Section
Topological algebra
Linearly topologized rings, adic rings, and completed tensor products.
Core idea
This section collects definitions for rings and tensor products that carry a compatible topology.
Let be a commutative topological ring. An open ideal is an ideal of definition if
is a neighborhood basis of . Equivalently, the given topology on is the -adic topology.
In the classical adic convention used for formal schemes, an ideal of definition is required to be finitely generated. A topological ring that has such an ideal is an adic ring.
An adic ring is a commutative topological ring whose topology is the -adic topology for some finitely generated ideal of definition . Thus the powers form a neighborhood basis of .
The ring is separated if
and it is complete if every compatible family of residue classes modulo the is represented by an element of . Equivalently, is complete and separated exactly when the canonical map
is an isomorphism. A complete adic ring means an adic ring that is complete and separated.
Let be a commutative ring, let have a decreasing filtration , and let have a decreasing filtration . Their completed tensor product for these filtrations is
It receives the ordinary tensor product through the compatible quotient maps. The symbol is therefore meaningful only after the filtrations or topologies being completed have been specified.