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## Mathematical Proceedings of the Cambridge Philosophical Society

- Mathematical Proceedings of the Cambridge Philosophical Society / Volume 124 / Issue 02 / September 1998, pp 215-229
- null null
- DOI: http://dx.doi.org/ (About DOI), Published online: 08 September 2000

## On the module of effective relations of a standard algebra
## AbstractLet A-module U_{1}. Let
V=A[] be the
polynomial ring with as many variables t (of
degree one) as t has elements and let
xf[ratio]V[rightward arrow]U be the graded free presentation
of U induced by the . For xn[gt-or-equal, slanted]2, we will
call the
module
of
effective
n-relations the A-module
E(U)_{n}=
ker f_{n}/V_{1}·
ker f_{n}. The minimum positive integer r[gt-or-equal, slanted]1
such
that the effective n-relations are zero for all n[gt-or-equal, slanted]r+1
is known to be an invariant of U. It is called the relation type
of U
and is denoted by rt(U). For an ideal I of A,
we define E(I)_{n}=
E([script R](I))_{n} and
rt(I)=rt([script
R](I)), where [script R](I)=
[oplus B: plus sign in circle]_{n[gt-or-equal, slanted]0}
I^{n}t^{n}[subset or is implied by]A
[t] is the Rees algebra of I.In this paper we give two descriptions of the E(U)_{n}=
H_{1}([], xU)_{n}
(see 2·4). Using these characterizations,
we show later some properties on the module of effective n-relations and
the relation
type of a graded algebra. Our approach has connections with several earlier
works
on the subject (see [2, 5–7, 9,
10, 13, 14]).(Revised December 17 1996) |