Algebraic K-theory originated in classical materials that connected class groups, unit groups and determinants, Brauer groups, and related things for rings of integers, fields, etc, and includes a lot of local-to-global principles. But that's the original motivation and not the way the work in the field is currently going - from your question it seems like you're asking about a motivation for "higher" algebraic K-theory.
From the perspective of homotopy theory, algebraic K-theory has a certain universality. A category with a symmetric monoidal structure has a classifying space, or nerve, that precisely inherits a "coherent" multiplication (an E_oo-space structure, to be exact), and such an object has a naturally associated group completion. This is the K-theory object of the category, and K-theory is in some sense the universal functor that takes a category with a symmetric monoidal structure and turns it into an additive structure. The K-theory of the category of finite sets captures stable homotopy groups of spheres. The K-theory of the category of vector spaces (with appropriately topologized spaces of endomorphisms) captures complex or real topological K-theory. The K-theory of certain categories associated to manifolds yields very sensitive information about differentiable structures.
One perspective on rings is that you should study them via their module categories, and algebraic K-theory is a universal thing that does this. The Q-construction and Waldhausen's S.-construction are souped up to include extra structure like universally turning a family of maps into equivalences, or universally splitting certain notions of exact sequence. But these are extra.
It's also applicable to dg-rings or structured ring spectra, and is one of the few ways we have to extract arithmetic data out of some of those.
And yes, it's very hard to compute, in some sense because it is universal. But it generalizes a lot of the phenomena that were useful in extracting arithmetic information from rings in the lower algebraic K-groups and so I think it's generally accepted as the "right" generalization.
This is all vague stuff but I hope I can at least make you feel that some of us study it not just because "it's there".
Can I be the first to recommend Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7 1971 29–56 (1971).
From the MR review: "In this important and elegant paper the author gives new elementary proofs of the structure theorems for the unoriented cobordism ring $N^\ast$ and the complex cobordism ring $U^\ast$, together with new results and methods. Everyone working in cobordism theory should read this paper."
The paper was revolutionary in (at least) two ways.
- The proofs are almost entirely geometric, with cobordism classes represented by proper oriented maps of manifolds. Quillen cites Grothendieck as inspiration for this, and such methods should appeal to algebraic geometers familiar with the Chow ring.
- Formal group methods are used to prove results in stable homotopy theory. It's hard to overestimate the impact this has had. Indeed almost all of the modern connections between homotopy theory and algebraic geometry seem to go through formal groups, drawing influence from Quillen's idea.
Best Answer
I've always liked the interpretation Quillen gave in his "On the group completion of a simplicial monoid" paper (Appendix Q in Friedlander-Mazur's "Filtrations on the homology of algebraic varieties"). Here is a somewhat revisionist version.
Associated to a monoidal category C, you can take its nerve NC, and the monoidal structure gives rise to a coherent multiplication (an A∞-space structure) on NC. (If you work a little harder you can actually convert it into a topological monoid.)
May showed in his paper "The geometry of iterated loop spaces" that an A∞-space structure on X is exactly the structure you need to produce a classifying space BX, and there is a natural map from X to the loop space Omega(BX) that is a map of A∞-spaces, and is a weak equivalence if and only if π0(X) was a group rather than a monoid using the A∞-monoid structure. In fact, Omega(BX) satisfies this property, and so you can think of it as a "homotopy theoretic" group completion of the coherent monoid X.
What Quillen showed was that you can recognize the homotopy theoretic group completion in the following way: the homotopy group completion of X has homology which is the localization of the homology ring of X by inverting the images of π0(X) in H0(X). Moreover, the connected component of the identity in the homotopy group completion is a connected H-space, so its fundamental group is abelian and acts trivially on the higher homotopy groups.
In particular, if X is the nerve of the category of finitely generated free modules over a ring R, then X is homotopy equivalent to a disjoint union of the classifying spaces BGLn(R), with monoidal structure induced by block sum. The monoid π0(R) is the natural numbers N, and so you can consider the map
to a union of copies of the infinite classifying space. This map induces the localization of H*(X), so the space on the right has to have the same homology as the homotopy group completion, but the problem is that the connected component of the identity on the right (BGL(R)) doesn't have an abelian fundamental group that acts trivially on the higher homotopy groups, so this can't be the homotopy group completion yet.
So this leads to the plus-construction: to find the homotopy group completion you're supposed to take BGL(R) and produce a new space, which has to have the same homology as BGL(R), and which has an abelian fundamental group (plus stuff on higher homotopy groups). This is what the plus-construction does for you.
Quillen's Q-construction contains within it the symmetric monoidal nerve construction (you can consider just the special exact sequences that involve direct sum inclusions and projections), but it's got the added structure that it "breaks" exact sequences for you. I wish I could tell you how Quillen came up with this, but this is the best I can do.