Free modules of a multigraded resolution from simplicial complexesFree modules of a multigraded resolution from simplicial complexes

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 Author Beecher, Amanda Source arXiv.org Content type Text File Format PDF Date of Submission 2009-10-21 Language English
 Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics Subject Keyword Mathematics - Commutative Algebra ♦ 05E40 ♦ math Abstract Let $R=\Bbbk [x_1,..., x_m]$ be a polynomial ring in $m$ variables over $\Bbbk$ with the standard $\mathbb{Z}^m$ 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 $\mathbb{Z}$ 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. Educational Use Research Learning Resource Type Article Page Count 18