Mathematics – Commutative Algebra
Scientific paper
2011-01-12
Mathematics
Commutative Algebra
To appear in Transactions of American Mathematical Society
Scientific paper
Let $R$ be a standard graded Noetherian algebra over an Artinian local ring. Motivated by the work of Achilles and Manaresi in intersection theory, we first express the multiplicity of $R$ by means of local $j$-multiplicities of various hyperplane sections. When applied to a homogeneous inclusion $A\subseteq B$ of standard graded Noetherian algebras over an Artinian local ring, this formula yields the multiplicity of $A$ in terms of that of $B$ and of local $j$-multiplicities of hyperplane sections along ${\rm Proj}\,(B)$. Our formulas can be used to find the multiplicity of special fiber rings and to obtain the degree of dual varieties for any hypersurface. In particular, it gives a generalization of Teissier's Pl\"{u}cker formula to hypersurfaces with non-isolated singularities. Our work generalizes results by Simis, Ulrich and Vasconcelos on homogeneous embeddings of graded algebras.
No associations
LandOfFree
Formulas for the multiplicity of graded algebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Formulas for the multiplicity of graded algebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Formulas for the multiplicity of graded algebras will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-264471