If R=S[Γ], use Problem 4 to prove that gl dim R ≥gldim S |...

Name: Problem 9, Chapter 8: A Course in Ring Theory
Uploaded: 2021-11-21T18:56:06-08:00
Duration: 1 min 25 s
Channel: Chai Santi
Description: If R=S[Γ], use Problem 4 to prove that gl dim R ≥gldim S

Key Concepts

Global Dimension

Global dimension is a homological invariant of a ring defined as the supremum of the projective dimensions of all its modules. It essentially measures the complexity of the module category over a ring by indicating the maximum length of projective resolutions needed to resolve any module.

Ring Extensions

Ring extensions, such as forming R = S[?], involve constructing a new ring by adjoining additional elements or structure to a base ring S. Investigating how homological properties, like global dimension, change under such extensions is critical, because extensions can potentially increase the complexity of module behavior over the new ring.

Projective Resolutions

A projective resolution of a module is an exact sequence of projective modules that provides a way to 'build' the module. The length of the shortest projective resolution (its projective dimension) is a key ingredient in determining the global dimension of the ring. In the context of ring extensions, one often compares the resolution lengths of modules when viewed over the base and extended rings.

Modules over Extended Rings

Considering modules over an extended ring, such as R = S[?], often involves viewing a module originally defined over S as an R-module. This comparison helps in obtaining lower bounds on the global dimension of R, since a module that requires a certain projective resolution length over S cannot have a shorter resolution when the module structure is extended to R.

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