Reverse search for monomial ideals
Let \(I\) be a monomial ideal minimally generated by \(x^{\mathbf a_1},\dots, x^{\mathbf a_r}\) in a polynomial ring \(S=k[x_1,\dots, x_n]\) over a field \(k\). For an element \(\mathbf b\) in \(\mathbb N^n\), set \[ K_\mathbf b=\{F\subset\{1,2,\dots,n\}\mid x^{\mathbf b-F}\in I\}, \] where we identify a subset \(F\) of \(\{1,2,\dots,n\}\) with the 0-1 vector whose support is \(F\). Then \(K_\mathbf b\) is a simplicial complex (or the null complex, i.e., \(\emptyset\not\in K_\mathbf b\), if \(x^\mathbf b\not\in I\)). It is known that the multigraded Betti numbers can be expressed as \[ \beta_{i,\mathbf b}(I)=\beta_{i+1,\mathbf b}(S/I)=\dim_k \tilde H_{i-1}(K_\mathbf b;k), \] where \(\tilde H_{i}(K_\mathbf b;k)\) is the reduced homology [see \textit{D. Bayer} and \textit{B. Sturmfels}, ``Cellular resolutions of monomial modules'', J. Reine Angew. Math. 502, 123--140 (1998; Zbl 0909.13011) or \textit{E. Miller} and \textit{B. Sturmfels}, Combinatorial commutative algebra. New York, NY: Springer (2005; Zbl 1066.13001 and Zbl 1090.13001)]. Since the reduced homologies of a simplicial complex whose geometric realization is contractible are all zero, in order to compute Betti numbers of \(I\), it is enough to consider \(\mathbf b\in\mathbb N^n\) such that \(K_\mathbf b\) is a non-cone, i.e., there is no vertex \(i\) such that \(K_\mathbf b=\text{star}_{K_\mathbf b}(\{i\})\). The authors call \(\mathbf b\in\mathbb N^n\) with \(K_\mathbf b\) is not a cone a corner of \(I\). Let us denote by \(\mathbf a\vee\mathbf a'\) the componentwise maximum of \(\mathbf a\) and \(\mathbf a'\) for \(\mathbf a, \mathbf a'\in\mathbb N^n\). One way to find all corners of \(I\) is first consider \(\mathbf a_1\vee\mathbf a_2\vee\cdots\vee\mathbf a_r\), the exponent vector of the least common multiple of the generators of \(I\), and use the full \(\text{lcm}\) lattice of the generators of \(I\). In this paper, the authors propose the following algorithm to find all the corners of \(I\). First consider \(\mathbf a_i\) \(i=1,2,\dots, r\), which is a corner since \(K_{\mathbf a_i}=\{\emptyset\}\). (Note that since \(\{\emptyset\}\neq\emptyset\), \(K_{\mathbf a_i}\) is not a null complex.) And given a corner \(\mathbf b\) of \(I\), search corners of the form \(\mathbf b\vee\mathbf a_i\) for some \(i\). By repeating this process, one gets all the corners of \(I\). This algorithm has advantage in terms of space complexity. Reviewer's remark: \(I=\langle x_1^3, x_2^3, x_4^3, x_2x_3x_4, x_1x_3x_4, x_3^2, x_1^2x_2^2\rangle\) and \(\mathbf b=(2,2,2,1)\) seems to be a counter example of Theorem 12 and Lemma 11.
scientific article; zbMATH DE number 5603617