On adjoint functors of the Heller operator

Matthias Künzer

December 16, 2011

Abstract

Given an abelian category A with enough projectives, we can form its stable category \underline{A} := A/Proj(A). The Heller operator Ω : \underline{A} \overline{}\underline{A} is characterised on an object X by a choice of a short exact sequence ΩX [figure] P \overline{}X in A with P projective. If A is Frobenius, then Ω is an equivalence, hence has a left and a right adjoint. If A is hereditary, then Ω is zero, hence has a left and a right adjoint. In general, Ω is neither an equivalence nor zero. In the examples we have calculated via Magma, it has a left adjoint, but in general not a right adjoint. If A has projective covers, then Ω preserves monomorphisms; this would also follow from Ω having a left adjoint. I do not know an example where Ω does not have a left adjoint.

Contents

0 Introduction  2
0.1 The question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.2 Monomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.3 Construction of a left adjoint to the Heller operator Ω . . . . . . . . . . . . . . . . . . . . . . . . 2
0.4 Two counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.6 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 The Heller operator Ω  3
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Preservation of monomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Relative projective covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Examples for adjoints of the Heller operator Ω  6
2.1 Example of a left adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 A list of indecomposables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Construction of a left adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Another example of a left adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 A list of indecomposables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Construction of a left adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Further examples of left adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Counterexample: no right adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

MSC 2010: 18E30.

0 Introduction

0.1 The question

Let E be an exact category. Let Proj(E) be its full additive subcategory of relative projectives, i.e. for P ∈ Ob E we have P ∈ Ob Proj(E) if and only if _{E}(P, −) maps pure short exact sequences of E to short exact sequences of abelian groups. Suppose that E has enough relative projectives, i.e. suppose that for any X ∈ Ob E, there exists a pure epimorphism P \overline{}X in E with P ∈ Ob Proj(E).

Write \underline{E} := E/ Proj(E). The Heller operator Ω : \underline{E} \overline{}\underline{E} is characterised on a given X ∈ Ob E by a choice of a pure short exact sequence

ΩX [figure] P \overline{}X

in E with P relatively projective. This then is extended to morphisms.

We ask whether Ω has a left adjoint; cf. Question 1. I do not know a counterexample.

If E is a Frobenius category, then Ω is an equivalence, thus has both a left and a right adjoint.

If E is hereditary, i.e. if Ω ≃ 0, then Ω has both a left and a right adjoint, viz. 0.

0.2 Monomorphisms

If a functor has a left adjoint, then it preserves monomorphisms. So first of all, we ask whether the Heller operator Ω : \underline{E} \overline{}\underline{E} preserves monomorphisms.

It turns out that if E is weakly idempotent complete and has relative projective covers in the sense of §1.3, then Ω maps monomorphisms even to coretractions; cf. Proposition 4.

0.3 Construction of a left adjoint to the Heller operator Ω

Let p ∈ [2, 997] be a prime. Let R := F_{p}[X] and π := X.

Let A := (R/π^{3})(e ^{\overline{}^{a}} f) ≃ (\begin{matrix}R/π^{3} & R/π^{3} \\ 0 & R/π^{3} \end{matrix}). Let E := mod-A.

Using Magma [1], we construct a left adjoint S : \underline{E} \overline{}\underline{E} to Ω. We do so likewise for certain factor rings of A. Cf. Propositions 6, 9 and 11.

Let now k be a field, R := k[X] and π := X. Let n ⩾ 1. An (R/π^{n})(e ^{\overline{}^{a}} f)-module is given by a morphism X ^{\overline{}^{f}} Y in mod-(R/π^{n}). The full subcategory of mod-(R/π^{n})(e ^{\overline{}^{a}} f) consisting of injective morphisms X ^{\overline{}^{f}} Y as modules has been intensely studied; it is of finite type if n ⩽ 5, tame if n = 6, wild if n ⩾ 7; cf. [7, (0.1), (0.6)].

0.4 Two counterexamples

The functor Ω : \underline{E} \overline{}\underline{E} does not have a right adjoint in general; cf. Remark 13.

Provided S ⊣ Ω exists, the composite Ω∘S : \underline{E} \overline{}\underline{E} is not idempotent in general; cf. Remark 12.

0.5 Acknowledgements

Sebastian Thomas asked whether there exists a category with set of weak equivalences (C, W) that carries the structure of a Brown fibration category, but whose Ω on C[W^{−1}] (^{1}) does not have a left adjoint. In our exact category context, this is Question 1. I thank Steffen König for help with monomorphisms, cf. §1.2. I thank Sebastian Thomas and Markus Kirschmer for help with Magma. I thank Markus Schmidmeier for help with mod-(R/π^{n})(e ^{\overline{}^{a}} f).

0.6 Notations and conventions

• Given a, b ∈ Z, we write [a, b] := { z ∈ Z : a ⩽ z ⩽ b }.

• Composition of morphisms is written naturally, ( ^{\overline{}^{a}} ^{\overline{}^{b}} ) = ^{\overline{}^{ab}} = ^{\overline{}^{a·b}} .

Composition of functors is written traditionally, ( ^{\overline{}^{F}} ^{\overline{}^{G}} ) = ^{\overline{}^{G∘F}} .

• In a category C, given X, Y ∈ Ob C, we write _{C}(X, Y) for the set of morphisms from X to Y.

• Given an isomorphism f, we write f^{−} for its inverse.

• In an additive category, a morphism of the form X ^{\overline{}^{(^{1 0} )}} X ⊕ Y, or isomorphic to such a morphism, is called split monomorphic; a morphism of the form X ⊕ Y ^{\overline{}^{( \begin{matrix}1 \\ 0 \end{matrix} )}} X, or isomorphic to such a morphism, is called split epimorphic.

