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The internal shift as a graded Eilenberg-Watts kernel

Example

Let $k$ be a field and $A$ a graded $k$ -algebra with $1_{A} \neq 0$ . For each $r \in Z$ the internal shift functor ${r} : {GrMod}_{0} (A) ⟶ {GrMod}_{0} (A), X ⟼ X {r},$ is $k$ -linear, right exact, coproduct preserving and coherently shift-compatible, and it is naturally isomorphic to the graded Eilenberg-Watts tensor functor $T_{A {r}} = A {r} \otimes_{A} -$ of Graded Eilenberg-Watts theorem with coherent shifts: the canonical isomorphisms $A {r} \otimes_{A} X ≅ X {r}$ of Graded associativity, units, and internal-shift tensor isomorphisms are natural degree-zero and compatible with outer actions, so the kernel attached to the internal shift by the theorem is the shifted regular bimodule $A {r}$ , and the standard comparisons $θ^{A {r}}$ of Graded tensor functors are k-linear, right exact, coproduct preserving and shift-coherent are the canonical shift comparisons of Internal shifts are autoequivalences and commute with the graded tensor product. The shift ${r}$ is the identity functor precisely when $r = 0$ ; the hypothesis $1_{A} \neq 0$ is used only for this criterion, since over the zero algebra the only graded left module is $0$ , the module category degenerates and every shift is the identity there. It is not the cochain (homological) shift $[1]$ of the bounded-complex page: $X [1]^{n} = X^{n + 1}$ , so $[1]$ lowers cochain placement by one and negates the differential, while ${r}$ raises internal degrees and introduces no sign and no differential; the published counterexample The internal and homological shifts are not interchangeable shows specifically that ${1}$ and $[1]$ are not naturally isomorphic on $K^{b} ({proj}^{g r} A_{m})$ for the Khovanov–Seidel algebra $A_{m}$ , $m \geq 1$ , so the internal and homological shifts must not be conflated. The example makes no choice.

Facts & Assumptions

Given: A field $k$ , a graded $k$ -algebra $A$ with $1_{A} \neq 0$ , an integer $r$ , graded left $A$ -modules $X, Y$ and the functors ${r}$ and $T_{A {r}} = A {r} \otimes_{A} -$ .

[L1]

The functor $Φ (M) = (T_{M}, θ^{M})$ classifies the $k$ -linear right exact coproduct-preserving coherently shift-compatible functors, its inverse attaches to $F$ the bimodule $F (A)$ with the reconstructed action, and $Φ (M) (A) = M \otimes_{A} A$ is identified with $M$ by the unit isomorphism (Graded Eilenberg-Watts theorem with coherent shifts).

[L2]

For a graded $(B, A)$ -bimodule $M$ the functor $T_{M}$ is $k$ -linear, right exact and coproduct preserving, the canonical comparisons $θ_{X, s}^{M}$ are the identity on elementary tensors, and $M \mapsto (T_{M}, θ^{M})$ is functorial (Graded tensor functors are k-linear, right exact, coproduct preserving and shift-coherent).

[L3]

The tensor-unit map $A \otimes_{A} X \to X$ , $a \otimes x \mapsto a x$ , and the shift isomorphism $M {r} \otimes_{A} X ≅ (M \otimes_{A} X) {r}$ are natural degree-zero isomorphisms compatible with outer actions (Graded associativity, units, and internal-shift tensor isomorphisms).

[L4]

The internal shift has $(X {r})_{d} = X_{d - r}$ with the same actions, is again a graded module, satisfies $X {0} = X$ , and distinct homogeneous pieces intersect trivially; it introduces no sign and no differential (Associative graded algebras, bimodules, and internal shifts).

[L5]

The internal shift is a strict autoequivalence with ${0} = i d$ and ${r} {s} = {r + s}$ , acting as the identity on underlying sets (Internal shifts are autoequivalences and commute with the graded tensor product).

[L7]

The cochain shift is $X [1]^{n} = X^{n + 1}$ with differential $- d_{X}^{n + 1}$ (Bounded bimodule tensor is associative, unital, and compatible with cones).

[L6]

In the signed tensor totalization of bounded cochain complexes the cochain sign depends only on cochain degree, and the internal $Z$ -grading is independent of cochain degree and contributes no additional sign (Bounded graded bimodule complexes and signed tensor totalization).

Verification

1.1L2L3

$T_{A {r}}$ is $k$ -linear, right exact and coproduct preserving with comparisons the identity on elementary tensors [L2]; the unit isomorphism $A \otimes_{A} X ≅ X$ and the shift isomorphism $A {r} \otimes_{A} X ≅ (A \otimes_{A} X) {r}$ of [L3] compose to a natural degree-zero isomorphism $A {r} \otimes_{A} X ≅ X {r}$ compatible with the outer actions, so $T_{A {r}} ≅ {r}$ as functors.

1.2L4L5

The shift ${r}$ is the identity functor precisely when $r = 0$ : for $r = 0$ this is ${0} = i d$ [L5]; conversely, if ${r}$ is the identity functor then ${r} (A) = A {r}$ equals $A$ , so the degree-zero pieces agree, $A_{- r} = (A {r})_{0} = A_{0}$ , and $1_{A} \in A_{- r} \cap A_{0}$ is nonzero by the standing hypothesis $1_{A} \neq 0$ , so $- r = 0$ because distinct homogeneous pieces intersect trivially [L4]. The excluded zero algebra is genuinely different: there the only graded left $A$ -module is $0$ , so every shift is the identity functor.

1.3L4L6L7

The functor ${r}$ is not the cochain shift $[1]$ : ${r}$ changes only the internal grading and inserts no sign or differential [L4], while the cochain shift changes cochain placement and, by the convention of the bounded-complex page, its sign is the cochain Koszul sign carried by the differential alone, with internal degrees contributing no cochain sign [L6, L7]; the two are different operations on different structures, and the published counterexample cited in the statement records their non-interchangeability in the bounded homotopy category, where it is not a prerequisite of the present computation.

2.1step 1.1L1L2L3L5

The kernel attached to the internal shift by the classification of [L1] is $A {r}$ : the bimodule attached to a tensor functor $T_{M}$ is recovered as $T_{M} (A) = M \otimes_{A} A ≅ M$ , and here $T_{A {r}} (A) = A {r} \otimes_{A} A ≅ A {r}$ by step 1.1 and [L3]; under the identification $T_{A {r}} ≅ {r}$ of step 1.1 the comparisons $θ^{A {r}}$ are the identity on elementary tensors [L2], which is the canonical comparison of the internal shift [L5].

3.1step 1.1step 1.2step 1.3step 2.1∎

Collecting steps 1.1, 1.2, 1.3 and 2.1: the internal shift is a $k$ -linear right exact coproduct-preserving coherent functor, it is naturally isomorphic to $T_{A {r}}$ , its kernel is the shifted regular bimodule $A {r}$ with the canonical comparisons, it is the identity exactly for $r = 0$ , and it is a different operation from the cochain shift $[1]$ ; all identifications used are the canonical ones, so no choice is made.

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