Polishchuk’s reduction of an algebro-geometric problem to commutative algebra

In the writing “The geometry of doughnuts and linear object on them” [1], we have seen Sir Micheal Atiyah’s classification of vector bundles on elliptic curves. For simplicity, we only described indecomposable vector bundles of degree 0 on an elliptic curve. In this writing, we would like to present Alexander Polishchuk’s conceptual approach to understand indecomposable vector bundles of degree 0 via a quite advanced technique, called Fourier-Mukai Transform. Postponing the discussion of Fourier-Mukai transform to another writing, we rather focus on Polishchuk’s conclusion and its interpretation.

(1) m.mathnet.ru/php/presentation.phtml?presentid=3714&option_lang=eng

Alexander Polishchuk is a working mathematician in the University of Oregan [2]. In his book “Abelian varieties, theta functions and the Fourier transform” [3], he discusses the equivalence of the category of semistable vector bundles on an elliptic curve to the category of coherent torsion sheaves on the same elliptic curve among many other things. I can hear you already complaining but do not be intimidated.. I will not overwhelm you with the technical details in that sentence, I will rather just illustrate the approach and lightly touch some of the objects.

What does it mean for a vector bundle on an elliptic curve to be semitable? Well, if you have been given the definition, it would not take much effort to see that they are simply direct sums of indecomposable vector bundles of degree 0 the star objects of our earlier writing [1].
What is a category? What is an equivalence of the ? Well, you can simply regard a category of objects as a structure consisting of a collection of objects of the same kind that encodes the relation between these objects. Categories are the rock stars of modern mathematics. Equivalence of categories is simply the situation of two categories having the same structure.

The suggested equivalence above means that if we want to understand the building blocks, namely the indecomposable vector bundles of degree zero, of the first category, then we should simply look at the latter category: coherent torsion sheaves on an elliptic curve. Vector bundles are algebro-geometric objects. Well, even if I have not described coherent torsion sheaves on an elliptic curve, you might say that they also sound very algebro-geometric. Where is commutative algebra hiding in this picture?

First of all, being a indecomposable object in a category is a property preserved under an equivalence. Then, the question should be what the indecomposable coherent torsion sheaves are? Here comes commutative algebra. They can be realized as modules of very nice rings where both rings and modules are the primary objects of commutative algebra. A well known theorem in commutative algebra the so-called the structure theorem of PIDs should then imply that indecomposable coherent torsion sheaves are parametrized by a pair: a positive integer and a point on the elliptic curve under consideration.

Well then, an elliptic curve has a distinguish point, namely the identity denoted simply by the letter e. What is the corresponding bundle to the pair: the natural number r and the point e? (Put drum sounds here) It is nothing other than the rank r Atiyah bundle that is denoted by F_r in our earlier writing. This distinguished algebro-geometric object F_r and more importantly its role as an object in an appropriate category is predicted by a theorem in commutative algebra, a theorem which most graduate students must have used in an algebra qualifying exam. This may seem like a very long road to get the conclusion given also that we have not touched on the Fourier-Mukai transform but the road we have taken is more conceptual hence easier to be digested by mathematicians. It seems appropriate to conclude with the following Atiyah’s quote:
“Algebra is the offer made by the devil to the mathematician. The devil says: “I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.”” [4]

References: [1] Irimagzi, Canberk. (26 January 2019) mozartcultures.com/10328-2/ Access date: 26 January 2019
[2] https://pages.uoregon.edu/apolish/ Access date:25 January 2019
[3] https://books.google.com/books?hl=en&lr=&id=G40LwgFdrXQC&oi=fnd&pg=PP1&dq=polishchuk+abelian+varieties&ots=e0zzmFJ0oJ&sig=fYGU9enkXmMMknXdlUfrfec-yGY#v=onepage&q=polishchuk%20abelian%20varieties&f=false Access date:25 January 2019
[4] https://en.m.wikiquote.org/wiki/Michael_Atiyah Access date:25 January 2019

Photos: (1) m.mathnet.ru/php/presentation.phtml?presentid=3714&option_lang=eng

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