• In exact categories, pure monomorphisms are denoted by [figure] , pure epimorphisms by \overline{}and pure squares, i.e. bicartesian squares with pure short exact diagonal sequence, by a box □ in the diagram.

• Given a ring A, an A-module is a finitely generated right A-module.

• Given a commutative ring A and a ∈ A, we often write A/a := A/(a) = A/aA.

• Given a noetherian ring A, we write \underline{mod}-A := \underline{mod-A} for the factor category of mod-A modulo the full additive subcategory of projectives. So in the language of §1.1 below, we consider the abelian category mod-A as an exact category with all short exact sequences declared to be pure and write \underline{mod}-A for its classical stable category.

1 The Heller operator Ω

1.1 Notation

Let E be an exact category in the sense of Quillen [5, p. 99] with enough relative projectives. We will use the notation of [4, §A.2] concerning pure short exact sequences, pure monomorphisms and pure epimorphisms. Let Proj(E) ⊆ E denote the full subcategory of relative projectives. Let

\underline{E} := E/ Proj(E)

denote the classical stable category of E. The residue class functor shall be denoted by

\begin{matrix}E & \overline{} & \underline{E} \\ (X ^{\overline{}^{f}} Y) & \overline{} & (X ^{\overline{}^{[f]}} Y) . \end{matrix}

1Cf. also [6, p. 210].

For each X ∈ Ob E, we choose a pure short exact sequence

(∗)  ΩX [figure: i_{X}] PX ^{\overline{}^{p_{X}}} X

with PX relatively projective. Let the Heller operator [3]

Ω : \underline{E} \overline{}\underline{E}

be defined on the objects by the choice just made. Suppose given a morphism X ^{\overline{}^{[f]}} Y in \underline{E}. Choose a morphism

[figure: ΩX; •; i_{X}; f′; PX; p_{X}; f̂; X; f; ΩY; •; i_{Y}; PY; p_{Y}; Y]

of pure short exact sequences. Let

Ω[f] := [f′] .

Different choices of pure short exact sequences (∗) yield mutually isomorphic Heller operators.

Question 1 Does Ω have a left adjoint?

I do not know a counterexample.

1.2 Preservation of monomorphisms

If Ω : \underline{E} \overline{}\underline{E} has a left adjoint, then it preserves monomorphisms. So if, for some E, the functor Ω did not preserve monomorphisms, then Ω could not have a left adjoint. Under certain finiteness assumptions, however, we will show that Ω maps monomorphisms to coretractions, so in particular to monomorphisms. This is to be compared to the case of E being Frobenius, where in the triangulated category \underline{E} all monomorphisms are split.

Lemma 2 Suppose that for X ∈ Ob E and for s ∈ _{E}(PX, PX) such that sp_{X} = p_{X} , the endomorphism s is an isomorphism.

Suppose given X ^{\overline{}^{f}} Y in E.

If [f] is a monomorphism, then Ω[f] is a coretraction.

In particular, Ω preserves monomorphisms.

Proof. Choose a morphism of pure short exact sequences as shown below. Insert a pullback (T, X, PY, Y) and the induced morphism PX ^{\overline{}^{v}} T, having vg = f̂ and vq = p_{X} . Insert a kernel j of q with jg = i_{Y} .

[figure: ΩX; •; i_{X}; f′; PX; p_{X}; f̂; v; X; f; T; q; g; ΩY; •; i_{Y}; •; j; PY; p_{Y}; Y]

We have f′j = i_{X}v, since f′jq = 0 = i_{X}p_{X} = i_{X}vq and f′jg = f′i_{Y} = i_{X} f̂ = i_{X}vg.

We have [q][f] = [gp_{Y}] = 0. Since [f] is monomorphic, we infer that [q] = 0. Hence there exists T ^{\overline{}^{u}} PX such that up_{X} = q. On the kernels, we obtain ΩY ^{\overline{}^{u′}} ΩX such that u′i_{X} = ju.

[figure: ΩX; •; i_{X}; f′; PX; p_{X}; f̂; v; X; f; T; q; g; u; ΩY; •; i_{Y}; •; j; u′; PY; p_{Y}; Y]

We have vup_{X} = vq = p_{X} . Hence vu is an isomorphism by assumption.

We obtain f′u′i_{X} = f′ju = i_{X}vu. So (f′u′, vu, id_{X}) is a morphism of pure short exact sequences. Hence f′u′ is an isomorphism. Thus f′ is a coretraction. We conclude that Ω[f] = [f′] is a coretraction.

1.3 Relative projective covers

Suppose E to be weakly idempotent complete; cf. [2, Def. 7.2].

A morphism S ^{\overline{}^{i}} M in E is called small if in each pure square

[figure: A; □; T; t; S; i; M]

in E, the morphism T ^{\overline{}^{t}} M is purely epimorphic; cf. [8, Def. 2.8.30]. In other words, i is small iff (\begin{matrix}i \\ t \end{matrix}) being purely epimorphic entails t being purely epimorphic for each morphism t with same target as i. E.g. i = 0 is small, for (\begin{matrix}0 \\ t \end{matrix}) = (\begin{matrix}0 \\ 1 \end{matrix}) t is purely epimorphic only if t is; cf. [2, Prop. 2.16].

If S [figure: i] M is small and split monomorphic, then there exists

[figure: 0; □; S′; •; i′; S; •; i; M ,]

forcing i′ to be an isomorphism and thus S to be isomorphic to 0.

