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The degree-zero projection is exact and cocontinuous but not a graded tensor functor

Statement refuted

Let $k$ be a field. An additive $k$ -linear functor $F : {GrMod}_{0} (k) \to {GrMod}_{0} (k)$ that is exact and preserves all coproducts and satisfies $F (k) = k$ need not be a graded tensor functor: it need not be isomorphic to $k \otimes_{k} -$ , and it need not admit coherent shift-comparison data at all. So right exactness, coproduct preservation and even the value on the regular module do not by themselves put a graded functor into the class classified by Graded Eilenberg-Watts theorem with coherent shifts; the shift-coherence hypothesis of Coherently shift-compatible functors and natural transformations is load-bearing.

Let $k$ be a field and let $A = B = k$ be the graded $k$ -algebra concentrated in degree $0$ ( $k_{0} = k$ and $k_{d} = 0$ for $d \neq 0$ ). Define $F : {GrMod}_{0} (k) ⟶ {GrMod}_{0} (k), F (X) : = X_{0},$ where $X_{0}$ is placed in degree $0$ (so $F (X)_{0} = X_{0}$ and $F (X)_{d} = 0$ for $d \neq 0$ ), with the evident $k$ -action. Then:

$F$ is $k$ -linear, exact (left and right exact) and preserves every coproduct, so it satisfies every hypothesis of the graded Eilenberg-Watts class except shift coherence;
$F (k) = k$ , so its value at the regular module coincides with that of the tensor functor $T_{k} = k \otimes_{k} -$ , which is the identity functor up to the unit isomorphism;
nevertheless $F$ is not isomorphic as a functor — hence not coherently isomorphic — to $T_{k}$ : on the internal shift $k {1}$ one has $F (k {1}) = (k {1})_{0} = k_{- 1} = 0$ , whereas $T_{k} (k {1}) = k \otimes_{k} k {1} ≅ k {1} \neq 0$ ;
consequently $F$ admits no coherent shift-comparison data at all: coherence would require a degree-zero isomorphism $F (k {1}) \to F (k) {1} = k {1}$ , impossible because $F (k {1}) = 0$ .

Hence the shift-coherence hypothesis in Coherently shift-compatible functors and natural transformations and Graded Eilenberg-Watts theorem with coherent shifts is load-bearing: right exactness, coproduct preservation and even the value $F (k) = k$ do not classify graded functors by graded bimodules. The witness is defined without choice. Hazrat's Example 2.3.9 distinguishes equivalences of graded module categories from shift-commuting equivalences; his Remark 2.3.4 instead concerns natural transformations. The present witness directly proves failure of shift compatibility without asserting that a graded tensor functor can fail it.

Facts & Assumptions

Given: A field $k$ , the graded $k$ -algebra $k$ concentrated in degree $0$ , graded left $k$ -modules $X, Y$ and a degree-zero $k$ -linear map $u : X \to Y$ , a family $(X_{i})_{i \in I}$ of graded left $k$ -modules, and the functor $F (X) = X_{0}$ , with $X_{0}$ placed in degree $0$ .

[L1]

Graded modules over the graded $k$ -algebra $k$ have homogeneous pieces $X_{d}$ with $X = ⨁_{d} X_{d}$ , degree-zero maps are the $k$ -linear maps with $u (X_{d}) \subseteq Y_{d}$ , and the internal shift has pieces $(X {r})_{d} = X_{d - r}$ (Associative graded algebras, bimodules, and internal shifts).

[L2]

The internal shift is an autoequivalence with ${0} = i d$ acting as the identity on underlying sets, and it preserves degreewise coproducts, kernels and cokernels (Internal shifts are autoequivalences and commute with the graded tensor product).

[L3]

Kernels, images, cokernels and finite biproducts in ${GrMod}_{0} (k)$ are computed in each homogeneous degree, and exactness is equivalent to exactness degreewise (Graded modules with degree-zero maps form an abelian category).

[L4]

The degreewise direct sum is the coproduct in ${GrMod}_{0} (k)$ , with $(⨁_{i} X_{i})_{d} = ⨁_{i} (X_{i})_{d}$ (Degreewise direct sums and homogeneous free covers in graded modules).

[L5]

The graded tensor functor $T_{k} = k \otimes_{k} -$ is $k$ -linear, right exact, coproduct preserving and coherently shift-compatible (Graded tensor functors are k-linear, right exact, coproduct preserving and shift-coherent).

[L6]

An additive functor preserves all small colimits if and only if it is right exact and coproduct preserving (Colimits of a graded additive functor equal right exactness plus coproduct preservation).

[L7]

A functor is right exact when it preserves every finite colimit existing in its source, and cocontinuous when it preserves all small colimits (Left exact and right exact functors, Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors).

