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Finite graded projectives are sums of shifted vertex projectives

Statement

Fix $m \geq 1$ and let $A_{m}$ be the Khovanov–Seidel type A algebra with vertex projectives $P_{i} = A_{m} e_{i}$ , $0 \leq i \leq m$ (Finite graded A_m-modules, internal shifts and the vertex projectives, The vertex modules S_i and their prime quotients). Every finitely generated graded projective left $A_{m}$ -module $P$ is isomorphic, as a graded module, to a finite direct sum $P ≅ ⨁_{i = 0}^{m} ⨁_{r \in Z} P_{i} {r}^{\oplus a_{i, r}}$ of internal shifts of the vertex projectives, with multiplicities $a_{i, r} \in N$ zero for all but finitely many pairs $(i, r)$ ; the multiplicities are uniquely determined by $P$ . Equivalently, the shifted modules $P_{i} {r}$ are the indecomposable objects, and their classes $[P_{i} {r}]$ , $0 \leq i \leq m$ , $r \in Z$ , form a free abelian basis of the split Grothendieck group of finite graded projectives, with no relations among distinct pairs.

Facts & Assumptions

Given: An integer $m \geq 1$ , the algebra $A_{m}$ with vertex idempotents $e_{i}$ , its path ideal $J$ generated by the arrows, the vertex projectives $P_{i} = A_{m} e_{i}$ with internal shift ${r}$ , and a finitely generated graded projective left $A_{m}$ -module $P$ .

[L1]

$J$ is homogeneous, $J^{3} = 0$ , $J$ is spanned by the paths of length at least one, and the quotient satisfies $A_{m} / J ≅ Z^{m + 1}$ with the classes of the vertex idempotents as basis (The Khovanov-Seidel path ideal).

[L2]

A graded left $A_{m}$ -module is finite graded projective exactly when it is a degree-zero direct summand of a finite direct sum of internal shifts of the regular module, $P \oplus Q ≅ A_{m} {r_{1}} \oplus \dots \oplus A_{m} {r_{n}}$ , with degree-zero inclusion and projection (Finite graded projectives are finite shifted-free summands).

[L3]

Every submodule of a finite free $Z$ -module is free, hence torsion-free and finitely generated if the ambient module is; a graded subgroup of a graded free abelian group that is a direct summand is again a graded free abelian group, and it has a homogeneous $Z$ -basis (A submodule of a free module of finite rank over a PID is free of no larger rank).

[L4]

$A_{m} = ⨁_{i = 0}^{m} P_{i}$ as a left module, $P_{i} {r}$ is the shift with $(P_{i} {r})_{d} = (P_{i})_{d - r}$ , the shift is an automorphism of the category, and a direct summand of a projective object is projective; a direct summand of a finitely generated module is finitely generated (Finite graded A_m-modules, internal shifts and the vertex projectives, A direct summand of a projective is projective).

[L5]

$A_{m}$ has the $Z$ -basis of $4 m + 1$ classes of vertices, arrows and returns, so $A_{m} / J$ kills exactly the non-vertex basis paths and the vertices $e_{0}, \dots, e_{m}$ are pairwise orthogonal idempotents with sum $1$ (The 4m+1 path basis, The Khovanov-Seidel path ideal).

Proof

technique · direct

1.1L1L2L3

$P / J P$ is a finite free graded abelian group. By [L2] there is a degree-zero isomorphism $P \oplus Q ≅ F : = A_{m} {r_{1}} \oplus \dots \oplus A_{m} {r_{n}}$ for some finitely generated graded projective $Q$ . Applying $(A_{m} / J) \otimes_{A_{m}} - = Z^{m + 1} \otimes_{A_{m}} -$ degreewise gives a degree-zero isomorphism of graded $Z^{m + 1}$ -modules $P / J P \oplus Q / J Q ≅ F / J F ≅ (Z^{m + 1}) {r_{1}} \oplus \dots \oplus (Z^{m + 1}) {r_{n}}$ ; in particular $P / J P$ is a direct summand, as a graded abelian group, of the finite free graded abelian group $F / J F$ whose homogeneous components are finite free $Z$ -modules. By [L3] each graded component $(P / J P)_{d}$ is a finitely generated torsion-free, hence free, $Z$ -module, and $P / J P$ has a homogeneous $Z$ -basis.

2.1step 1.1L1L5

The vertex decomposition of $P / J P$ . Since $J$ acts as $0$ on $P / J P$ , the latter is a graded module over $A_{m} / J ≅ ⨁_{i} Z e_{i}$ : the projections $e_{i}$ are orthogonal idempotents with sum $1$ [L5], so as a graded abelian group $P / J P = ⨁_{i} e_{i} (P / J P)$ , and a homogeneous basis of $P / J P$ is the disjoint union of homogeneous bases of the graded free abelian groups $e_{i} (P / J P)$ . Write $a_{i, r}$ for the rank of the degree- $r$ part $(e_{i} (P / J P))_{r}$ of the $i$ -th summand; the $a_{i, r}$ are the multiplicities of the Cartesian basis of [L3] and are zero for all but finitely many $(i, r)$ .

