Extending the idea of compressed algebra to arbitrary socle-vectors
✍ Scribed by Fabrizio Zanello
- Publisher
- Elsevier Science
- Year
- 2003
- Tongue
- English
- Weight
- 224 KB
- Volume
- 270
- Category
- Article
- ISSN
- 0021-8693
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✦ Synopsis
In this paper we will study artinian quotients A = R/I of the polynomial ring R = k[x 1 , . . . , x r ], where k is a field of characteristic zero, the x i 's all have degree 1 and I is a homogeneous ideal of R. These rings are often referred to as standard graded artinian algebras. Before explaining the main results of this work, we establish some of the notation we will use: the h-vector of A is h(A) = h = (h 0 , . . . , h e ), where h i = dim k A i and e is the last index such that dim k A e > 0.
Since we may suppose that I does not contain non-zero forms of degree 1, r = h 1 is defined to be the embedding dimension (emb.dim., in brief) of A.
The socle of A is the annihilator of the maximal homogeneous ideal m = (x 1 , . . . , x r ) ⊂ A, namely soc(A) = {a ∈ A | am = 0}. Since soc(A) is a homogeneous ideal, we define the socle-vector of A as s(A) = s = (s 0 , . . . , s e ), where s i = dim k soc(A) i . Note that s e = h e > 0.
We will say that an h-vector h is admissible for the pair (r, s) if there exists an algebra A with emb.dim.(A) = r, s(A) = s and h(A) = h. When the pair (r, s) is clear from the context, we will simply say that h is admissible.
A natural question which arises is the following: what are the admissible h-vectors for a given pair (r, s)?
This problem has been considered in several different guises: e.g., there are several papers which treat the question of determining the possible h-vectors of Gorenstein algebras, i.e., finding all the admissible h's which correspond to a fixed r and s =