The revolution for homological algebra pretty much started with
Eilenberg-MacLane - who wrote an articlebook that did the constructions
necessary for the very topological versions of homological algebra - but
without ever involving the actual topological spaces.
The point is that the way you do algebraic topology is that you tend to
set up a functor Top → R-ChMod that assigns a chain complex of R-modules
to each (nice enough) topological space, and then you add functors
R-ChMod → R-Mod that extract informations from these. Typical examples
are cellular chain complexes with coefficients somewhere nice for the
first functor, and then homology or cohomology for the second functor -
depending on what viewpoint is the most obvious.
The revolution was that we simply throw out that first functor.
In order to study (co)homology, we don't really need to care that there
was a topological space somewhere to begin with. We only, really, need a
nice enough category of chain complexes (if it's abelian, then that's
fine - we get the long exact sequences in homology and other niftiness
easily then, but if it's not, triangulated will do...) and we study
certain types of functors from these to module categories.
Low order Ext
The area where this is most notable is in representation theory. This
field comes in several flavours: group representations, where we study
kG-Mod for some (sometimes finite) group G; Lie algebra representations,
where we study g-Mod for some Lie algebra g; quiver representations,
where we study kQ-Mod for some (finite) quiver algebra kQ - and so on.
One question that tends to crop up here, and with a high degree of
importance for the non-homological algebraists around me - is what
happens if we know only parts of our group? Can we say something about
the entire group based on that?
It turns out that we can. There are very neat correspondences between
the lower order Ext groups over kG and the behaviour of G itself. I'm
going to stick to group representations here, since that's the area I
know best - however, this is something that pops up analogously all over
the place.
Extensions
Suppose you have some R-module K that you know embeds, in some specific
way, into some larger R-module M. And suppose you find the quotient
L=M/K in some manner. What could, then, M be? One obvious answer is
[tex]G=K\oplus L[/tex], but is this enough? This ends up depending on
Ext1R(L,K), with each element of this particular Ext
group indexing precisely one such extension.
This is at the core of Maschke's theorem, by the way, which says that if
the characteristic of the field k doesn't divide the group order |G|,
then by a specific construction, the only extensions possible for
any kG-modules are the split extension - the one where we just take
the direct sum.
This all leads to a wealth of useful information and ideas in
representation theory. For instance, there is a way to describe modules
proposed by Dave Benson and some co-authors, where you draw diagrams
with each vertex being a simple module, occupying that spot in a
composition series, and the edges being taken from the relevant
Ext1.
Invariants and coinvariants
Suppose you have a group acting on a vector space. This can be taken
extremely physical - quantum mechanics is all about this kind of
situation, or so I've heard. Then it might be interesting to figure out
the invariant subspace: {a|ga=a for all g in G}. This is Ext0.
Or we might want a basis for the complement: representatives for every
way that things can move. This is the coinvariant vector space, defined
as A/(ga-a), and this is just Tor0.
Simples, projectives and the stable module category
Simple modules are nice. They don't have invariant subspaces. In the
best of all worlds - which is when Maschke holds - simple modules are
precisely the irreducible modules. However, when Maschke doesn't hold,
we can have non-trivial Ext1, and thus we can build larger
modules out of simples by a kind of gluing: they aren't just a nice
direct sum of simples, but something ickier.
Thus, unless Maschke holds, there will be weird things happening in the
module category.
These weird things, though, are controllable. To be specific, we can
consider the smallest possible irreducible modules. These will end up
being building blocks, and for nice enough worlds, these will also end
up corresponding closely to the simples - in the way that we can
allocate a simple to an irreducible projective in a bijective manner.
So ... what is this projective I keep throwing around? Take a free
module. This is a direct sum of a finite number of copies of the ring R.
This will have direct summands. By picking apart all summands into
further direct summands, at some point we hit bottom: we cannot pick
anything apart any longer. This is, by the theorem of Krull-Schmidt, a
well-defined state of being. We can permute things, but in essence, a
module is just its decomposition into irreducibles.
