I'd like to know what $$\prod_{k=1}^n (1-x^k)$$
evaluates to (assuming there is a simple closed form) and what it "is" in the context of commutative algebra (of which I knew little and recall less).
I'm sure I've seen this in the past but don't know where to place it. LaTeX search doesn't help.
Well, one has $$\prod_{n\geq1}(1-x^k) = \sum_{-\infty\leq n\leq\infty}(-1)^nx^{(3n^2-n)/2}.$$ This is a consequence of Jacobi's triple product identity.
You asked about a finite product, but from this equality you can tell what are the coefficients in the expanded finite product.
The context for this identities is, among others, the theory of partitions. I am sure you will find a proof of this in Andrews' excellent The Theory of Partitions, along with related information.