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The two-sided projective bimodules U_i and their tensor functors

Definition

Fix $m \geq 1$ , let $A_{m}$ be the Khovanov–Seidel type A algebra with its internal grading, and let $A_{m} -mod$ be the category of Finite graded A_m-modules, internal shifts and the vertex projectives with vertex projectives $P_{i} = A_{m} e_{i} (paths ending at i),_{i} P = e_{i} A_{m} (paths beginning at i) .$ For $1 \leq i \leq m$ put $U_{i} : = P_{i} \otimes_{Z}_{i} P,$ the tensor product over $Z$ of the left $A_{m}$ -module $P_{i}$ and the right $A_{m}$ -module $_{i} P$ , with the left $A_{m}$ -action $a \cdot (x \otimes y) : = (a x) \otimes y$ on the first factor and the right $A_{m}$ -action $(x \otimes y) \cdot a : = x \otimes (y a)$ on the second; the two actions commute because they act on different factors. Its grading is the total internal degree of Graded balanced tensor product and homogeneous Hom: an elementary tensor of homogeneous elements of degrees $p$ and $q$ has degree $p + q$ , which agrees with that construction applied to the right module $_{i} P$ and the left module $P_{i}$ under the canonical flip of the commutative ground ring $Z$ . The claims that $U_{i}$ is a graded $(A_{m}, A_{m})$ -bimodule, that it is finite graded projective as a left module and as a right module, and that it is flat as an underlying right $A_{m}$ -module are proved below, so the notation denotes.

The functors. For $1 \leq i \leq m$ let $U_{i} (-) : = U_{i} \otimes_{A_{m}} - : A_{m} -mod ⟶ A_{m} -mod, U_{i} (M) : = U_{i} \otimes_{A_{m}} M,$ the functor on graded left $A_{m}$ -modules and degree-zero maps obtained from the bimodule structure, with the left $A_{m}$ -module structure on $U_{i} \otimes_{A_{m}} M$ coming from the left action on $U_{i}$ . The claims that $U_{i} (-)$ is a functor, that it is exact and that it carries finite graded projectives to finite graded projectives are proved below.

Convention. The index range is $1 \leq i \leq m$ as in the source; the case $i = 0$ is excluded, so the neighboring vertices are $i - 1, i + 1$ when $i < m$ , and only $m - 1$ when $i = m$ . Internal shifts on $U_{i}$ are those of $A_{m} -mod$ in each variable, and the homological shift $[1]$ of The bounded projective homotopy category C_m and the two shifts never acts on $U_{i}$ here.

Facts & Assumptions

Given: An integer $m \geq 1$ , the algebra $A_{m}$ with vertex idempotents $e_{i}$ and internal grading, the vertex projectives $P_{i} = A_{m} e_{i}$ and $_{i} P = e_{i} A_{m}$ , and an index $1 \leq i \leq m$ .

[L1]

$A_{m} -mod$ is the abelian category of finitely generated graded left $A_{m}$ -modules and degree-zero maps; $A_{m} = ⨁_{i} P_{i} = ⨁_{i}_{i} P$ ; 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 $e_{i}$ ; the internal shift is an automorphism and exactness is degreewise (Finite graded A_m-modules, internal shifts and the vertex projectives).

[L2]

$A_{m}$ has a $Z$ -basis of $4 m + 1$ classes: the $m + 1$ vertices, the $2 m$ arrows and the returns $(i ∣ i - 1 ∣ i)$ for $1 \leq i \leq m$ ; the product is left-to-right concatenation, so a path lies in $P_{j}$ exactly when it ends at $j$ and in $_{j} P$ exactly when it begins at $j$ (The 4m+1 path basis, Integral path ring of a finite quiver).

[L3]

For graded modules $M$ (right) and $N$ (left) over a graded ring the balanced tensor $M \otimes_{A} N$ carries the total-degree grading in which a homogeneous elementary tensor has degree the sum of the degrees of its factors, and if $M$ is a graded $(B, A)$ -bimodule and $N$ a graded left $A$ -module then $M \otimes_{A} N$ is a graded left $B$ -module by the outer action, symmetrically on the right, and the two outer actions commute (Graded balanced tensor product and homogeneous Hom, Associative graded algebras, bimodules, and internal shifts).

