Let P=k[x1...xn] be a poly over a field. Suppose that R=k[x1...xn]/I. The canonical module of R is $ω_R=Ext^{n−dim(R)}_P(R,P)$.
The question is that is there any upper bound for the min num of generators of ωR in terms of r=the min num of gen of I? and\or the embedding dim of R?
One may add more reasonable conditions to R, e.g. assume that R is Cohen-Macaulay?
and\or
Assume that I is almost complete intersection i.e. r=codim(I)+1.
A desired upper bound would be (r \choose 2) i.e. μ(ω_R)≤(r\choose 2).
Recall that min num gen ωR denoted by μ(ωR) is also called the type of the ring R denoted by r(R). And the latter is one iff R is Gorenstein , this is a result of P.Roberts.
Let's also remind that in the CM case the question is tantamount to ask an upper bound for the last betti number in the min free resolution of R over P.
Hi Hamid,
I believe there is no general bound for the type of R in terms of dim(R) or edim(R), even in the CM case. In the attached paper, the authors construct monomial curves in A4 with arbitrarily large type. I do not know whether there are bounds in terms of the number of generators of I, or in the almost complete intersection case.
Best,
Alessio
http://link.springer.com/article/10.1007%2FBF01169580?LI=true
If I understand your question correctly, I think the answer is no. For instance,
almost complete intersections I (which define C-M quotients) are linked to Gorenstein
ideals J, and the canonical module of R/I is J/(regular seq). But Gorensteins can
have arbitrarily large numbers of generators. For explicit examples, try codim 3 Pfaffians.
I agree. There are numerical semigroup rings (1-dimensional CM) with 4 generators with socle generated by an arbitrary number of elements. Generators of the socle correspond to generators of the canonical module.
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