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The graded Grothendieck group is free on the vertex-projective classes

Statement

Let $G (A_{m}) = K_{0} (C_{m})$ be the graded Grothendieck group of The graded Grothendieck group of A_m. Then $G (A_{m})$ is a free $Z [q, q^{- 1}]$ -module with basis $[P_{0}], [P_{1}], \dots, [P_{m}] .$ Equivalently, the comparison isomorphism $K_{0} (C_{m}) ≅ K_{0}^{s p l i t} ({proj}^{g r} A_{m})$ sends the class $[X]$ of a bounded complex $X$ of finite graded projectives to its Euler class $\sum_{n} (- 1)^{n} [X^{n}]$ , the split Grothendieck group is free abelian on the classes $[P_{i} {r}]$ , and the internal-shift rule $[P_{i} {r}] = q^{r} [P_{i}]$ identifies it with $⨁_{i = 0}^{m} Z [q, q^{- 1}] [P_{i}] .$

Facts & Assumptions

Given: An integer $m \geq 1$ , the category $C_{m} = K^{b} ({proj}^{g r} A_{m})$ with its triangulation and the equivalence $Θ : C_{m} \to D^{b} (A_{m} -mod)$ , the graded Grothendieck group $G (A_{m}) = K_{0} (C_{m})$ , and the split Grothendieck group of the finite graded projectives.

[L1]

$G (A_{m}) = K_{0}^{t r i} (C_{m})$ is the free abelian group on the isomorphism classes of $C_{m}$ modulo the triangle relations $[Y] = [X] + [Z]$ , with $[X \oplus Y] = [X] + [Y]$ , $[X [1]] = - [X]$ and a $Z [q, q^{- 1}]$ -module structure with $[X {r}] = q^{r} [X]$ (The graded Grothendieck group of A_m, Homological and internal shifts on K_0(C_m)).

[L2]

$Θ : C_{m} \to D^{b} (A_{m} -mod)$ is exact, full, faithful and essentially surjective; every bounded complex of finitely generated graded $A_{m}$ -modules is isomorphic in $D^{b} (A_{m} -mod)$ to the image of an object of $C_{m}$ , so every such complex is perfect, and $Θ$ induces an isomorphism $K_{0} (C_{m}) ≅ K_{0}^{t r i} (D_{p e r f} (A_{m}))$ (The bounded projective comparison for the derived category, The bounded projective homotopy category C_m and the two shifts).

[L3]

For any unital ring the degree-zero inclusion of finitely generated projectives induces an isomorphism $K_{0}^{s p l i t} ({Proj}_{f g}) \to K_{0}^{t r i} (D_{p e r f})$ whose inverse sends a perfect object represented by a bounded finite-projective complex $P$ to $\sum_{n} (- 1)^{n} [P^{n}]$ ; the same holds in the graded setting with degree-zero maps (Triangle K0 of perfect complexes equals split K0 of finite projectives).

[L4]

Every finitely generated graded projective left $A_{m}$ -module is isomorphic to a finite direct sum $⨁_{i, r} P_{i} {r}^{\oplus a_{i, r}}$ with unique multiplicities, and the classes $[P_{i} {r}]$ are linearly independent in the split Grothendieck group of the additive category of finite graded projectives (Finite graded projectives are sums of shifted vertex projectives, Split Grothendieck group of an additive category).

Proof

technique · direct

1.1L1L2L3

$K_{0} (C_{m})$ is the split Grothendieck group of finite graded projectives. By [L2] the functor $Θ$ is an exact equivalence onto the perfect objects, so it induces a bijection on isomorphism classes preserving cones and shifts and hence an isomorphism of abelian groups $K_{0} (C_{m}) \to K_{0}^{t r i} (D_{p e r f} (A_{m}))$ ; by [L3] this group is identified with $K_{0}^{s p l i t} ({proj}^{g r} A_{m})$ through the Euler class $\sum_{n} (- 1)^{n} [X^{n}]$ . Composition gives the displayed comparison isomorphism.

1.2L4

Freeness of the split group. By [L4] every finite graded projective is a finite direct sum of shifts of the $P_{i}$ with unique multiplicities, so the split Grothendieck group is free abelian with basis the classes $[P_{i} {r}]$ , $0 \leq i \leq m$ , $r \in Z$ ; there are no relations among distinct pairs by uniqueness.

2.1step 1.2L1

The $Z [q, q^{- 1}]$ -action on the basis. The internal shift is an automorphism of $C_{m}$ commuting with $Θ$ , so its induced operator $q$ on $K_{0} (C_{m})$ is invertible and $q^{r} [P_{i}] = [P_{i} {r}]$ by [L1]; hence the free abelian group $⨁_{i, r} Z [P_{i} {r}]$ acquires the $Z [q, q^{- 1}]$ -module structure of $⨁_{i = 0}^{m} Z [q, q^{- 1}] [P_{i}]$ , with the $q$ -action shifting the basis.

3.1step 1.1step 1.2step 2.1∎

Conclusion. $G (A_{m})$ is a free $Z [q, q^{- 1}]$ -module with basis $[P_{0}], \dots, [P_{m}]$ , and the Euler-class comparison identifies it with the split Grothendieck group of finite graded projectives; the freeness uses the explicit classification of finite graded projectives and no finite-dimensional-field-algebra structure theorem. No choice principle is used.

Depends on

Used by

Decategorifying a generator on the vertex-projective basis Example
Decategorification is the unreduced Burau action Proposition

Dependency tree · two levels

52 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, J. Amer. Math. Soc. 15 (2002) 203-271, Section 2e.1 (standard reference, not scraped)
The Stacks Project, More on Algebra, Lemma 15.121.2 (K_0 of perfect complexes versus split K_0 of finite projectives) (standard reference, not scraped)