Are you dreading the social responsibilities that come with this time of the year? You can spice up the conversations with your relatives and friends using the following fun facts in algebraic geometry. Whenever a conversational lull comes up, drop one of these and things will turn fun fun fun again!

Direct products of quasicoherent sheaves are not exact

The category of quasicoherent sheaves on a scheme is a nice Grothendieck abelian category. But one thing that is a bit weird is that direct products are not necessarily exact in there (unless the scheme is affine). The most down-to-earth example is the projective line over a field, an example attributed to Bernhard Keller you can read about in Henning Krause's paper The stable derived category of a noetherian scheme, published in Compositio Mathematica. There is also a MathOverflow question about this property of the category of quasicoherent sheaves, where Leonid Positselski answers a question about a paper by himself. I like it when something like that happens.

The projective line is a projective variety, which means that by Serre's theorem we can study its category of quasicoherent sheaves using the category of graded modules over the graded ring $k[x,y]$ modulo the category of torsion modules. If we were looking at coherent sheaves a torsion graded module is something which is finite-dimensional over the field, to study quasicoherent sheaves we take direct limits of these, or equivalently we say that they are annihilated by all elements of sufficiently high degree in the ring.

Hence if we want to compute direct products of quasicoherent sheaves, we might do this using graded modules. But the category of torsion modules is not closed under torsion modules: you can take a direct product of torsion modules where you keep on increasing the degree of the elements that annihilate it. This graded module is no longer torsion...

You can make this explicit for $k[x,y]$, as is done in the Krause paper, by looking at the epimorphism coming from the evaluation morphism $$ \mathcal{O}_{\mathbb{P}^1}(-n)\otimes\mathrm{Hom}(_{\mathbb{P}^1}(\mathcal{O}_{\mathbb{P}^1}(-n),\mathcal{O}_{\mathbb{P}^1})\to\mathcal{O}_{\mathbb{P}^1}\to 0. $$

In the language of graded modules we are tensoring the twist of $k[x,y]$ by $n$ with the degree $n$ part of this graded ring (if you ever wondered what is on my Stacks project t-shirt, it is precisely this result we are using now).

If you take the product over all $n\geq 0$ of these morphisms of graded modules, you observe that the cokernel is no longer torsion, by the reasoning explained above. Hence the direct product of the associated quasicoherent sheaves is no longer an epimorphism!

Now we know that direct products are not exact in one of the easiest non-affine schemes. Another easy non-affine scheme is the affine plane minus the origin. As suggested by Leonid in his MathOverflow answer, you can have a fun game during your festivities, in which you try to prove that direct products are not exact for this scheme!