DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Laurent-series target and pole projection for subtraction

Definition

Let $K$ be a commutative unital $Q$ -algebra and let $ϵ$ be a formal indeterminate. The Laurent-series algebra $A = K ((ϵ))$ consists of series $\sum_{n \geq N} a_{n} ϵ^{n}$ for some integer $N$ , with coefficients in $K$ . Put $A_{-} = ϵ^{- 1} K [ϵ^{- 1}]$ (finite sums of strictly negative powers) and $A_{+} = {\sum_{n \geq 0} a_{n} ϵ^{n} : a_{n} \in K}$ (formal series of nonnegative powers). Then $A = A_{-} \oplus A_{+}$ as a $K$ -module, and both summands are subalgebras, with $1 \in A_{+}$ . Define $R : A \to A_{-}$ by retaining exactly the strictly negative powers; $1 - R$ is the projection onto $A_{+}$ . The parameter $ϵ$ is formal and carries no physical or empirical interpretation in this definition.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Dominique Manchon, Hopf algebras, from basics to applications to renormalization, full lecture notes, arXiv:math/0408405v2