is an example of distinguished triangle in derived category – gives rise to apropriate long exact Hom_Tsequence. I hope to write few words on that in a next article.)

Word about a proof of a Fact: It’s easy to see that every element of classic Extⁱ_A(X,Y) gives rise to an element of Hom_DA(X[k],Y[k+i]), for clarity let’s put k=0. Indeed, element of Ext group can be represented as an extension

0-> Y=V^-i -> V^-i+1 ->…-> V⁰-> V¹=X -> 0

This extension clearly defines element of Hom_DA(X,Y[i]), that is a diagram

X <-d^-1– V -> Y

where V is a 0 -> V^-i->…-> V⁰-> 0. G&M prove that this assignment is an isomorphism.

Let’s focus for a moment on derived functor of a general additive left-exact functor F: A -> B. We’d like to obtain a DF: D⁺A -> D⁺B. Naively, we could take object of DA, represent it as a chain complex, and act on this chain complex with F. This is bad idea, because F in general doesn’t preserve quasiismorphisms – if it does, then I believe it’s exact (I’m sure there is an implication exactness => preserves quasiisomorphisms)(although, notice that F defines a functor on homotopy category KA, because it is additive, so it preserves chain homotopy). So we woudn’t end up with a functor, as functor must preserve isomorphisms (and quasiisomorphisms become isomorphisms in derived category).

What we’ll do instead is motivated by classical derived functors.

We say that class R⊆ObA is adapted to the functor F if following conditions hold:

i) R is closed with respect to taking finite sums

ii) Every object of A might be embedded in some object of R

iii) F takes bounded from the left, exact chain complexes consisting of objects of R into exact sequences (for right exact F one would have to take here complexes bounded from the right).

Now, to make long story short, It’s pretty simple to see that F defines a functor G from K⁺(R) localized in quasiisomorphisms to DB, where K⁺(C) is a full subcategory of K⁺A consisting of these chain complexes whose all objects are from C. One proves that actually K⁺(C) localized in quasiisomorphisms is equivalent to DA, so we can choose an equivalence Ω: D⁺A ->”K⁺(C) localized in quasiisomorphisms” and say that DF is a composition G∘Ω.

This definitions depends obviously on a choice of adapted class C and of equivalence Ω. That’s why one for a second forgets what we’ve done and adopts following definition for a derived functor:

Definition: Given F: A->B as usually, derived functor is a pair consisting of

1) a functor DF: D⁺A -> D⁺B which takes exact triples into exact triples

2) a natural transformation ε_F of obvious functors

K⁺A -> K⁺B -> DB and K⁺A -> DA -> DB

that is universal with these properties (it there is some other pair (G, ε) then there is a unique transformation of functors

K⁺A —-> DA —DF—> DB and K⁺A —-> DA —G—> DB

such that suitable diagram commutes.)

Remark about 1): DA, and D⁺A, are not abelian, but we call a triple exact if it comes from an exact triple in ChA (or Ch⁺A).

It takes some trouble to prove that what we’ve constructed above is a derived functor, after equipping it with suitable ε_F.(5 pages in G&M. I don’t even guess what’s trivial and what’s not.)

But, modulo some checking, we’ve done what we wanted to :-)

(There’s a famous exercise in Lang’s Algebra, in the chapter concerning homological algebra:”Take any Homological Algebra notebook and prove all the theorems by yourself”. There is also a famous footnote in polish translation:”We advise a reader to omit this exercise at a first reading”)

I hope to read rest of G&M’s chapter on derived categories, using the “homological momentum” that the exam gave me. Maybe then I’ll write some (shorter) notes here.