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Finite graded A_m-modules, internal shifts and the vertex projectives

Definition

Fix $m \geq 1$ and let $A_{m}$ be the Khovanov–Seidel type A algebra of Khovanov–Seidel type A algebra, graded by internal degree, with the vertex idempotents $e_{i} : = (i)$ and the generators $u_{i} : = (i ∣ i + 1)$ , $d_{i} : = (i + 1 ∣ i)$ .

The finite module category. $A_{m} -mod$ is the category whose objects are the finitely generated graded left $A_{m}$ -modules and whose morphisms are the degree-zero $A_{m}$ -linear maps $f : M \to N$ , that is $f (M_{d}) \subseteq N_{d}$ for every $d$ , with composition of maps. Throughout this page every left module, right module and bimodule over $A_{m}$ is graded and every homomorphism is degree-zero unless stated otherwise. Finite generation is in the sense of Finite graded projective modules, and graded modules, degree-zero maps and internal shifts are those of Associative graded algebras, bimodules, and internal shifts.

Internal shift. For $r \in Z$ and a graded left $A_{m}$ -module $M$ the internal shift $M {r}$ is the graded module with $(M {r})_{d} = M_{d - r} (d \in Z),$ carrying the same left $A_{m}$ -action as $M$ . The shift ${1}$ moves the grading upwards by one: an element of $M_{d}$ is an element of $(M {1})_{d + 1}$ .

Vertex projectives. Put $P_{i} : = A_{m} e_{i},_{i} P : = e_{i} A_{m} (0 \leq i \leq m),$ equipped with the grading in which a homogeneous element of $A_{m}$ of degree $d$ lies in degree $d$ . Because paths are written left to right and the product is concatenation, $P_{i}$ consists of the classes of the paths ending at $i$ and $_{i} P$ of the classes of the paths beginning at $i$ . The algebra splits as $A_{m} = ⨁_{i = 0}^{m} P_{i} = ⨁_{i = 0}^{m}_{i} P$ as a left, respectively right, module over itself; the decomposition into vertex projectives is displayed as proved below.

Claims proved below. $A_{m} -mod$ is an abelian category; the internal shift ${r}$ is an automorphism of $A_{m} -mod$ ; each $P_{i}$ is a finite graded projective left $A_{m}$ -module and each $_{i} P$ a finite graded projective right $A_{m}$ -module, generated by $e_{i}$ .

Notation. The source writes $P_{i} = A_{m} (i)$ for the summand $A_{m} e_{i}$ , and ${r}$ for the internal shift; the parentheses in $A_{m} (i)$ are part of that summand notation and never denote a shift. On this page only the braces ${r}$ denote the internal shift, and only the brackets $[r]$ of The bounded projective homotopy category C_m and the two shifts denote the homological shift, which is not defined yet.

Facts & Assumptions

Given: An integer $m \geq 1$ , the algebra $A_{m}$ with its internal degree, the vertex idempotents $e_{i} = (i)$ and the vertex projectives $P_{i} = A_{m} e_{i}$ and $_{i} P = e_{i} A_{m}$ .

[F1]

$A_{m} = Z Γ_{m} / I_{m}$ is a ring, graded by internal degree with $A_{m} = ⨁_{d \geq 0} (A_{m})_{d}$ , the classes $e_{i} = (i)$ and $u_{i} = (i ∣ i + 1)$ are homogeneous of degree $0$ and $d_{i} = (i + 1 ∣ i)$ is homogeneous of degree $1$ , and the generators of $I_{m}$ vanish in $A_{m}$ (Khovanov–Seidel type A algebra).

[F2]

In $Z Γ_{m}$ the vertex paths are orthogonal idempotents, $(u) (v) = δ_{u v} (u)$ , with $\sum_{j = 0}^{m} (j) = 1$ , and the endpoint formulas $(u) p = p$ for $s (p) = u$ and $p (v) = p$ for $t (p) = v$ hold, the products being $0$ otherwise; passing to the quotient $A_{m}$ , the images $e_{j}$ satisfy the same identities (Integral path ring of a finite quiver).

[L3]

$A_{m}$ is a free abelian group of rank $4 m + 1$ with basis the classes of the $m + 1$ vertices, the $2 m$ arrows and the $m$ returns $(i ∣ i - 1 ∣ i)$ , $1 \leq i \leq m$ (The 4m+1 path basis).

[L4]

A graded left $A$ -module $M$ is an $A$ -module with $M = ⨁_{d} M_{d}$ and $A_{i} M_{d} \subseteq M_{i + d}$ ; a degree-zero map is $A$ -linear with $f (M_{d}) \subseteq N_{d}$ ; ${GrMod}_{0} (A)$ is the category of graded left $A$ -modules and degree-zero maps; the internal shift $M {r}$ has $(M {r})_{d} = M_{d - r}$ and is again graded; a graded submodule is a submodule with $S = ⨁_{d} (S \cap M_{d})$ (Associative graded algebras, bimodules, and internal shifts).

