Direkter Summand/Polynomring/Standardgraduiert/Differentiell ausdehnbar/Signatur/Fakt/en

Let ${}R\subseteq S=K[Y_{1},\ldots ,Y_{r}]$ be an ${}\mathbb {N}$-graded ${}K$-subalgebra of the standard-graded polynomial ring. Suppose that this inclusion is a direct summand which is also differentially extensible.

Then

${}R_{\geq n}={\mathfrak {m}}^{\langle n\rangle }\,$

and

${}R_{n}={\mathfrak {m}}^{\langle n\rangle }/{\mathfrak {m}}^{\langle n+1\rangle }\,,$

where ${}{\mathfrak {m}}=R_{+}$. The differential signature of ${}R$ equals its degree (as a graded ring), it is a rational number and the defining sequence converges.