[L8]

The tensor-unit map $k \otimes_{k} X \to X$ , $c \otimes x \mapsto c x$ , is a natural isomorphism (The regular module is a tensor unit: $R \otimes_{R} N ≅ N$ and $M \otimes_{R} R ≅ M$ ).

[L9]

A field has $0 \neq 1$ and a commutative multiplication (Field).

[L10]

A functor is additive when it induces group homomorphisms on hom-groups, equivalently $F (f + g) = F f + F g$ , and $k$ -linear when those maps are $k$ -linear (Additive functor, k-linear categories and k-linear functors).

[L11]

A functor between abelian categories is exact when it is additive, left exact and right exact (Exact functor between abelian categories), and an additive functor between abelian categories is exact if and only if it preserves kernels and cokernels (An additive functor is exact exactly when it preserves kernels and cokernels).

[L12]

The graded Eilenberg-Watts class consists of the $k$ -linear right exact coproduct-preserving coherently shift-compatible functors, and coherently shift-compatible functors carry isomorphisms $θ_{X, r} : F (X {r}) \to F (X) {r}$ (Graded Eilenberg-Watts theorem with coherent shifts, Coherently shift-compatible functors and natural transformations).

Proof

Given: A field $k$ , the graded $k$ -algebra $k$ concentrated in degree $0$ , the functor $F (X) = X_{0}$ in degree $0$ , a degree-zero $k$ -linear $u : X \to Y$ , and a family $(X_{i})_{i \in I}$ .

Proof technique: direct.

1.1L1

$F$ is a functor: for a degree-zero $k$ -linear $u : X \to Y$ one has $u (X_{0}) \subseteq Y_{0}$ , so the restriction $F (u) : = u ∣_{X_{0}}$ is a degree-zero $k$ -linear map $F (X) \to F (Y)$ ; restrictions preserve identities and composites, so $F$ is a functor.

2.1step 1.1L10

$F$ is additive and $k$ -linear: on hom-groups the assignment $u \mapsto u ∣_{X_{0}}$ satisfies $(u + v) ∣_{X_{0}} = u ∣_{X_{0}} + v ∣_{X_{0}}$ and $(λ u) ∣_{X_{0}} = λ u ∣_{X_{0}}$ , so the induced maps are $k$ -linear [L10].

2.2step 1.1L4

$F$ preserves every coproduct: the identity map gives $F (⨁_{i} X_{i}) = (⨁_{i} X_{i})_{0} = ⨁_{i} (X_{i})_{0} = ⨁_{i} F (X_{i})$ with the same coordinate inclusions, so the canonical comparison is an isomorphism [L4].

2.3step 1.1L1L2L8L9

On the regular module $F (k) = k_{0} = k$ ; on the internal shift $F (k {1}) = (k {1})_{0} = k_{- 1} = 0$ because $k$ is concentrated in degree $0$ [L1], while $T_{k} (k {1}) = k \otimes_{k} k {1} ≅ k {1} \neq 0$ by the unit isomorphism [L8], since $(k {1})_{1} = k_{0} = k \neq 0$ ; a natural isomorphism $F ≅ T_{k}$ would therefore induce an isomorphism $0 \to k {1}$ , which is impossible [L9].

3.1step 2.1L3L11

$F$ preserves kernels and cokernels: for degree-zero $u : X \to Y$ the degreewise descriptions give $F (\ker u) = (\ker u)_{0} = \ker (u ∣_{X_{0}}) = \ker (F u)$ and $F (coker u) = (coker u)_{0} = Y_{0} / u (X_{0}) = coker (F u)$ with the induced maps [L3], so $F$ preserves the kernel and cokernel of every morphism; hence $F$ is exact, and in particular left and right exact, by [L11].

4.1step 2.1step 2.2step 2.3step 3.1L2L5L6L7L9L12∎

No coherent shift-comparison data exist for $F$ : by [L12] such data would include a degree-zero isomorphism $θ_{k, 1} : F (k {1}) \to F (k) {1} = k {1}$ , but $F (k {1}) = 0$ while $(k {1})_{1} = k \neq 0$ by steps 2.3 and [L2, L9], so no isomorphism exists. More generally, if $F ≅ T_{M}$ for a graded $(k, k)$ -bimodule $M$ , evaluation at $k$ and [L8] give $M ≅ F (k) = k$ , whereas $T_{M} (k {1}) ≅ M {1} \neq 0$ by [L5], contradicting $F (k {1}) = 0$ . With steps 2.1, 2.2 and 3.1 the functor $F$ is additive, right exact and coproduct preserving, hence cocontinuous by [L6, L7]; therefore $F$ is a $k$ -linear exact cocontinuous functor with $F (k) = k$ that is not isomorphic to any graded tensor functor [L5] and does not lie in the coherent class, and the witness uses no choice.

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