For a cup of coffee more

As forces and probabilities greater than I decide my location in the next 48 hours, let me briefly profess my love for Samantha Morton as Serpent Queen. It is beautiful to watch a sly smile creep across her lips as schemes come to fruition, and perhaps more importantly, how hardships are quickly incorporated into yet another scheme.

Indeed, isn’t it most interesting when things fail? When we invariably hit our nose, the rock rolls down the hill again. For we can scheme how to roll it up next time. Blood in the streets and all that.

One’s thoughts, I assume, are then invariably drawn to commutative algebra and its many Vanishing theorems. And when things vanish, good things happen, as Catherine would witness (or you know, let the witness vanish). Indeed, equalities follow, beautiful expressions abound. We all love it. Alas, I like when things go wrong. It is simply more fun when the problem fights back. Or as my also friend Gil once said, inequalities are harder to prove than equalities. And so, let me ask a question as an example here, concerning the flatness condition.

Gerd Gotzmann at investigated in 1978 the following question. Consider R a polynomial ring over a noetherian one (a field is fine for this). And consider moreover I a graded ideal within it.

Now, if I has some given size in degree d, then it cannot be too small in degree d+1. After all, multiplication does happen. In fact, there is a rather nice lower bound that is a bit complicated to write down explicitly (it was worked out by Macaulay), but it is attained by ideals that are lexicographically first. One can also think of this in another way, as in this case the quotient R/I grows maximally after degree d, meaning that almost every product is then distinct.

So let us say that s(j) (of course this silently depends on the dimension if I in degree d, as well as R)is the size of such a lexicographically first ideal in degree j, so that

and .

We then have the following result:

Theorem (Gotzmann) If I is generated in degree d, then for every j at least d, we have

So what happens if things go just a little wrong. For this, let us introduce an error E.

Question Is there a function P[E,d](j) such that, if I is generated in degree d, and

and

then for every j at least d, we have

?

In other words, how stable is Gotzmann’s result (notice that P does not depend on R, otherwise the answer is trivial)? How stable are other results? This reminds me of the beautiful program in quantitative topology that another friend, Misha Gromov, had pushed into the limelight. Let us call it quantitative algebra. As for the above question, I know it only in the case of monomial ideals for now.

(Btw: tis the season, at least as far as supermarkets go. Glühwein 🙂 )