[L4]

Let $A, B$ be graded $k$ -algebras and $M$ a graded $(B, A)$ -bimodule with $Φ_{M} = M \otimes_{A} -$ . If $M$ is flat as an underlying right $A$ -module then $Φ_{M}$ is exact on graded left $A$ -modules; if $M$ is finite graded projective as a left $B$ -module then $Φ_{M}$ carries every finite graded projective left $A$ -module to a finite graded projective left $B$ -module; neither hypothesis implies the other (Bimodule tensor exactness and preservation of finite projectives have separate hypotheses).

[L5]

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

[L6]

A graded module is finite graded projective when it is graded projective and generated by finitely many homogeneous elements (Finite graded projective modules).

[F7]

Every projective left or right module over an arbitrary unital ring is flat on its appropriate side, and this implication requires no Axiom of Choice (Projective left and right modules are flat over an arbitrary ring).

[L8]

Every direct summand of a projective object in an abelian category is projective (A direct summand of a projective is projective).

Proof

technique · direct

1.1

The two factors are finitely generated free abelian groups. By [L2] the classes of the paths ending at $i$ form a $Z$ -basis of $P_{i}$ and the classes of the paths beginning at $i$ form a $Z$ -basis of $_{i} P$ ; since there are finitely many basis paths, both $P_{i}$ and $_{i} P$ are free abelian groups of finite rank and the tensor product $P_{i} \otimes_{Z}_{i} P$ is a graded abelian group in the total-degree grading of [L3].

L2L3

2.1

$U_{i}$ is a graded $(A_{m}, A_{m})$ -bimodule. The left action $a \cdot (x \otimes y) = (a x) \otimes y$ is well defined because $P_{i}$ is a left $A_{m}$ -module and additive in each variable, and it is homogeneous because $A_{m}$ is graded with $(A_{m})_{l} (P_{i})_{d} \subseteq (P_{i})_{l + d}$ for all internal degrees $l, d$ , with the vertex index $i$ fixed; the right action $(x \otimes y) \cdot a = x \otimes (y a)$ is well defined because $_{i} P$ is a right $A_{m}$ -module, and homogeneous for the same reason on the second factor. The two actions commute because the first acts on the first tensor factor and the second on the second, and both are unital; so $U_{i}$ is a graded $(A_{m}, A_{m})$ -bimodule by [L3] and the bimodule convention of [L3].

step 1.1L3

2.2

$U_{i}$ is finite graded projective on the left. Let $y_{1}, \dots, y_{R}$ be the homogeneous basis of $_{i} P$ of step 1.1 with degrees $d_{1}, \dots, d_{R}$ ; an element of $U_{i}$ is a finite sum $\sum_{j} x_{j} \otimes y_{j}$ with $x_{j} \in P_{i}$ , and the assignment $(x_{1}, \dots, x_{R}) \mapsto \sum_{j} x_{j} \otimes y_{j}$ defines an $A_{m}$ -linear bijection $⨁_{j = 1}^{R} P_{i} {d_{j}} \to U_{i}$ that is degree-zero for the total grading of [L3], since $P_{i} {d_{j}}$ has $(P_{i} {d_{j}})_{e} = (P_{i})_{e - d_{j}}$ and $x \otimes y_{j}$ has degree $\deg x + d_{j}$ . The module $P_{i}$ is finite graded projective by [L1], its internal shift is finite graded projective by [L5], and a finite direct sum of finite graded projectives is finite graded projective by [L5] again, applied to the direct-summand characterization; hence $U_{i} ≅ ⨁_{j} P_{i} {d_{j}}$ is a finite graded projective left $A_{m}$ -module, in particular finitely generated by [L6].