Given S̃ ^{\overline{}^{j}} S ^{\overline{}^{i}} M with S ^{\overline{}^{i}} M small, then S̃ ^{\overline{}^{ji}} M is small. In fact, given t such that (\begin{matrix}ji \\ t \end{matrix}) is a pure epimorphism, the factorisation (\begin{matrix}ji \\ t \end{matrix}) = (\begin{matrix}j & 0 \\ 0 & 1 \end{matrix}) (\begin{matrix}i \\ t \end{matrix}) shows that (\begin{matrix}i \\ t \end{matrix}) is a pure epimorphism; cf. [2, Cor. 7.7]. Thus t is purely epimorphic by smallness of i.

A relative projective cover of X ∈ Ob E is a pure epimorphism P ^{\overline{}^{p}} X in E such that P is relatively projective and such that Kern p [figure] P is small; cf. [8, 2.8.31].

We say that E has relative projective covers if for each X ∈ Ob E, there exists a relative projective cover P ^{\overline{}^{p}} X.

Lemma 3 Suppose given a relative projective cover P ^{\overline{}^{p}} X in E. Suppose given P ^{\overline{}^{s}} P such that sp = p. Then s is an isomorphism.

Proof. We complete to a pure short exact sequence K [figure: k] P ^{\overline{}^{p}} X. We obtain a morphism

[figure: K; •; k; g; □; P; p; s; X; K; •; k; P; p; X .]

of pure short exact sequences. Since the left hand side quadrangle is a pure square, we conclude that s is purely epimorphic by smallness of K [figure: k] P. Hence s is split epimorphic by relative projectivity of P ; cf. [2, Rem. 7.4]. Let L [figure: ℓ] P be a kernel of s. Since ℓ factors over the small morphism k, it is small as well. Since ℓ is split monomorphic, we have L ≃ 0. Thus s is an isomorphism.

Proposition 4 Suppose that the exact category E is weakly idempotent complete and has relative projective covers.

Then Ω : \underline{E} \overline{}\underline{E} maps each monomorphism to a coretraction. In particular, Ω preserves monomorphisms.

Proof. We may use relative projective covers to construct Ω in (∗). Then Lemma 3 allows us to apply Lemma 2.

2 Examples for adjoints of the Heller operator Ω

Let R be a principal ideal domain, with a maximal ideal generated by an element π ∈ R.

Let

A := (R/π^{3})(e ^{\overline{}^{a}} f) .

I.e. A is the path algebra of e ^{\overline{}^{a}} f over the ground ring R/π^{3}. It has primitive idempotents e and f, and a ∈ eAf.

An object in mod-A is given by a morphism X \overline{}Y in mod-(R/π^{3}). A morphism in mod-A is given by a commutative quadrangle in mod-(R/π^{3}).

2.1 Example of a left adjoint

2.1.1 A list of indecomposables

Define the following objects in mod-A.

P_{1} := (R/π^{3} ^{^{1}\overline{}}R/π^{3})  P_{2} := (0 \overline{}R/π^{3})
X_{1} := (R/π ^{^{1}\overline{}}R/π)  X_{14} := (R/π^{3} ^{\overline{}^{π^{2}}} R/π^{3})
X_{2} := (R/π^{2} ^{^{1}\overline{}}R/π^{2})  X_{15} := (R/π^{3}  \overline{}0)
X_{3} := (R/π^{2} ^{^{1}\overline{}}R/π)  X_{16} := (R/π^{2} ⊕ R/π^{3} ^{\overline{}^{(\begin{matrix}1 & π \\ 1 & 0 \end{matrix} )}} R/π ⊕ R/π^{3})
X_{4} := (R/π^{3} ^{^{1}\overline{}}R/π^{2})  X_{17} := (R/π ⊕ R/π^{3} ^{\overline{}^{(\begin{matrix}π & π^{2} \\ 1 & 0 \end{matrix} )}} R/π^{2} ⊕ R/π^{3})
X_{5} := (R/π^{2} ^{\overline{}^{π}} R/π^{3})  X_{18} := (R/π ⊕ R/π^{3} ^{\overline{}^{(\begin{matrix}0 & π^{2} \\ 1 & π \end{matrix} )}} R/π ⊕ R/π^{3})
X_{6} := (R/π ^{\overline{}^{π}} R/π^{2})  X_{19} := (R/π ^{\overline{}^{π^{2}}} R/π^{3})
X_{7} := (R/π^{2} ^{\overline{}^{(^{1 π})}} R/π ⊕ R/π^{3}) X_{20} := (R/π^{3} ^{^{1}\overline{}}R/π)
X_{8} := (R/π ⊕ R/π^{3} ^{\overline{}^{(\begin{matrix}π \\ 1 \end{matrix} )}} R/π^{2}) X_{21} := (R/π \overline{}0)
X_{9} := (R/π^{3} ^{\overline{}^{π}} R/π^{3})  X_{22} := (0 \overline{}R/π)
X_{10} := (R/π^{2}  \overline{}0)  X_{23} := (R/π^{3} ^{\overline{}^{π}} R/π^{2})
X_{11} := (0 \overline{}R/π^{2})  X_{24} := (R/π^{2} ^{\overline{}^{π^{2}}} R/π^{3})
X_{12} := (R/π ⊕ R/π^{3} ^{\overline{}^{(\begin{matrix}π^{2} \\ π \end{matrix} )}} R/π^{3}) X_{25} := (R/π^{2} ^{\overline{}^{π}} R/π^{2})
X_{13} := (R/π^{3} ^{\overline{}^{(^{1 π})}} R/π ⊕ R/π^{3})

A matrix inspection yields the

Lemma 5

(1) For each projective indecomposable A-module P, there exists a unique i ∈ [1, 2] such that P ≃ P_{i}.

(2) For each nonprojective indecomposable A-module X, there exists a unique i ∈ [1, 25] such that X ≃ X_{i}.

