DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

Monomials on an index set as finitely supported exponent families

Definition

Let $I$ be a set. A monomial on $I$ is a function $a : I \to N$ (A function is a relation $f$ with $(a, b) \in f$ and $(a, c) \in f$ implying $b = c$ ; $f : A \to B$ , the value $f (a)$ , domain and codomain, The natural numbers $N$ (von Neumann)) whose support

$supp (a) : = {i \in I : a_{i} \neq 0}$

is finite in the sense of Finite, countably infinite, countable, uncountable. The set of all such exponent families is denoted $M (I)$ . The zero monomial is the constant-zero function, and the sum $a + b$ is defined pointwise using natural-number addition (Addition of natural numbers). It again has finite support because $supp (a + b) \subseteq supp (a) \cup supp (b)$ .

We write $x^{a}$ for the formal monomial indexed by $a$ . If $I = \emptyset$ , there is exactly one function $I \to N$ , so $M (I)$ consists only of the zero monomial.

Depends on

Used by

Quasi-separatedness in pushforward cannot be omitted Counterexample
The polynomial ring R[xᵢ:i∈ I] as finitely supported coefficient families on monomials Definition
Cohomology of O(d) on projective space Theorem
Finite convolution makes R[xᵢ:i∈ I] a commutative ring containing R Theorem

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

U. Thiel, Commutative Algebra, Section 1.4 (standard reference, not scraped)