Mayer‐Vietoris Presentations Over Colimits of Rings

with two 'vertices' T,8, depicted 2vJL_»$. The choice of sign in (1) then amounts to an orientation of this edge. We can extend the above, as follows. Informally, a 'graph of rings' is a graph whose vertices and edges are labelled with rings, and for each edge e, and any vertex v of e, there is a homomorphism from the ring B(e) corresponding to e, to the ring B(v) corresponding to v. By generators-andrelations arguments, this diagram of ring homomorphisms has a colimit, B say. Given any orientation of the graph (that is, of each edge) we can, as above, construct a complex (2) 0