2.1.2 Construction of a left adjoint

Our aim in this section is to computationally verify the

Proposition 6 Suppose given a prime p ∈ [2, 997]. Suppose that R = F_{p}[X] and π = X. Then the Heller operator Ω : \underline{mod}-A \overline{}\underline{mod}-A has a left adjoint.

For ease of Magma input, we have used that

A ≃ F_{p}( [figure: e; a; u; f; v] ) /(u^{3}, v^{3}, ua − av)

as F_{p}-algebras. To reduce the calculation of this adjoint functor to the proof of the representability of certain functors, we use

Lemma 7 ([9, 16.4.5, 4.5.1]) Suppose given categories C and D and a functor C ^{\overline{}^{F}} D. Suppose that

\begin{matrix}C & \overline{}^{^{D}(Y,F (−))} & Sets \\ (X ^{\overline{}^{x}} X′) & \overline{} & (_{D}(Y, F X) ^{\overline{}^{^{D}(Y, F x)}} _{D}(Y, F X′)) \end{matrix}

is representable for each Y ∈ Ob D. Then F has a left adjoint. More precisely, given a map Ob C ^{\overline{}^{γ}} Ob D and an isomorphism

D(Y, F (−)) \overline{}^{φ_{Y}}_{∼} _{C}(Y γ, −)

for Y ∈ Ob D, there exists a left adjoint C ^{\overline{}^{G}} D to F, i.e. G ⊣ F, such that, writing

εY := (1_{Y γ})(φ_{Y} (Y γ))^{−} : Y \overline{}F (Y γ)

for Y ∈ Ob D, we have

G(Y ⟶^{y} Y′) = (Y γ ⟶^{(y · εY′)(φ_{Y} (Y′γ))} Y′γ)

for Y ^{\overline{}^{y}} Y in D.

Thus in order to construct the left adjoint to Ω on \underline{mod}-A, it suffices to show that the functor \underline{mod}-A(X_{i} , Ω(−)) is representable i ∈ [1, 25]. We shall do so by an actual construction of an isotransformation from a Hom-functor. Suppose given i ∈ [1, 25]. Such an isotransformation is necessarily of the form

\underline{mod}-A(SX_{i} , Y) \overline{}_{\underline{mod}-A}(X_{i} , ΩY)
[f] \overline{}[ϵ_{i}] · Ω[f]

for some SX_{i} ∈ Ob \underline{mod}-A and some A-linear map ϵ_{i} : X_{i} \overline{}ΩSX_{i}, where Y ∈ Ob \underline{mod}-A.

So it suffices to find an A-module SX_{i} and an A-linear map ϵ_{i} : X_{i} \overline{}ΩSX_{i} such that the induced map

(∗∗) \underline{mod}-A(SX_{i} , X_{j}) \overline{}_{\underline{mod}-A}(X_{i} , ΩX_{j}) [f] \overline{}[ϵ_{i}] · Ω[f]

is an isomorphism for j ∈ [1, 25].

In particular, given an automorphism α of X_{i} in \underline{mod}-A, an automorphism β of SX_{i} in \underline{mod}-A and a valid such morphism [ϵ_{i}] , then α · [ϵ_{i}] · Ωβ is another valid such morphism, sometimes of a simpler shape.

To show that a guess for SX_{i} is in fact the sought-for representing object, we make use of the fact that _{\underline{mod}-A}(X_{i} , ΩSX_{i}) is finite, so that we have only a finite set of candidates for ϵ_{i}. Then to check whether the candidate-induced maps (∗∗) are isomorphisms, is also feasible via Magma, using in particular its commands ProjectiveCover, AHom and PHom; cf. [1].

We obtain

SX_{1} = X_{2}  ΩSX_{1} = X_{1}  SX_{14} = X_{5}  ΩSX_{14} = X_{19}
SX_{2} = X_{1}  ΩSX_{2} = X_{2}  SX_{15} = 0  ΩSX_{15} = 0
SX_{3} = X_{2}  ΩSX_{3} = X_{1}  SX_{16} = X_{2} ⊕ X_{19}  ΩSX_{16} = X_{1} ⊕ X_{5}
SX_{4} = X_{1}  ΩSX_{4} = X_{2}  SX_{17} = X_{1} ⊕ X_{5}  ΩSX_{17} = X_{2} ⊕ X_{19}
SX_{5} = X_{19}  ΩSX_{5} = X_{5}  SX_{18} = X_{6}  ΩSX_{18} = X_{7}
SX_{6} = X_{7}  ΩSX_{6} = X_{6}  SX_{19} = X_{5}  ΩSX_{19} = X_{19}
SX_{7} = X_{6}  ΩSX_{7} = X_{7}  SX_{20} = X_{2}  ΩSX_{20} = X_{1}
SX_{8} = X_{1}  ΩSX_{8} = X_{2}  SX_{21} = 0  ΩSX_{21} = 0
SX_{9} = X_{19}  ΩSX_{9} = X_{5}  SX_{22} = X_{11}  ΩSX_{22} = X_{22}
SX_{10} = 0  ΩSX_{10} = 0  SX_{23} = X_{7}  ΩSX_{23} = X_{6}
SX_{11} = X_{22}  ΩSX_{11} = X_{11}  SX_{24} = X_{5}  ΩSX_{24} = X_{19}
SX_{12} = X_{19}  ΩSX_{12} = X_{5}  SX_{25} = X_{7}  ΩSX_{25} = X_{6}
SX_{13} = X_{6}  ΩSX_{13} = X_{7}

