Multigraded Betti numbers without computing minimal free resolutions
Let \(\mathbf k\) be a field of characteristic 0 and \(R=\mathbf k[x_1,\dots,x_n]\) a polynomial ring over \(\mathbf k\). It is well known that for two ideals \(I_1\) and \(I_2\) of \(R\), there is an exact sequence \[ 0\to I_1\cap I_2\to I_1\oplus I_2\to I_1+I_2\to 0. \] In particular, if \(J\) is a monomial ideal minimally generated by monomials \(m_1,\dots,m_s\), then there is an exact sequence \[ 0\to\widetilde J\to J'\oplus\langle m_s\rangle\to J\to 0, \] where \(\widetilde J=\langle\text{lcm}(m_1,m_s),\dots,\text{lcm}(m_{s-1},m_s)\rangle\) and \(J'=\langle m_1,\dots,m_{s-1}\rangle\). From this exact sequence, one can construct the long exact sequence of Koszul homologies with respect to \(x_1,\dots,x_n\), which the author calls the Mayer-Vietoris sequence. For a given monomial ideal \(I\) with minimal monomial generating system \(m_1,\dots,m_r\) with specified order, the author defines the Mayer-Vietoris tree \(MVT(I)\) as follows. The root of \(MVT(I)\) is \(I\). If \(J\) is a node of \(MVT(I)\) with generating system \(n_1,\dots,n_s\), then the children \(\widetilde J\) on the left and \(J'\) on the right. One assigns position indices to every node in the following way. \(I\) has position 1 and if \(J\) has position \(p\), then \(\widetilde J\) has position \(2p\) and \(J'\) has position \(2p+1\). The nodes with even position and the root are called relevant nodes. If a relevant node is in position \(p\), the number of zeros in the binary expansion of \(p\) is called the dimension of this node. Let \(\mu\) be an element of \(\mathbb N^n\) and \(\beta_{i,\mu}(I)\) the multigraded Betti number of \(I\). Set \(\overline\beta_{i,\mu}(I)=1\) if \(x^\mu\) appears as a generator of a relevant node exactly once and \(i\) is the dimension of the relevant node and \(\overline\beta_{i,\mu}(I)=0\) otherwise. Let \(\widehat\beta_{i,\mu}(I)\) be the number of times \(x^\mu\) appears as the generator of some relevant node of dimension \(i\). Then \[ \overline\beta_{i,\mu}(I)\leq\beta_{i,\mu}(I)\leq\widehat\beta_{i,\mu}(I). \] The author defines that \(MVT(I)\) is of type B1 if the equality holds for the left inequality for any \(i\in\mathbb N\) and \(\mu\in\mathbb N^n\) and is of type B2 if it does for the right. \(MVT(I)\) is of type A if it is both of type B1 and B2. The author shows that the Mayer-Vietoris tree of a monomial ideal which is a prime power is of type B2, of a stable monomial ideal is of type B2, of a monomial ideal \(I\) whose Scarf complex gives a minimal free resolution of \(R/I\) is of type B1, of a Valla ideal is of type B2 and of a Ferrers ideal is of type A. It follows that the Mayer-Vietoris tree is a powerful tool to compute Betti numbers of these ideals.
scientific article; zbMATH DE number 5659688