Weak dimension¶

The least upper bound of the flat dimensions of all modules over a ring, measuring how far the ring is from making every module flat.

Version: v1 · 2026-09-08 · History
Domain-specific #: 7456
Origin domain: homological algebra
Subdomain: homological algebra

Core Idea¶

The weak or weak global dimension of a ring is the supremum of flat dimensions of its left modules, equivalently under standard conventions the largest degree in which an appropriate Tor group can be nonzero. Flat resolutions replace a module by successively simpler flat modules; the earliest uniform length that suffices for every module determines the ring-wide dimension. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application¶

Weak dimension belongs to homological algebra and is useful where the analyst can specify the typed homological algebra carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the ring side and module category are fixed and the stated supremum of flat dimensions or equivalent Tor-vanishing threshold is used consistently. The scope is broad within that domain but bounded by the need for the ring side and module category are fixed and the stated supremum of flat dimensions or equivalent Tor-vanishing threshold is used consistently. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the ring side and module category are fixed and the stated supremum of flat dimensions or equivalent Tor-vanishing threshold is used consistently the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Weak dimension can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Weak dimension. Weak dimension compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed homological algebra carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the ring side and module category are fixed and the stated supremum of flat dimensions or equivalent Tor-vanishing threshold is used consistently independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of homological algebra because they reuse the typed homological algebra carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Flat resolutions replace a module by successively simpler flat modules; the earliest uniform length that suffices for every module determines the ring-wide dimension., and type the carrier, state every parameter and convention in the definition, test that the ring side and module category are fixed and the stated supremum of flat dimensions or equivalent Tor-vanishing threshold is used consistently, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Weak dimension Domain-specific

Parents (1) — more general patterns this builds on

Weak dimension is a kind of Dimension Prime

The proposed strict upward parent is prime:dimension.

Hierarchy path (1) — routes to 1 parentless root

Weak dimension → Dimension

Neighborhood in Abstraction Space¶

Weak dimension sits in a crowded region of the domain-specific corpus (6^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Homological Ring & Scheme Invariants (13 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08