Tor Groups of the Stanley-Reisner Ring of a Matroid

Abstract: We introduce the Tor groups $ \mathop{Tor}{\bullet}^{S{\mathrm{M}}^{\circ}} \left( \mathbb{C}[\Sigma_{\mathrm{M}}], \mathbb{C} \right){\bullet} $ for a loopless matroid $\mathrm{M}$ as a way to study the extra relations occurring in the linear ideal of the Feichtner-Yuzvinsky presentation of the Chow ring $ A^{\bullet}(\mathrm{M}) $. This extends the definition of the Chow ring of a matroid since $ \mathop{Tor}{0}^{S_{\mathrm{M}}^{\circ}} \left( \mathbb{C}[\Sigma_{\mathrm{M}}], \mathbb{C} \right)_{\bullet} \cong A^{\bullet}(\mathrm{M}) $. Our main tool in studying these groups is to recognize them as cohomology of the toric variety associated to the Bergman fan of the matroid. With this geometric approach, we show that these Tor groups fit into a long exact sequence arising from the matroidal flips of Adiprasito, Huh, and Katz, extending the short exact sequence in the case of Chow rings. Using this long exact sequence we give a recursive formula for the Hilbert series of the Tor algebra of a uniform matroid.