Let A be a commutative ring. A graded A-algebra
U = [oplus ]n[ges ]0 is a standard A-algebra
if U0 = A and U = A[U1]
is generated as an A-algebra by the elements of U1. A graded
U-module F = [oplus ]n[ges ]0Fn
is a standard U-module if F is generated as a U-module
by the elements of F0, that is,
Fn = UnF0 for all n [ges ] 0. In
particular, Fn = U1Fn−1 for all
n [ges ] 1. Given I, J, two ideals of A, we consider the following standard
algebras: the Rees algebra of I, [Rscr ](I) =
[oplus ]n[ges ]0Intn =
A[It] ⊂ A[t], and the multi-Rees algebra of I and
J, [Rscr ](I, J) =
[oplus ]n[ges ]0([oplus ]p+q=nIpJqupvq) =
A[Iu, Jv] ⊂ A[u, v].
Consider the associated graded ring of I, [Gscr ](I) =
[Rscr ](I) [otimes ] A/I =
[oplus ]n[ges ]0In/In+1, and the
multi-associated graded ring of I and J, [Gscr ](I, J) =
[Rscr ](I, J) [otimes ] A/(I+J) =
[oplus ]n[ges ]0([oplus ]p+q=
nIpJq/(I+J)IpJq).
We can always consider the tensor product of two standard A-algebras U =
[oplus ]p[ges ]0Up and V =
[oplus ]q[ges ]0Vq as a standard A-algebra
with the natural grading U [otimes ] V =
[oplus ]n[ges ]0([oplus ]p+q=nUp [otimes ]
Vq). If M is an A-module, we have the standard modules: the Rees module
of I with respect to M, [Rscr ](I; M) =
[oplus ]n[ges ]0InMtn =
M[It] ⊂ M[t] (a standard [Rscr ](I)-module), and the
multi-Rees module of I and J with respect to M, [Rscr ](I, J; M) =
[oplus ]n[ges ]0([oplus ]p+q=nIpJqMupvq) =
M[Iu, Jv] ⊂ M[u, v] (a standard
[Rscr ](I, J)-module). Consider the associated graded module of M with respect to I,
[Gscr ](I; M) = [Rscr ](I; M) [otimes ] A/I =
[oplus ]n[ges ]0InM/In+1M
(a standard [Gscr ](I)-module), and the multi-associated graded module of M with respect to I and J,
[Gscr ](I, J; M) = [Rscr ](I, J; M) [otimes ]
A/(I+J) =
[oplus ]n[ges ]0([oplus ]p+q=
nIpJqM/
(I+J)IpJqM)
(a standard [Gscr ](I, J)-module). If U, V are two standard A-algebras,
F is a standard U-module and G is a standard V-module, then F [otimes ]
G = [oplus ]n[ges ]0([oplus ]p+q=
nFp [otimes ] Gq) is a standard
U [otimes ] V-module.
Denote by π[ratio ][Rscr ](I) [otimes ] [Rscr ](J; M) → [Rscr ](I, J;
M) and σ[ratio ][Rscr ](I, J; M) → [Rscr ](I+J; M)
the natural surjective graded morphisms of standard [Rscr ](I) [otimes ] [Rscr ](J)-modules. Let
ϕ[ratio ][Rscr ](I) [otimes ] [Rscr ](J; M) → [Rscr ](I+J; M) be
σ∘π. Denote by &πmacr;[ratio ][Gscr ](I) [otimes ] [Gscr ](J; M) →
[Gscr ](I, J; M) and &σmacr;[ratio ][Gscr ](I, J; M) →
[Gscr ](I+J; M) the tensor product of π and σ by A/(I+J);
these are two natural surjective graded morphisms of standard [Gscr ](I) [otimes ] [Gscr ](J)-modules. Let
&ϕmacr;[ratio ][Gscr ](I) [otimes ] [Gscr ](J; M) → [Gscr ](I+J;
M) be &σmacr;∘&πmacr;. The first purpose of this paper is to prove the following theorem.