and

[figure: (; X_{1}; ϵ_{1}; ΩSX_{1}; ); =; (; R/π; 1; 1; R/π; 1; R/π; 1; R/π; ); (; X_{14}; ϵ_{14}; ΩSX_{14}; ); =; (; R/π^{3}; π^{2}; 1; R/π^{3}; 1; R/π; π^{2}; R/π^{3}; ); (; X_{2}; ϵ_{2}; ΩSX_{2}; ); =; (; R/π^{2}; 1; 1; R/π^{2}; 1; R/π^{2}; 1; R/π^{2}; ); (; X_{15}; ϵ_{15}; ΩSX_{15}; ); =; (; R/π^{3}; 0; 0; 0; ); (; X_{3}; ϵ_{3}; ΩSX_{3}; ); =; (; R/π^{2}; 1; 1; R/π; 1; R/π; 1; R/π; ); (; X_{16}; ϵ_{16}; ΩSX_{16}; ); =; (; R/π^{2} ⊕ R/π^{3}; ^{( \begin{matrix}1 & π \\ 1 & 0 \end{matrix} )}; ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ); R/π ⊕ R/π^{3}; ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ); R/π ⊕ R/π^{2}; _{( \begin{matrix}1 & 0 \\ 0 & π \end{matrix} )}; R/π ⊕ R/π^{3}; )]

[figure: (; X_{4}; ϵ_{4}; ΩSX_{4}; ); =; (; R/π^{3}; 1; 1; R/π^{2}; 1; R/π^{2}; 1; R/π^{2}; ); (; X_{17}; ϵ_{17}; ΩSX_{17}; ); =; (; R/π ⊕ R/π^{3}; ^{( \begin{matrix}π & π^{2} \\ 1 & 0 \end{matrix} )}; ( \begin{matrix}π & 1 \\ 1 & 0 \end{matrix} ); R/π^{2} ⊕ R/π^{3}; ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ); R/π^{2} ⊕ R/π; ( \begin{matrix}1 & 0 \\ 0 & π^{2} \end{matrix} ); R/π^{2} ⊕ R/π^{3}; )]

[figure: (; X_{5}; ϵ_{5}; ΩSX_{5}; ); =; (; R/π^{2}; π; 1; R/π^{3}; 1; R/π^{2}; π; R/π^{3}; ); (; X_{18}; ϵ_{18}; ΩSX_{18}; ); =; (; R/π ⊕ R/π^{3}; ^{( \begin{matrix}0 & π^{2} \\ 1 & π \end{matrix} )}; ( \begin{matrix}π \\ 1 \end{matrix} ); R/π ⊕ R/π^{3}; ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ); R/π^{2}; _{(_{1 π})}; R/π ⊕ R/π^{3}; )]

[figure: (; X_{6}; ϵ_{6}; ΩSX_{6}; ); =; (; R/π; π; 1; R/π^{2}; 1; R/π; π; R/π^{2}; ); (; X_{19}; ϵ_{19}; ΩSX_{19}; ); =; (; R/π; π^{2}; 1; R/π^{3}; 1; R/π; π^{2}; R/π^{3}; )]

[figure: (; X_{7}; ϵ_{7}; ΩSX_{7}; ); =; (; R/π^{2}; ^{(^{1 π})}; 1; R/π ⊕ R/π^{3}; ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ); R/π^{2}; _{(_{1 π})}; R/π ⊕ R/π^{3}; ); (; X_{20}; ϵ_{20}; ΩSX_{20}; ); =; (; R/π^{3}; 1; 1; R/π; 1; R/π; 1; R/π; )]

[figure: (; X_{8}; ϵ_{8}; ΩSX_{8}; ); =; (; R/π ⊕ R/π^{3}; ^{( \begin{matrix}π \\ 1 \end{matrix} )}; ( \begin{matrix}π \\ 1 \end{matrix} ); R/π^{2}; 1; R/π^{2}; 1; R/π^{2}; ); (; X_{21}; ϵ_{21}; ΩSX_{21}; ); =; (; R/π; 0; 0; 0; )]

[figure: (; X_{9}; ϵ_{9}; ΩSX_{9}; ); =; (; R/π^{3}; π; 1; R/π^{3}; 1; R/π^{2}; π; R/π^{3}; ); (; X_{22}; ϵ_{22}; ΩSX_{22}; ); =; (; 0; R/π; 1; 0; R/π; )]
[figure: (; X_{10}; ϵ_{10}; ΩSX_{10}; ); =; (; R/π^{2}; 0; 0; 0; ); (; X_{23}; ϵ_{23}; ΩSX_{23}; ); =; (; R/π^{3}; π; 1; R/π^{2}; 1; R/π; π; R/π^{2}; )]

[figure: (; X_{11}; ϵ_{11}; ΩSX_{11}; ); =; (; 0; R/π^{2}; 1; 0; R/π^{2}; ); (; X_{24}; ϵ_{24}; ΩSX_{24}; ); =; (; R/π^{2}; π^{2}; 1; R/π^{3}; 1; R/π; π^{2}; R/π^{3}; )]

( \xymatrix{X_{12} \ar[d]_{ϵ_{12}} \\ ΩSX_{12}} ) = ( \xymatrix{R/π ⊕ R/π^{3} \ar[r]^{^{( \begin{matrix}π^{2} \\ π \end{matrix} )}} \ar[d]_{( \begin{matrix}π \\ 1 \end{matrix} )} & R/π^{3} \ar[d]^{1} \\ R/π^{2} \ar[r]^{π} & R/π^{3}} )  ( \xymatrix{X_{25} \ar[d]_{ϵ_{25}} \\ ΩSX_{25}} ) = ( \xymatrix{R/π^{2} \ar[r]^{π} \ar[d]_{1} & R/π^{2} \ar[d]^{1} \\ R/π \ar[r]^{π} & R/π^{2}} ) .

