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The polynomial ring as finitely supported coefficient families on monomials
Definition
Let be a commutative ring (Commutative ring) and let be a set. The set consists of the functions
with finite support, where is the monoid of Monomials on an index set as finitely supported exponent families. We write such a function as the formal sum . Addition is pointwise. Multiplication is the convolution
where only pairs in contribute and the sum is the finite sum of A finite sum in a commutative monoid indexed by an arbitrary finite set. The constant is the coefficient family supported at the zero monomial with value , and the indeterminate is supported at the exponent family that is at and elsewhere.
The convolution is well defined and these operations make the displayed set a commutative ring by Finite convolution makes a commutative ring containing ↗.
Depends on
Used by
- Quasi-separatedness in pushforward cannot be omitted Counterexample
- Algebraic independence in a field extension Definition
- Standard étale algebra Definition
- Standard smooth presentations and locally standard smooth maps Definition
- All twists on the projective line Example
- Fitting ideals of a diagonal two-by-two presentation Example
- Hilbert polynomial of projective space Example
- The diagonal bimodule k[x] is projective on both sides but not over its enveloping algebra Example
- The square-root standard etale chart Example
- Artin's ideal generated by f(x_f) for all monic nonconstant f∈ F[x] is proper Lemma
- Base change of standard smooth presentations Lemma
- Coprime polynomial factorisations lift after an etale localisation Lemma
- Differentials of a polynomial quotient and the Jacobian cokernel Lemma
- Field extension preserves the graded pieces and the total length of a zero-dimensional projective quotient Lemma
- High-degree section module is finite graded Lemma
- Hypersurface cohomology sequence Lemma
- Standard smooth algebras are finitely presented and flat Lemma
- 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
15 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)