tensor product of algebras
Algebras and modules
Model category presentations
Geometry on formal duals of algebras
Let be a commutative ring. The category of associative algebras over is the category
of rings under . If is a commutative rig, we can do the same with
The tensor product of -algebras has as underlying -module just the tensor product of modules of the underlying modules, . On homogeneous elements the algebra structure is given by
We write also for the tensor product of algebras.
For commutative -algebras, the tensor product is the coproduct in :
hence the pushout in Comm Ring? (or Comm Rig?)
Relation to tensor product of categories of modules
For an associative algebra over a field , write Mod for its category of modules of finite dimension. Then the tensor product of algebras corresponds to the Deligne tensor product of abelian categories :
See at tensor product of abelian categories for more.
Revised on March 23, 2016 03:22:29