Relative multiplicities of graded algebras

Abstract

Without any finiteness assumption we define a sequence of relative multiplicities for a pair A ⊂ B of standard graded Noetherian algebras that extends the notion of relative multiplicities of Simis, Ulrich and Vasconcelos and unifies them with the j-multiplicity of ideals introduced by Achilles and Manaresi as well as the jmultiplicity of modules defined by Ulrich and Validashti. Using our relative multiplicities, we give numerical criteria for integrality and birationality of the extension A ⊂ B.

Blowups and Fibers of Morphisms

“…Results of this type were obtained before by Simis et al. [49], Validashti [52], Xie [53] and by Jeffries et al. [32].…”

Formulas for the multiplicity of graded algebras

“…Since B 1 B and m have the same radical, 0 : G B 1 G = 0 : G mG . Therefore j d (A 1 B) = e(0 : G mG ) = e(G) − e ∞ (A, B) = re(A) (see also [21]).…”

“…Thus when dim B/A 1 B > 1, some terms are added to the right hand side of Equation ( 1) to make it closer to e(B). But Validashti [21] also gave an example to show that Inequality (2) can be strict. In Theorem 4.1, we give the extra terms on the right hand side of Inequality (2) required to yield an equality for arbitrary dimensions of B/A 1 B.…”

“…It is well-known that if R has infinite residue field, every ideal has reduction sequences or, if the ideal has maximal analytic spread, super-reductions. For concepts and results about analytic spread, (j-)multiplicities and (super-) reduction sequences, see [1], [2], [3], [5], [15], [16] and [21]. Now we will define the j-multiplicity for ideals that have possibly non-maximal analytic spread.…”

Numerical criteria for integral dependence

“…The sum on the right-hand side gives rise to the Hilbert polynomial of m (B 1 G(M)), where G(M) denotes the associated graded module of M with respect to the B-ideal A 1 B, endowed with the 'internal grading', see [9, 2•3 and 3•1]. This polynomial has degree at most dim M − 1 and its normalized coefficient in degree dim M − 1 is called the relative j-multiplicity j (A|B, M) of A and B with coefficients in M, see [10]. Thus the sequence used in the definition of the ε-multiplicity is bounded and the limit superior is finite,…”