[L5]

A graded left $A$ -module $P$ is graded projective when every degree-zero epimorphism $q : E ↠ M$ and every degree-zero $f : P \to M$ admit a degree-zero lift $\tilde{f}$ with $q \tilde{f} = f$ ; $P$ is finitely generated when $P = A p_{1} + \dots + A p_{n}$ for finitely many homogeneous $p_{j}$ , equivalently when the underlying module is finitely generated; $P$ is finite graded projective when it is both (Finite graded projective modules).

[L6]

For every unital associative $Z$ -graded algebra $A$ the category ${GrMod}_{0} (A)$ is abelian; kernels, images, cokernels and finite biproducts are computed in each homogeneous degree; a sequence is exact precisely when it is exact degreewise (Graded modules with degree-zero maps form an abelian category).

[L7]

A graded left $A$ -module $P$ is finite graded projective if and only if $P$ is a degree-zero direct summand of a finite direct sum of internal shifts $A {r_{1}} \oplus \dots \oplus A {r_{n}}$ ; in particular every finite direct sum of shifts of $A$ is a projective object of ${GrMod}_{0} (A)$ , and no arbitrary-index choice is used (Finite graded projectives are finite shifted-free summands).

[F8]

$Z$ is a principal ideal domain and hence Noetherian (Every principal ideal domain is Noetherian).

[L9]

Every finitely generated left module over a left Noetherian ring is Noetherian, that is, each of its submodules is finitely generated (Finitely generated modules over a left Noetherian ring are Noetherian).

[L10]

A unital ring $R$ is left Noetherian when its left regular module $_{R} R$ is Noetherian, and a module is Noetherian when each of its submodules is finitely generated; the submodules of $_{R} R$ are exactly the left ideals (Left and right Noetherian rings, Noetherian modules: every submodule is finitely generated).

Proof

technique · direct

1.1

$A_{m}$ is left and right Noetherian. By [L3] the abelian group $A_{m}$ is free of rank $4 m + 1$ over $Z$ , hence finitely generated as a $Z$ -module; $Z$ is Noetherian by [F8], so [L9] makes $A_{m}$ Noetherian as a left $Z$ -module, that is, every $Z$ -submodule of $A_{m}$ is finitely generated as a $Z$ -module. A left ideal of $A_{m}$ is a $Z$ -submodule, so it is generated as a $Z$ -module, hence as a left ideal, by finitely many elements, and [L10] gives that $A_{m}$ is left Noetherian; the same computation with right ideals, $A_{m}$ being finitely generated as a right $Z$ -module as well, makes $A_{m}$ right Noetherian.

L3F8L9L10

1.2

The category and its ambient category. By [L4] and [L5] the objects of $A_{m} -mod$ are the graded left $A_{m}$ -modules that are finitely generated over $A_{m}$ , and its morphisms are the degree-zero $A_{m}$ -linear maps, so $A_{m} -mod$ is the full subcategory of ${GrMod}_{0} (A_{m})$ on the finitely generated objects; by [L6] the ambient category ${GrMod}_{0} (A_{m})$ is abelian, with degreewise kernels, cokernels and finite biproducts.

L4L5L6

1.3

$P_{i}$ is a degree-zero direct summand of $A_{m}$ . By [F2] the identities $(u) (v) = δ_{u v} (u)$ and $\sum_{j} (j) = 1$ hold in $A_{m}$ , so with $e_{j} = (j)$ one has $A_{m} = ⨁_{j} A_{m} e_{j}$ : every $x \in A_{m}$ equals $x \cdot 1 = \sum_{j} x e_{j}$ with $x e_{j} \in A_{m} e_{j}$ , and if $\sum_{j} x_{j} = 0$ with $x_{j} \in A_{m} e_{j}$ then $x_{i} = x_{i} e_{i} = (\sum_{j} x_{j}) e_{i} = 0$ . The inclusion $ι_{i} : P_{i} \to A_{m}$ and the map $p_{i} (x) : = x e_{i}$ are $A_{m}$ -linear, and $p_{i}$ is degree-zero because $e_{i}$ is homogeneous of degree $0$ by [F1]; as $p_{i} ι_{i} = {i d}_{P_{i}}$ and $ι_{i} p_{i}$ is the projection onto the $i$ -th summand, $P_{i}$ is a degree-zero direct summand of $A_{m} = A_{m} {0}$ , a finite direct sum of zero shifts of $A_{m}$ , and $P_{i}$ is generated over $A_{m}$ by $e_{i}$ .