[figure: (; X_{13}; ϵ_{13}; ΩSX_{13}; ); =; (; R/π^{3}; ^{(^{1 π})}; 1; R/π ⊕ R/π^{3}; ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ); R/π^{2}; _{(_{1 π})}; R/π ⊕ R/π^{3}; )]

Remark 8 Keep the assumptions of Proposition 6. We have (Ω ∘ S)^{2} Y ≃ (Ω ∘ S)Y for Y ∈ Ob \underline{mod}-A.

The unit of the adjunction S ⊣ Ω at an A-module X ^{\overline{}^{f}} Y is represented by a factorisation

[figure: X; f; f̄; Y; I_{f}; •; ḟ; Y]

over an image I_{f} of the module-defining morphism f. I do not know why.

2.2 Another example of a left adjoint

Recall that R is a principal ideal domain, with a maximal ideal generated by an element π ∈ R.

2.2.1 A list of indecomposables

Let

B := A/(π^{2}a) = (R/π^{3})(e ^{\overline{}^{a}} f)/(π^{2}a) .

Indecomposable nonprojective B-modules become indecomposable nonprojective A-modules via restriction along the residue class map A \overline{}B. We list the 24 representatives of isoclasses of indecomposable nonprojective B-modules in the numbering used in §2.1.1 as follows.

\begin{matrix}X_{1} , X_{2} , X_{3} , X_{5} , X_{6} , X_{7} , X_{8} , X_{9} , X_{10} , X_{11} , X_{12} , X_{13} , X_{14} , X_{15} , X_{16} , \\ X_{17} , X_{18} , X_{19} , X_{20} , X_{21} , X_{22} , X_{23} , X_{24} , X_{25} \end{matrix}

2.2.2 Construction of a left adjoint

Our aim in this section is to computationally verify the

Proposition 9 Suppose given a prime p ∈ [2, 997]. Suppose that R = F_{p}[X] and π = X. Then the Heller operator Ω : \underline{mod}-B \overline{}\underline{mod}-B has a left adjoint.

We proceed analogously to §2.1.

We obtain

SX_{1} = X_{2}  ΩSX_{1} = X_{21}  SX_{14} = X_{2} ⊕ X_{9}  ΩSX_{14} = X_{21} ⊕ X_{11}
SX_{2} = X_{1}  ΩSX_{2} = X_{3}  SX_{15} = X_{2}  ΩSX_{15} = X_{21}
SX_{3} = X_{1}  ΩSX_{3} = X_{3}  SX_{16} = X_{1} ⊕ X_{17}  ΩSX_{16} = X_{3} ⊕ X_{25}
SX_{5} = X_{17}  ΩSX_{5} = X_{25}  SX_{17} = X_{1} ⊕ X_{2} ⊕ X_{9} ΩSX_{17} = X_{21} ⊕ X_{3} ⊕ X_{11}
SX_{6} = X_{2} ⊕ X_{13} ΩSX_{6} = X_{21} ⊕ X_{22} SX_{18} = X_{1} ⊕ X_{2} ⊕ X_{9} ΩSX_{18} = X_{21} ⊕ X_{3} ⊕ X_{11}
SX_{7} = X_{1} ⊕ X_{9}  ΩSX_{7} = X_{3} ⊕ X_{11}  SX_{19} = X_{2} ⊕ X_{9}  ΩSX_{19} = X_{21} ⊕ X_{11}
SX_{8} = X_{1} ⊕ X_{2}  ΩSX_{8} = X_{21} ⊕ X_{3}  SX_{20} = X_{1}  ΩSX_{20} = X_{3}
SX_{9} = X_{17}  ΩSX_{9} = X_{25}  SX_{21} = X_{2}  ΩSX_{21} = X_{21}
SX_{10} = X_{2}  ΩSX_{10} = X_{21}  SX_{22} = X_{13}  ΩSX_{22} = X_{22}
SX_{11} = X_{9}  ΩSX_{11} = X_{11}  SX_{23} = X_{17}  ΩSX_{23} = X_{25}
SX_{12} = X_{2} ⊕ X_{17} ΩSX_{12} = X_{21} ⊕ X_{25} SX_{24} = X_{2} ⊕ X_{9}  ΩSX_{24} = X_{21} ⊕ X_{11}
SX_{13} = X_{1} ⊕ X_{9} ΩSX_{13} = X_{3} ⊕ X_{11} SX_{25} = X_{17} ΩSX_{25} = X_{25}

