The graded Grothendieck group of A_m

Fix $m\ge 1$, let $A_m$ be the Khovanov–Seidel type A algebra, and let $C_m=K^b({\mathrm{proj}}^{gr}{A}_m)$ be the bounded homotopy category of finite graded projective left $A_m$-modules of The bounded projective homotopy category C_m and the two shifts, with its distinguished triangles and its homological shift $[1]$. The graded Grothendieck group of $A_m$ is the triangulated Grothendieck group $G(A_m):=K_0(C_m)=K_0^{\mathrm{t}\mathrm{r}\mathrm{i}}(K^b({\mathrm{proj}}^{gr}{A}_m))$ of Grothendieck group of an essentially small triangulated category applied to the small coded presentation of The triangulated K_0 of the Khovanov–Seidel projective category: the free abelian group on the set of isomorphism classes of objects of $C_m$ modulo the subgroup generated by the relations $[Y]=[X]+[Z]$ for every distinguished triangle $X\to Y\to Z\to X[1]$ of $C_m$. Write $[X]$ for the class of $X$.

Class rules. The definition is the triangulated one, so:

• the cone triangle $X\oplus Y\to Y\to X[1]\to X[1]\oplus Y[1]$ of a biproduct gives $[X\oplus Y]=[X]+[Y]$;
• the rotated triangle $X\to 0\to X[1]\to X[1]$ gives $[X[1]]=-[X]$, the statement of Homological and internal shifts on K_0(C_m);
• the internal shift $\{1\}$, which is an exact automorphism of $C_m$ distinct from $[1]$, makes $G(A_m)$ into a module over $\mathbb{Z}[q,q^{-1}]$ with $[X\{r\}]=q^r[X]$, again by Homological and internal shifts on K_0(C_m).

The two shifts are never identified: $[1]$ is homological and contributes a sign $-1$, $\{1\}$ is internal and contributes a unit $q$.

Comparison with perfect complexes. The composite $\mathrm{\Theta }\text{\hspace{0.17em}⁣}:C_m=K^b({\mathrm{proj}}^{gr}{A}_m)⟶D^b(A_m\text{-mod})$ of inclusion and localization is exact, full, faithful and essentially surjective by The bounded projective comparison for the derived category; the graded clause of Triangle K0 of perfect complexes equals split K0 of finite projectives identifies the target's triangulated Grothendieck group with $K_0^{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}$ of the finite graded projectives. Consequently $G(A_m)$ is canonically isomorphic to the Grothendieck group of the graded perfect complexes of Perfect complexes over a ring and its graded version on the bounded projective model, and $[X]$ corresponds to the Euler characteristic ${\sum }_n(-1{)}^n[P^n]$ of any bounded finite graded projective complex representing $X$.

Scope and warnings. This is emphatically the triangulated $K_0$ of projective complexes and not the short-exact $G_0$ of the abelian category of all finitely generated graded $A_m$-modules, and no identification of the two is claimed; $G_0$ may a priori be a quotient of $G(A_m)$. No structure theorem for finite-dimensional algebras over a field is applied to the integral algebra $A_m$. The freeness of $G(A_m)$ on the shifted vertex-projective classes is not asserted here: it is the content of The graded Grothendieck group is free on the vertex-projective classes, which uses the classification of finite graded projectives and the comparison above.

Notation used below. Since the internal shift is an automorphism, write $q^r[P_i]:=[P_i\{r\}]$, so that $G(A_m)$ is generated as a $\mathbb{Z}[q,q^{-1}]$-module by the classes of the vertex projectives $P_0,\dots ,P_m$; the next lemma on this page shows these generate freely.