Fruitless Schemes

Alexander Grothendieck was one of the most influential mathematicians of the twentieth century, and his central achievement, so I hear, was the notion of scheme. What the hell is that, you might ask, or at least I did, and in a moment of moral weakness, I thought I might try to find out. Wikipedia, sad to say, was little help:

To be technically precise, a scheme is a topological space together with commutative rings for all of its open sets, which arises from gluing together spectra (spaces of prime ideals) of commutative rings along their open subsets.

That didn't help me much, even though I sort of know what a commutative ring and a topological space are. However, the further reading had this article: Can one explain schemes to biologists?, an obituary for Grothendieck that attempted to do just that. I'm not sure how biologists did, but I flunked.

His best known work is his attack on the geometry of schemes and varieties by finding ways to compute their most important topological invariant, their cohomology.

There is the respect that makes makes calamity...

The trouble is that I don't understand even the simplest examples of cohomology. If the biologists do, they understand more algebraic topology than I do - of course I don't understand much at all.

PS - I think that a scheme is sort of a generalization of a variety, which, again, I think, is the solution set of a set of polynomial equations. Don't hold me to that.