On the minimal free resolution of the Rees algebra of tri-generated bivariate monomial ideals

Abstract

Let $I$ be a monomial ideal in two variables generated by three monomials and let $\mathcal{R}(I)$ be its Rees ideal. We describe an algorithm to compute the minimal generating set of $\mathcal{R}(I)$. Based on the data obtained by this algorithm, we build a graph that encodes the minimal free resolution of $\mathcal{R}(I)$. We explicitly describe the modules and differentials on the minimal free resolution of $\mathcal{R}(I)$.

Figures (5)

• Figure 1: Step by step of the construction of $\mathrm{Graph}(15,13,9,6)$. The thick nodes correspond to lower generators.
• Figure 2: $\overline{\mathrm{Graph}}(15,13,9,6)$.
• Figure 3: The nodes $g_h$, $g_j$, $g_k$, $g_\ell$ appearing in the the construction of the syzygy $\mathbf{s}_{(j,k,\ell)}^{(2)}$ in the proof of Theorem \ref{['thm:SecondSyzygies']} are situated as illustrated here. There are two cases. Note that $g_j$, $g_k$ must belong to opposite types (upper vs. lower generators); the only difference between the two cases is whether $g_\ell$ is of the same type as $g_k$ (seen on the left) or of the same the type as $g_j$ (seen on the right). Without loss of generality we assumed both $g_h$ and $g_k$ are lower generators, there exists the other possibility of both being upper generators in which case the images would be flipped upside down.
• Figure 4: Illustration of the construction of the syzygy $\mathbf{s}_{(j,k,\ell)}^{(2)}$ in the proof of Theorem \ref{['thm:SecondSyzygies']}. Only the terms in bold and coloured in red are divisible by $X_1$, note that these terms can only come from $v(a,b)$ with $a<b$ and $(a,b)\neq (1,2)$. In the proof, it is shown that both $v(h,k)v(j,k)=\mathop{\mathrm{lt}}\nolimits(g_k)$ and $v(j,\ell)v(\ell,k)=\mathop{\mathrm{lt}}\nolimits(g_k)$; hence, $v(h,k)v(j,k)-v(j,\ell)v(\ell,k)=0$. Analogously both $v(h,k)v(k,j)=\mathop{\mathrm{lt}}\nolimits(g_j)$ and $v(k,\ell)v(\ell,j)=\mathop{\mathrm{lt}}\nolimits(g_j)$; so, $-v(h,k)v(k,j)+v(k,\ell)v(\ell,j)=0$. These two identities relate the middle rows of the matrix with columns $\mathbf{s}_{(h,k)}^{(1)},\ldots,\mathbf{s}_{(k,\ell)}^{(1)}$ to the column representing $\mathbf{s}_{(j,k,\ell)}^{(2)}$. Moreover, it is clearly to be seen that after a multiplication of these matrices, the top and bottom entries vanish. There remain only sums of products of entries that are not divisible by $X_1$ (which must reduce to zero as explained at the end of proof of Theorem \ref{['thm:SecondSyzygies']}).
• Figure :

Theorems & Definitions (45)

• Example 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Corollary 3.1
• Lemma 3.4
• proof