3.1step 2.1L1L4L5

A surjection from the proposed direct sum. Choose, for every $i$ and $r$ , a homogeneous $Z$ -basis ${[x_{i, r, s}]}_{s = 1}^{a_{i, r}}$ of the degree- $r$ part of $e_{i} (P / J P)$ and lift each $x_{i, r, s}$ to a homogeneous element ${\hat{x}}_{i, r, s} \in P$ of degree $r$ that lies in $e_{i} P$ ; since $P = ⨁_{i} e_{i} P$ this is possible component by component. The lifts assemble into a degree-zero $A_{m}$ -linear map $φ ⁣ : F^{'} : = ⨁_{i, r} P_{i} {r}^{\oplus a_{i, r}} \to P$ with $φ (e_{i} \otimes 1_{r, s}) = {\hat{x}}_{i, r, s}$ , the map on a summand being $a e_{i} \mapsto a {\hat{x}}_{i, r, s}$ . This is well defined because ${\hat{x}}_{i, r, s} = e_{i} {\hat{x}}_{i, r, s}$ , so $a e_{i} = 0$ implies $a {\hat{x}}_{i, r, s} = 0$ . Reducing modulo $J$ gives the isomorphism $F^{'} / J F^{'} \to P / J P$ determined by the chosen bases, because $P_{i} / J P_{i} ≅ Z$ is concentrated at the vertex $i$ and $J$ kills the generators; hence $φ$ is surjective by the nilpotent Nakayama argument: if $C : = coker φ$ then $C = J C$ , so $C = J^{3} C = 0$ because $J^{3} = 0$ .

3.2step 2.1L1L5

Uniqueness of the multiplicities. Suppose $⨁_{i, r} P_{i} {r}^{a_{i, r}} ≅ ⨁_{i, r} P_{i} {r}^{b_{i, r}}$ , and reduce modulo $J$ : since $P_{i} / J P_{i} ≅ Z$ placed at internal degree $0$ with the vertex $i$ acting by $1$ , an isomorphism of the direct sums induces, for every $i$ and $r$ , an isomorphism of graded abelian groups between the degree- $r$ parts of the $i$ -th vertex components, which are the free abelian groups of ranks $a_{i, r}$ and $b_{i, r}$ ; hence $a_{i, r} = b_{i, r}$ .

4.1step 3.1L1L4

The surjection splits, and is an isomorphism. The module $P$ is projective, so the surjection $φ$ splits: there is a degree-zero $A_{m}$ -linear $ψ : P \to F^{'}$ with $φ ψ = {i d}_{P}$ , and then $F^{'} ≅ P \oplus K$ with $K : = \ker φ$ a direct summand of $F^{'}$ , hence finitely generated graded projective by [L4]. Applying $(A_{m} / J) \otimes_{A_{m}} -$ to $F^{'} ≅ P \oplus K$ and comparing with the isomorphism $F^{'} / J F^{'} ≅ P / J P$ of step 3.1 gives $K / J K = 0$ , so $K = J K = J^{2} K = J^{3} K = 0$ ; hence $φ$ is injective and therefore an isomorphism $F^{'} ≅ P$ .

5.1step 4.1step 3.2∎

Conclusion. Every finitely generated graded projective $P$ is isomorphic to $⨁_{i, r} P_{i} {r}^{\oplus a_{i, r}}$ by step 4.1, the multiplicities are finite in number by step 2.1 and unique by step 3.2. Equivalently, the comparison with the split Grothendieck group sends the class of such a sum to the finite sum $\sum a_{i, r} [P_{i} {r}]$ , so the classes $[P_{i} {r}]$ are linearly independent over $Z$ by step 3.2, and each $P_{i}$ is indecomposable: for $i = 0$ its endomorphism ring is $e_{0} A_{m} e_{0} = Z e_{0}$ , whose only idempotents are $0, 1$ . For $1 \leq i \leq m$ the corner is $e_{i} A_{m} e_{i} = Z e_{i} \oplus Z (i ∣ i - 1 ∣ i)$ with $(i ∣ i - 1 ∣ i)^{2} = 0$ , an idempotent $a + b r$ satisfies $a^{2} = a$ and $b (2 a - 1) = 0$ in $Z$ , so only $a = b = 0$ and $a = 1, b = 0$ occur, and a decomposition $P_{i} = X \oplus Y$ would give a nontrivial idempotent. No choice principle is used: all bases are finite and chosen explicitly from the finitely many graded pieces.

Depends on

Used by

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

Dependency tree · two levels

26 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, Sections 1b and 2e.1 (standard reference, not scraped)