Ghost Sheaf At The Nodes Of A Stable Log Curve

Let \(C\) be a smooth log curve over \(S = \Spec(Q\to k)\) a log point with \(Q\) a fine saturated monoid, and let \(q \in C\) be a double point of \(C\) where two irreducible components \(C_1\) and \(C_2\) meet, and let \(\eta_1\) and \(\eta_2\) be the generic points of \(C_1\) and \(C_2\) respectively. This discussion will also work for a self node once you work in a small enough etale neighborhood of the node.

According to Kato, the stalk of the ghost sheaf at \(q\) is \[\overline{\mathcal M}_{C, q} \cong Q\oplus_{\mathbb N}\mathbb N^2.\] This is the pushout of the diagram

and since every monoid here is integral, \(Q\oplus_{\mathbb N}\mathbb N^2\) can be viewed as the pairs \((m, (a,b)) \) in \(Q\oplus \mathbb N^2\) with \((m_1,(a_1,b_1))\) glued to \((m_2, (a_2,b_2))\) whenever \(a_2 - a_1 = b_2 - b_1\) and \(m_1 - m_2 = (a_2 - a_1)\rho_q\). In other words, we can write every element of \(Q\oplus_\mathbb N\mathbb N^2\) as either \((m, (a,0))\) or \((m, (0, b))\) for some \(a,b\in \mathbb N\).

Alternatively, we can view it as a submonoid of \(Q\oplus Q\) in the following way. We have generization maps \(\chi_i:\overline{\mathcal M}_{C, q}\to \overline{\mathcal M}_{C, \eta_i}\)for \(i=1,2\) which, more explicitly, are maps \(\chi_i:Q\oplus_\mathbb N \mathbb N^2 \to Q\) given by \[ ~\chi_1(m, (a,b)) = m + a\rho_q\] and \[\chi_2(m, (a,b)) = m + b\rho_q.\] Summing these gives us a morphism \(\iota:Q\oplus_\mathbb N\mathbb N^2 \to Q\) defined \[\iota(m, (a,b)) = (m + a\rho_q, m+b\rho_q).\] If \(\iota(m, (a,b)) = 0\) then \(m + a\rho_q = m + b\rho_q = 0 \implies a = b\) since \(\rho_q \neq 0\). This is enough to show that \(\iota\) is injective since all monoids are integral, so \(Q\oplus_\mathbb N\mathbb N^2\) is isomorphic to the submonoid of \(Q\oplus Q\) given by \[\{(m_1,m_2) \in Q\oplus Q ~ \mid~ m_1 - m_2 = n\cdot \rho_q \text{ for some } n\in \mathbb Z\}.\] In fact, if \(m_1 - m_2 = n\cdot \rho_q\),