FTP, name these sets of zeroes of non-empty sets of polynomial equations, the basic object of study in algebraic geometry.
In order to have finite limits on these objects from the categorical point of view, nilpotents must be allowed in the sheaf of rings.  Grothendieck generalized them to obtain his theory of schemes, but lost some of their content in the process.  These mathematical objects are reduced schemes of finite type over a field and kinds of these objects include the Abelian, Albanese, Chow, and the Brauer-Severi.  FTP, name these sets of zeroes of non-empty sets of polynomial equations, the basic object of study in algebraic geometry.
In order to have finite limits on these objects from the categorical point of view, nilpotents must be allowed in the sheaf of rings.  
algebraic varieties