F1F2L4

1.4

The internal shift is an automorphism of $A_{m} -mod$ . Let $M$ be a finitely generated graded left $A_{m}$ -module and $r \in Z$ . By [L4] the shift $M {r}$ has $(M {r})_{d} = M_{d - r}$ and is a graded left $A_{m}$ -module; if $M = A x_{1} + \dots + A x_{n}$ with homogeneous $x_{j}$ , then each $x_{j}$ lies in $(M {r})_{d_{j} + r}$ for its degree $d_{j}$ in $M$ and $M {r} = A x_{1} + \dots + A x_{n}$ , so $M {r}$ is an object of $A_{m} -mod$ . A degree-zero map $f : M \to N$ satisfies $f (M_{d - r}) \subseteq N_{d - r}$ for every $d$ , so the same underlying map is a degree-zero map $M {r} \to N {r}$ : the shift is a functor, additive because addition of maps is unchanged, and ${r} {s} = {r + s}$ on objects and on morphisms, so ${r}$ has the two-sided inverse ${- r}$ and is an automorphism of $A_{m} -mod$ .

2.1

Closure of the finitely generated objects under kernels, cokernels and finite sums. Let $f : M \to N$ be a degree-zero map of finitely generated graded left $A_{m}$ -modules. By [L6] the kernel, image and cokernel of $f$ are graded modules, with $\ker f$ a graded submodule of $M$ ; since $M$ is a finitely generated left module over the left Noetherian ring $A_{m}$ of step 1.1, [L9] makes $\ker f$ finitely generated. The image $im f ≅ M / \ker f$ and the cokernel $coker f = N / im f$ are quotients of finitely generated modules, hence finitely generated, and the direct sum $M \oplus N$ is generated by the union of finite homogeneous generating families of $M$ and $N$ , while the zero module is generated by the empty family.

step 1.1L6L9

2.2

$P_{i}$ is finite graded projective. By step 1.3 the module $P_{i}$ is a degree-zero direct summand of $A_{m} = A_{m} {0}$ , a finite direct sum of internal shifts of $A_{m}$ , and $P_{i}$ is generated by the single element $e_{i}$ ; [L7] therefore makes $P_{i}$ a finite graded projective left $A_{m}$ -module in the sense of [L5].

step 1.3L5L7

2.3

$_{i} P$ is finite graded projective on the right. The opposite ring $A_{m}^{o p}$ is a graded $Z$ -algebra with the same homogeneous pieces $(A_{m}^{o p})_{i} = (A_{m})_{i}$ , because the graded algebra axiom $A_{j}^{o p} A_{i}^{o p} \subseteq (A_{m}^{o p})_{i + j}$ reads $A_{i} A_{j} \subseteq (A_{m})_{i + j}$ in $A_{m}$ ; a graded right $A_{m}$ -module is precisely a graded left $A_{m}^{o p}$ -module with the same degree-zero maps, and the mirrored computation of step 1.3, using [F2] and the right module structure, gives $A_{m} = ⨁_{j} e_{j} A_{m}$ as a direct sum of right submodules, so $_{i} P = e_{i} A_{m}$ is a degree-zero direct summand of $A_{m}^{o p} {0}$ , generated by $e_{i}$ . Applying [L7] to the graded algebra $A_{m}^{o p}$ makes $_{i} P$ a finite graded projective right $A_{m}$ -module in the sense of [L5].

step 1.3F2L5L7

3.1

$A_{m} -mod$ is abelian. By steps 1.2 and 2.1 the full subcategory $A_{m} -mod$ of the abelian category ${GrMod}_{0} (A_{m})$ contains a zero object, is closed under binary biproducts, and contains a kernel and a cokernel of each of its morphisms: if $f g = 0$ for a morphism $g$ of the subcategory, the factorization of $g$ through $\ker f$ asserted by [L6] has codomain the finitely generated object $\ker f$ , so it is a morphism of the subcategory and exhibits $\ker f \to M$ as a kernel in $A_{m} -mod$ , and dually $coker f$ is a cokernel in $A_{m} -mod$ . Hom-sets are abelian groups under addition of maps and composition is additive, so $A_{m} -mod$ is additive; and the canonical comparison $coim f \to im f$ , an isomorphism in ${GrMod}_{0} (A_{m})$ by [L6], is a morphism of $A_{m} -mod$ because the subcategory is full. Hence $A_{m} -mod$ is abelian.

step 1.2step 2.1L6

4.1

Conclusion. $A_{m} -mod$ is the abelian category of the finitely generated graded left $A_{m}$ -modules and degree-zero maps (steps 1.2 and 3.1); the internal shift ${r}$ is an automorphism of it (step 1.4); $P_{i} = A_{m} e_{i}$ is a finite graded projective left $A_{m}$ -module and $_{i} P = e_{i} A_{m}$ is a finite graded projective right $A_{m}$ -module, each generated by the homogeneous idempotent $e_{i}$ (steps 2.2 and 2.3); and $A_{m}$ decomposes as $⨁_{i} P_{i}$ on the left and as $⨁_{i}_{i} P$ on the right (steps 1.3 and 2.3). The source's notation $P_{i} = A_{m} (i)$ refers to this summand; the internal shift of this page is written ${r}$ , so no claim about a shift is being made by that notation.

step 1.2step 1.3step 2.2step 2.3step 1.4step 3.1∎

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Sources

Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, §1b and §§2a-2c, printed pp. 3-4 and 9-11 (standard reference, not scraped)