Gabriel's reconstruction theorem and birational geometry

A famous theorem of Gabriel states that two varieties are isomorphic if and only if their categories of coherent sheaves are equivalent. On the other hand, two varieties are birational if and only if their function fields are isomorphic. In this talk I will discuss an elementary result which interpolates between these two theorems. Namely, taking the quotient of Coh(X) by sheaves supported in codimension c (or bigger) controls the isomorphism type of X in codimension c. This is joint work with Roberto Pirisi.