Directsum

Directsum is a construction in linear algebra and module theory that describes how a larger object can be assembled from subobjects with minimal overlap. It is used to express when a space or module can be decomposed into a direct sum of subspaces or submodules.

Internal direct sum: Let V be a vector space (or module) and {U_i}_{i∈I} a family of subspaces (or submodules). If every v ∈ V can be written uniquely as a finite sum v = Σ_{i∈F} u_i with u_i ∈ U_i and F a finite subset of I, then V is the direct sum of the U_i, denoted V = ⊕_{i∈I} U_i. Equivalently, V = Σ_i U_i and the intersections are trivial in a suitable sense, such as U_i ∩ (Σ_{j≠i} U_j) = {0} for all i.

Finite case and dimension: If I is finite, then dim V = Σ_i dim U_i. The decomposition is often called an internal direct sum when the subobjects lie inside V.

External direct sum: Given a family {M_i} of modules, the external direct sum ⊕ M_i is a submodule of the product ∏ M_i consisting of all tuples with only finitely many nonzero components. For a finite index set, the external direct sum coincides with the external direct product, i.e., ⊕ M_i ≅ ∏ M_i.

Direct sum vs direct product: The direct product ∏ M_i allows arbitrary families (no restriction to finite support), whereas the direct sum restricts to elements with finite nonzero coordinates. For finite index sets, these notions agree.

Universal property: The direct sum has a universal property: given maps f_i: A → M_i, there exists a unique map f: A → ⊕ M_i with the projection to each M_i equal to f_i. This makes the direct sum a canonical way to assemble independent morphisms.