and

[figure: (; X_{1}; ϵ_{1}; ΩSX_{1}; ); =; (; R/π; 1; 1; R/π; R/π; 0; ); (; X_{14}; ϵ_{14}; ΩSX_{14}; ); =; (; R/π^{3}; π^{2}; 1; R/π^{3}; 1; R/π; 0; R/π^{2}; )]
[figure: (; X_{2}; ϵ_{2}; ΩSX_{2}; ); =; (; R/π^{2}; 1; 1; R/π^{2}; 1; R/π^{2}; 1; R/π; ); (; X_{15}; ϵ_{15}; ΩSX_{15}; ); =; (; R/π^{3}; 1; 0; R/π; 0; )]
[figure: (; X_{3}; ϵ_{3}; ΩSX_{3}; ); =; (; R/π^{2}; 1; 1; R/π; 1; R/π^{2}; 1; R/π; ); (; X_{16}; ϵ_{16}; ΩSX_{16}; ); =; (; R/π^{2} ⊕ R/π^{3}; ^{( \begin{matrix}1 & π \\ 1 & 0 \end{matrix} )}; ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ); R/π ⊕ R/π^{3}; ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ); R/π^{2} ⊕ R/π^{2}; _{( \begin{matrix}1 & 0 \\ 0 & π \end{matrix} )}; R/π ⊕ R/π^{2}; )]
[figure: (; X_{17}; ϵ_{17}; ΩSX_{17}; ); =; (; R/π ⊕ R/π^{3}; ^{( \begin{matrix}π & π^{2} \\ 1 & 0 \end{matrix} )}; ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ); R/π^{2} ⊕ R/π^{3}; ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ); R/π ⊕ R/π^{2}; _{( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )}; R/π ⊕ R/π^{2}; )]
[figure: (; X_{5}; ϵ_{5}; ΩSX_{5}; ); =; (; R/π^{2}; π; 1; R/π^{3}; 1; R/π^{2}; π; R/π^{2}; ); (; X_{18}; ϵ_{18}; ΩSX_{18}; ); =; (; R/π ⊕ R/π^{3}; ^{( \begin{matrix}0 & π^{2} \\ 1 & π \end{matrix} )}; ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ); R/π ⊕ R/π^{3}; ( \begin{matrix}1 & −π \\ 0 & 1 \end{matrix} ); R/π ⊕ R/π^{2}; _{( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )}; R/π ⊕ R/π^{2}; )]
[figure: (; X_{6}; ϵ_{6}; ΩSX_{6}; ); =; (; R/π; π; 1; R/π^{2}; 1; R/π; 0; R/π; ); (; X_{19}; ϵ_{19}; ΩSX_{19}; ); =; (; R/π; π^{2}; 1; R/π^{3}; 1; R/π; 0; R/π^{2}; )]
[figure: (; X_{7}; ϵ_{7}; ΩSX_{7}; ); =; (; R/π^{2}; ^{(^{1 π})}; 1; R/π ⊕ R/π^{3}; ( \begin{matrix}1 & −π \\ 0 & 1 \end{matrix} ); R/π^{2}; _{(_{1 0} )}; R/π ⊕ R/π^{2}; ); (; X_{20}; ϵ_{20}; ΩSX_{20}; ); =; (; R/π^{3}; 1; 1; R/π; 1; R/π^{2}; 1; R/π; )]

[figure: (; X_{8}; ϵ_{8}; ΩSX_{8}; ); =; (; R/π ⊕ R/π^{3}; ^{( \begin{matrix}π \\ 1 \end{matrix} )}; ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ); R/π^{2}; 1; R/π ⊕ R/π^{2}; _{( \begin{matrix}0 \\ 1 \end{matrix} )}; R/π; ); (; X_{21}; ϵ_{21}; ΩSX_{21}; ); =; (; R/π; 1; 0; R/π; 0; )]

[figure: (; X_{9}; ϵ_{9}; ΩSX_{9}; ); =; (; R/π^{3}; π; 1; R/π^{3}; 1; R/π^{2}; π; R/π^{2}; ); (; X_{22}; ϵ_{22}; ΩSX_{22}; ); =; (; 0; R/π; 1; 0; R/π; )]
[figure: (; X_{10}; ϵ_{10}; ΩSX_{10}; ); =; (; R/π^{2}; 1; 0; R/π; 0; ); (; X_{23}; ϵ_{23}; ΩSX_{23}; ); =; (; R/π^{3}; π; 1; R/π^{2}; 1; R/π^{2}; π; R/π^{2}; )]

[figure: (; X_{11}; ϵ_{11}; ΩSX_{11}; ); =; (; 0; R/π^{2}; 1; 0; R/π^{2}; ); (; X_{24}; ϵ_{24}; ΩSX_{24}; ); =; (; R/π^{2}; π^{2}; 1; R/π^{3}; 1; R/π; 0; R/π^{2}; )]

( \xymatrix{X_{12} \ar[d]_{ϵ_{12}} \\ ΩSX_{12}} ) = ( \xymatrix{R/π ⊕ R/π^{3} \ar[r]^{^{( \begin{matrix}π^{2} \\ π \end{matrix} )}} \ar[d]_{( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )} & R/π^{3} \ar[d]^{1} \\ R/π ⊕ R/π^{2} \ar[r]^{_{( \begin{matrix}0 \\ π \end{matrix} )}} & R/π^{2}} )  ( \xymatrix{X_{25} \ar[d]_{ϵ_{25}} \\ ΩSX_{25}} ) = ( \xymatrix{R/π^{2} \ar[r]^{π} \ar[d]_{1} & R/π^{2} \ar[d]^{1} \\ R/π^{2} \ar[r]^{π} & R/π^{2}} ) .

[figure: (; X_{13}; ϵ_{13}; ΩSX_{13}; ); =; (; R/π^{3}; ^{(^{1 π})}; 1; R/π ⊕ R/π^{3}; ( \begin{matrix}1 & −π \\ 0 & 1 \end{matrix} ); R/π^{2}; _{(_{1 0} )}; R/π ⊕ R/π^{2}; )]

Cf. §2.1.

Remark 10 Keep the assumptions of Proposition 9. We have (Ω ∘ S)^{2} Y ≃ (Ω ∘ S)Y for Y ∈ Ob \underline{mod}-B. Given an R/π^{3}-module X, we write X̄ := X/π^{2}X and Ann_{π} X̄ := { x̄ ∈ X̄ : πx̄ = 0 }. The unit of the adjunction S ⊣ Ω at a B-module X ^{\overline{}^{f}} Y is represented by the composite

[figure: X; f; Y; X̄; f̄; Ȳ; X̄/π Kern f̄; Ȳ /(Ann_{π} X̄)f̄ ,]

where the vertical maps are the respective residue class maps, and the middle and lower horizontal maps are the induced maps. I do not know why.

