Let
R be a positively graded ring such that
R is finitely generated as an
R0-algebra and
R0 is
Artinian. Note that
R has finite
Krull dimension d. Let
M be a finitely generated
R-module and
FM(
t) its
Hilbert–Poincaré series. This series is a rational function of the form :\frac{P(t)}{(1-t)^d}, where P(t) is a polynomial. By definition, the multiplicity of
M is :\mathbf{e}(M) = P(1). The series may be rewritten :F(t) = \sum_1^d {a_{d-i} \over (1 - t)^d} + r(t). where
r(
t) is a polynomial. Note that a_{d-i} are the coefficients of the Hilbert polynomial of
M expanded in binomial coefficients. We have :\mathbf{e}(M) = a_0. As Hilbert–Poincaré series are additive on exact sequences, the multiplicity is additive on exact sequences of modules of the same dimension. The following theorem, due to Christer Lech, gives a priori bounds for multiplicity. {{math_theorem :e(I) \le d! \deg(R) \lambda(R/\overline{I}).| name = Lech }} == See also ==