$M$-TF equivalences on the real Grothendieck groups

For an abelian length category $\mathcal{A}$ with only finitely many isoclasses of simple objects, we have the wall-chamber structure and the TF equivalence on the dual real Grothendieck group $K_0(\mathcal{A}{)}_{\mathbb{R}}^{\ast }={\mathrm{Hom}}_{\mathbb{R}}(K_0(\mathcal{A}{)}_{\mathbb{R}},\mathbb{R})$, which are defined by semistable subcategories and semistable torsion pairs in $\mathcal{A}$ associated to elements $\theta \in K_0(\mathcal{A}{)}_{\mathbb{R}}^{\ast }$. In this paper, we introduce the $M$-TF equivalence for each object $M\in \mathcal{A}$ as a systematic way to coarsen the TF equivalence. We show that the set $\mathrm{\Sigma }(M)$ of closures of $M$-TF equivalence classes is a rational generalized fan in $K_0(\mathcal{A}{)}_{\mathbb{R}}^{\ast }$ which is finite and complete. More precisely, we show that $\mathrm{\Sigma }(M)$ is the normal generalized fan of the Newton polytope $\mathrm{N}(M)$ in $K_0(\mathcal{A}{)}_{\mathbb{R}}$. When $\mathcal{A}$ is the category of finitely generated modules over a finite dimensional algebra $A$, $\mathrm{\Sigma }(M)$ can be regarded as a completion of a certain coarsening of the $g$-fan of $A$.