2009/10/21 by Beecher, Amanda
#05E40 #13-02 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.0910.4112
Let R=\Bbbk [x1,..., xm] be a polynomial ring in m variables over \Bbbk with the standard ℤm grading and L a multigraded Noetherian R-module. When \Bbbk is a field, Tchernev has an explicit construction of a multigraded free resolution called the T-resolution of L over R. Despite the explicit canonical description, this method uses linear algebraic methods, which makes the structure hard to understand. This paper gives a combinatorial description for the free modules, making the T-resolution clearer. In doing so, we must introduce an ordering on the elements. This ordering identifies a canonical generating set for the free modules. This combinatorial construction additionally allows us to define the free modules over ℤ instead of a field. Moreover, this construction gives a combinatorial description for one component of the differential. An example is computed in the first section to illustrate this new approach.