step 1.1L1L5L6

2.3

$U_{i}$ is finite graded projective on the right. The mirrored argument with a homogeneous basis $x_{1}, \dots, x_{S}$ of $P_{i}$ and the assignment $(y_{1}, \dots, y_{S}) \mapsto \sum_{k} x_{k} \otimes y_{k}$ identifies $U_{i}$ with $⨁_{k}_{i} P {d_{k}^{'}}$ as a graded right $A_{m}$ -module, where $d_{k}^{'}$ is the degree of $x_{k}$ ; each $_{i} P$ is finite graded projective on the right by [L1], its shifts are again finite graded projective by the opposite-ring instance of [L5], and finite direct sums are finite graded projective, so $U_{i}$ is a finite graded projective right $A_{m}$ -module.

step 1.1L1L5

3.1

$U_{i}$ is flat as an underlying right $A_{m}$ -module. By [L5] and step 2.3 the right $A_{m}$ -module $U_{i}$ is a degree-zero direct summand of a finite direct sum of internal shifts of the right regular module $A_{m}$ ; a shift $A_{m} {r}$ has the same underlying ungraded module as $A_{m}$ , which is free, so the underlying ungraded right module of $U_{i}$ is a direct summand of a finite free right $A_{m}$ -module and is therefore projective by [L8]; by [F7] a projective right module over an arbitrary unital ring is flat, so the underlying right $A_{m}$ -module of $U_{i}$ is flat.

step 2.3L5L8F7

4.1

The functor $U_{i} (-)$ is exact and preserves finite graded projectives. Tensoring over $A_{m}$ with the graded $(A_{m}, A_{m})$ -bimodule $U_{i}$ gives a functor on graded left $A_{m}$ -modules which is additive and degree-zero on maps, and the result $U_{i} \otimes_{A_{m}} M$ is a graded left $A_{m}$ -module by the outer-action clause of [L3], with degreewise kernels and exactness by [L1]; by step 3.1 the hypothesis of clause 1 of [L4] holds for the graded algebra pair $(A_{m}, A_{m})$ and the bimodule $U_{i}$ , so $U_{i} (-)$ is exact, and by step 2.2 the hypothesis of clause 2 of [L4] holds, so $U_{i} (-)$ carries finite graded projective left $A_{m}$ -modules to finite graded projective left $A_{m}$ -modules. For a finitely generated graded $M$ , take a surjection from a finite sum of shifts of $A_{m}$ onto $M$ . Exactness gives a surjection from the corresponding finite sum of shifts of $U_{i}$ onto $U_{i} \otimes_{A_{m}} M$ ; since $U_{i}$ is finite on the left, the target is finitely generated. Thus the functor restricts to the stated finite module category. Both clauses are used as stated, and neither is inferred from the other; the two-sided projectivity of $U_{i}$ is exactly the conjunction of steps 2.2 and 2.3.

step 2.2step 2.3step 3.1L1L3L4

5.1

Conclusion. For every $1 \leq i \leq m$ the module $U_{i} = P_{i} \otimes_{Z}_{i} P$ is a graded $(A_{m}, A_{m})$ -bimodule (step 2.1) that is finite graded projective as a left module (step 2.2) and as a right module (step 2.3) and flat as an underlying right module (step 3.1); consequently the tensor functor $U_{i} (-) = U_{i} \otimes_{A_{m}} -$ is a well-defined endofunctor of $A_{m} -mod$ which is exact and preserves finite graded projectives (step 4.1). No choice principle is used: the bases of step 1.1 are finite, the shifts of steps 2.2 and 2.3 are indexed by those finite bases, and [F7] is choice-free.

step 2.1step 2.2step 2.3step 3.1step 4.1F7∎

Depends on

Used by

The Khovanov–Seidel bimodule maps βᵢ and γᵢ Definition
The twist complexes Rᵢ and Rᵢ⁻¹ Definition
Bounded two-sided projective bimodule complexes act on Cₘ Lemma
Corner computations: the Uᵢ satisfy the Temperley-Lieb relations Theorem

Dependency tree · two levels

32 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, §2b, printed pp. 11-12 (standard reference, not scraped)