Algebra/Standard-grduiert/Multiplizität/Differentielle Signatur/Abschätzung/Fakt/en

Let ${}R$ denote a standard-graded ring over ${}K$ of dimension ${}d$ and of multiplicity ${}e$. Suppose that there exists ${}\alpha \in \mathbb {R} _{\geq 0}$ such that

${}\operatorname {Hom} _{R}(\operatorname {Sym} ^{n}(\Omega _{R{|}K}),R)_{\ell }=0\,$

for all ${}\ell <-\alpha n$.

Then the differential signature of ${}R$ is bounded from above by ${}e\alpha ^{d}$.