This chapter gives a “look under the hood” at the algorithm that actually lets us perform computations over a polynomial ring. In order to work with polynomials, we need to be able to answer the ideal membership question. For example, there is no chance of writing down a minimal free resolution if we cannot even find a minimal set of generators for an ideal. How might we do this? If R = k[x], then the Euclidean algorithm allows us to solve the problem. What makes things work is that there is an invariant (degree), and a process which reduces the invariant. Then ideal membership can be decided by the division algorithm. When we run the univariate division algorithm, we “divide into” the initial (or lead) term. In the multivariate case we'll have to come up with some notion of initial term – for example, what is the initial term of x2y + y2x? It turns out that this means we have to produce an ordering of the monomials of R = k[x1, …, xn]. This is pretty straightforward. Unfortunately, we will find that even once we have a division algorithm in place, we still cannot solve the question of ideal membership. The missing piece is a multivariate analog of the Euclidean algorithm, which gave us a good set of generators (one!) in the univariate case. But there is a simple and beautiful solution to our difficulty; the Buchberger algorithm is a systematic way of producing a set of generators (a Gröbner basis) for an ideal or module over R so that the division algorithm works.

Although the title of this book is “Computational Algebraic Geometry”, it could also be titled “Snapshots of Commutative Algebra via Macaulay 2”. The aim is to bring algebra, geometry, and combinatorics to life by examining the interplay between these areas; it also provides the reader with a taste of algebra different from the usual beginning graduate student diet of groups and field theory. As background the prerequisite is a decent grounding in abstract algebra at the level of [56]; familiarity with some topology and complex analysis would be nice but is not indispensable. The snapshots which are included here come from commutative algebra, algebraic geometry, algebraic topology, and algebraic combinatorics. All are set against a backdrop of homological algebra. There are several reasons for this: first and foremost, homological algebra is the common thread which ties everything together. The second reason is that many computational techniques involve homological algebra in a fundamental way; for example, a recurring motif is the idea of replacing a complicated object with a sequence of simple objects. The last reason is personal – I wanted to give the staid and abstract constructs of homological algebra (e.g. derived functors) a chance to get out and strut their stuff. This is said only half jokingly – in the first class I ever had in homological algebra, I asked the professor what good Tor was; the answer that Tor is the derived functor of tensor product did not grip me.

In this chapter we give a quick introduction to sheaves, Čech cohomology, and divisors on curves. The first main point is that many objects, in mathematics and in life, are defined by local information–imagine a road atlas where each page shows a state and a tiny fraction of the adjacent states. If you have two different local descriptions, how can you relate them? In the road map analogy, when you switch pages, where are you on the new page? Roughly speaking, a sheaf is a collection of local data, and cohomology is the mechanism for “gluing” local information together. The second main point is that geometric objects do not necessarily live in a fixed place. They have a life of their own, and we can embed the same object in different spaces. For an algebraic curve C, it turns out that the ways in which we can map C to ℙn are related to studying sets of points (divisors) on the curve. If the ground field is ℂ, the maximum principle tells us that there are no global holomorphic functions on C, so it is natural to consider meromorphic functions. Hence, we'll pick a bunch of points on the curve, and study functions on C with poles only at the points. Sheaves and cohomology enter the picture because, while it is easy to describe a meromorphic function locally, it is hard to get a global understanding of such things.