So, anything that is a direct sum of a free module is a projective. We
can lose projectivity by taking quotients - so if we add relations, we
may well get lost. But as long as we just look for direct summands,
we're pretty much home free. Now, the irreducible projectives have to be
summands of the ring R itself, so they end up actually being (left)
ideals in the ring. And each of them corresponds intimately to a simple
module.
One trick that's very beloved among the people who worry about these
things is to get rid of anything projective, and look at the stable
module category. In this, we just quotient away anything projective -
morphism sets are taken modulo morphisms that factor through a
projective... This way, we only have the "essential", or as it is known
to the experts of the field "difficult" information left. Then
Extn(M,N)=Hom(Ω:sup:nM,N), where Ωn is the nth
syzygy - see below for more on this.
So, homological algebra lets us understand the stable module category,
which in turn lets us understand the parts that are essential to the
module category structure.
Resolutions
I just promised you I'd tell you about syzygies. First off, some
personal information - because readers always love that!
If you find me on IRC, on EFNet or on Freenode, you'll find me under the
nick Syzygy-. The - is there because there is someone who's been using
Syzygy for years and years on EFNet and because I'm not deliberately
trying to be a bastard if I can help it. The rest of the nick is there
to a certain extent because I like the way I write it in longhand.
And to a certain extent because it is an epitome of why homological
algebra is interesting in my eyes.
Suppose we are interested in a finitely presented module, which we might
be for many reasons, including being interested in algebraic geometry
and in solving systems of polynomial equations. We might then just
figure out what relations hold within a set of generators, which gives
us a generating set, and some relation set.
These, relations, though are far from guaranteed to be the whole story.
It's probable that there are non-trivial relations between the
relations. What do we do? We figure out what these are. They span the
first syzygy module of the module we started with, denoted by ΩM.
But this is unlikely to be free, so we can keep on going.
This way, we get a sequence of modules, all of which are free - since we
just choose a generating set in each step - and with maps between them
adding all the extra relations. But this is nothing other than a free
resolution of the starting module. And here comes the candy that hooked
me for this discipline: studying modules over their resolutions is the
same thing as studying what chain complexes are, deep down, which in
turn is the same thing as studying homological algebra.
Want to figure out what a module map means for the family of syzygies?
What you really want is a chain map in the chain complex category. But
some of these maps - or even portions of maps - will not carry relevant
information. So we factor those away, and we get a slightly weirder
category. But here, equality doesn't quite mean what it should, so we
add in more equality relations. And suddenly, we live in a derived
category - and in here, the Hom sets are Ext groups, and the tensor
products are Tor groups.
Number theory, geometry, and computation!
To continue this tour de force, consider the theorems in vector calculus
relating various triple and double integrals. (note - I never dealt with
this. I rode on technicalities to root out calculus from my curriculum
so it would fit more algebra....) These theorems, in the end, only state
that [tex]\partial^2=0[/tex], which is at the core of what homological
algebra is all about.
If we formalize this particular recognition a bit, and tug at the
corners, we end up with de Rham cohomology, which deals with what you
can do with differential forms on manifolds (layman speak: things you
can integrate. The f(x)dx after the integral sign is a typical
differential form) - and this is one of the many many places where
cohomology rather than homology ends up being the "right" way to view
things just because you started out as a geometer instead of .. well ..
topologist or algebraist.
The same kind of thing happens in algebraic geometry as well. You start
out happily with your varieties, you conclude that as soon as things get
interesting, the nice and pretty concepts of coordinate rings don't hold
up, and you're forced to go to coordinate sheaves. And then you try to
figure out what you can do with sheaves of functions on a variety - and
before you know it, you reconstructed sheaf cohomology. This, by the
way, a quick look at wikipedia told me, lets you define euler
characteristics for varieties in a way consistent with the classical
uses of it.
I am no geometer, and I'm not the person to tell you about the
intricacies of these things. If you understand them, though, I'd love to
figure them out at some point.
The discussion of Khovanov homology is a slightly (though not very)
similar thing to this. Again, I have no real idea, and am treading on
thin ice here.
So, alpheccar. Is this what you asked for? Please tell me what more you
want covered, and I'll write up some more! This was fun writing!