A characterization of zero-dimensional annihilators of tableaus

[摘要] In the theory of linear recursive sequences over a field K there is a natural action of the polynomial ring K[x] on sequences (the end-off left shift), and every ideal of K[x] is the annihilator with respect to this action of some sequence s. This is a direct consequence of the fact that K[x] is a principal ideal domain. In dimension n greater than or equal to 2 the analogue of a sequence is a tableau, and there is an analogous action of K[x(1),x(2),...,x(n)] on tableaus. However, a given ideal of K[X] = K[x(1),x(2),...,x(n)] is not necessarily the annihilator of some tableau. We will give an example in Section 2. Our main result is to characterize an important subset of those ideals which are annihilators. More specifically, in the theory of linear recursive sequences, an ideal I of K[x] and a finite set of initial conditions (the initial fill) completely determine a sequence annihilated by I. This is a consequence of the fact that K[x]/I is a finite-dimensional K-vector space. In order for a tableau to be completely determined by an ideal I and a finite set of initial conditions, we must have that K[X]/I is a finite-dimensional K-vector space, i.e., the only primes containing I are maximal, i.e., I is a 0-dimensional ideal, In particular, if m is maximal and I is m-primary then I is 0-dimensional. Using methods of primary decomposition, we will be able to reduce to this case. Following a suggestion of B. Sturmfels, we show that a 0-dimensional ideal is the annihilator of some tableau if and only if it is Gorenstein. (C) 1997 Academic Press.

[发布日期] 1997-07-15