2.3 Further examples of left adjoints

Let

\begin{aligned}C_{1} & := & A/(π^{2}f) & = & (R/π^{3})(e ^{\overline{}^{a}} f)/(π^{2}f) \\ C_{2} & := & A/(πf) & = & (R/π^{3})(e ^{\overline{}^{a}} f)/(πf) \\ C_{3} & := & A/(π^{2}e, π^{2}f) & = & (R/π^{2})(e ^{\overline{}^{a}} f) \\ C_{4} & := & A/(πa) & = & (R/π^{3})(e ^{\overline{}^{a}} f)/(πa) \\ C_{5} & := & A/(πa, π^{2}f) & = & (R/π^{3})(e ^{\overline{}^{a}} f)/(πa, π^{2}f) \\ C_{6} & := & A/(π^{2}e) & = & (R/π^{3})(e ^{\overline{}^{a}} f)/(π^{2}e) \\ C_{7} & := & A/(πe) & = & (R/π^{3})(e ^{\overline{}^{a}} f)/(πe) \\ C_{8} & := & A/(π^{2}e, πa) & = & (R/π^{3})(e ^{\overline{}^{a}} f)/(π^{2}e, πa) \end{aligned}

Proposition 11 Suppose given a prime p ∈ [2, 997]. Suppose that R = F_{p}[X] and π = X. Then the Heller operator Ω : \underline{mod}-C_{j} \overline{}\underline{mod}-C_{j} has a left adjoint for j ∈ [1, 8].

Remark 12 Keep the assumptions of Proposition 11. We have (Ω ∘ S)^{2} Y ≃ (Ω ∘ S)Y for Y ∈ Ob \underline{mod}-C_{j} for j ∈ [1, 8] ∖ \{5\}. For j = 5, we have

\begin{aligned}(Ω ∘ S)X_{10} & = & X_{10} ⊕ X_{21} \\ (Ω ∘ S)X_{21} & = & X_{21} \end{aligned}

in the notation of §2.1.1, i.e.

\begin{aligned}(Ω ∘ S)(R/π^{2} \overline{}0) & = & (R/π^{2} \overline{}0) ⊕ (R/π \overline{}0) \\ (Ω ∘ S)(R/π \overline{}0) & = & (R/π \overline{}0) . \end{aligned}

2.4 Counterexample: no right adjoint

Recall from §2.3 that C_{3} = (R/π^{2})(e ^{\overline{}^{a}} f). As representatives of isoclasses of nonprojective C_{3}-modules we obtain, in the notation of §2.1.1,

Y_{1} := X_{1} = (R/π ^{\overline{}^{1}} R/π)  Y_{5} := X_{21} = (R/π \overline{}0)
Y_{2} := X_{3} = (R/π^{2} ^{\overline{}^{1}} R/π)  Y_{6} := X_{22} = (0 \overline{}R/π)
Y_{3} := X_{6} = (R/π ^{\overline{}^{π}} R/π^{2})  Y_{7} := X_{25} = (R/π^{2} ^{\overline{}^{π}} R/π^{2}) .
Y_{4} := X_{10} = (R/π \overline{}0)

Remark 13 Suppose that R = F_{3}[X] and π = X. The functor Ω : \underline{mod}-C_{3} \overline{}\underline{mod}-C_{3} does not have a right adjoint.

Proof. Magma yields

H := ( dim_{F_{3}}(_{\underline{mod}-C_{3}}(Y_{i} , Y_{j})) )_{i,j} = (\begin{matrix}1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 2 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 1 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}) ∈ (Z_{⩾0})^{7×7}

and

H′ := ( dim_{F_{3}}(_{\underline{mod}-C_{3}}(ΩY_{i} , Y_{j})) )_{i,j} = (\begin{matrix}1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}) ∈ (Z_{⩾0})^{7×7} .

Assume that Ω has right adjoint T : \underline{mod}-C_{3} \overline{}\underline{mod}-C_{3} .

Write T Y_{j} ≃ ⨁_{k∈[1,7]} Y_{k}^{⊕u_{k,j}} for j ∈ [1, 7], where U := (u_{k,j})_{k,j} ∈ (Z_{⩾0})^{7×7}. We obtain

H′ = ( dim_{F_{3}}(_{\underline{mod}-C_{3}}(ΩY_{i} , Y_{j})) )_{i,j}
= ( dim_{F_{3}}(_{\underline{mod}-C_{3}}(Y_{i} , T Y_{j})) )_{i,j}
= ( dim_{F_{3}}(_{\underline{mod}-C_{3}}(Y_{i} , ⨁_{k∈[1,7]} Y_{k}^{⊕u_{k,j}})) )_{i,j}
= ( ∑_{k∈[1,7]} dim_{F_{3}}(_{\underline{mod}-C_{3}}(Y_{i} , Y_{k})) · u_{k,j})_{i,j}
= H · U .

So every column of H′ is a linear combination of columns in H with coefficients in Z_{⩾0} . However, the third column of H′ would afford a coefficient ∈ Z_{>0} at the first, third or fifth column of H because its first entry equals 1. But then its second entry would also be in Z_{>0} , because these columns of H all have second entry equal to 1. But this second entry equals 0. We have arrived at a contradiction.

References

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[2] Bühler, T., Exact Categories, Exp. Math. 28 (1), p. 1–69, 2010.

[3] Heller, A., The loop-space functor in Homological Algebra, Trans. Am. Math. Soc. 96, p. 382–394, 1960.

[4] Künzer, M., Heller triangulated categories, Homology, Homotopy and Applications 9 (2), p. 233–320, 2007.

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[9] Schubert, H., Categories, Springer, 1972.

Matthias Künzer

Lehrstuhl D für Mathematik

RWTH Aachen Templergraben 64 D-52062 Aachen

kuenzer@math.rwth-aachen.de www.math.rwth-aachen.de/∼kuenzer