Baker – Akhiezer modules, Krichever sheaves, and commuting rings of partial differential operators

In this work we give a review of several results about commutative subrings of partial differential operators. We show that $n$-dimensional commutative ring of partial differential operators with scalar (not matrix) coefficients (with certain mild conditions) corresponds to a Baker – Akhiezer module on the spectral algebraic variety. We also show that there is a family of coherent torsion free sheaves of special type. The existence of such sheaves gives a strong restriction on the structure of the spectral variety, in particular, it is possible to find the selfintersection index of a divisor at infinity.