Characterisations of Purity in a Locally Finitely Presented Additive Category

Home , Direct limit

arXiv:1702.07184v1 [math.RT] 23 Feb 2017 hrceiain fprt nalclyfiieypresented finitely locally a in purity of Characterisations ceiain fprt nalclyfiieypeetdaddi presented finitely locally a in purity of acterisations rosfrseiccocsof choices specific for proofs pijcieadijcieojcsin objects injective and fp-injective eko htFpinj that know we in fprt namdl category module a in purity of tions ento 2. Definition l aeoisadfntr etoe nti ae r addi are paper this in mentioned functors limits and categories direct All over functors Extending 1 rsredrc iis ewiefp write We limits. direct preserve bla rusad o ml category small a f for gotten and, be groups can abelian which categories, presented finitely locally ersnal functor representable ento 1. Definition n Flat and hoe 1. Theorem fp eeecssc s[]ad[] u eofrtefloigdefi following the 3. offer Definition we but [8], and [7] as such references em 1. Lemma sflyfihu n etit oa equivalence an to restricts and faithful fully is product Proof. nisomorphism an hc sntrlin natural is which C nti hr oe ewl iea fcetfntra proof functorial efficient an give will we note, short this In ewl s esrpout ffntr.Frti,teuncomf the this, For functors. of products tensor use will We sseeal ml n vr beti ietlmto finit of limit direct a is object every and small skeletally is e 7 rpsto .]o aeti sa xriei h cal the in exercise an as this take or 1.1] Proposition [7, See ( G A, ⊗ Let Ab A 3 o n oal ntl rsne category presented finitely locally any For [3] Let Let Let F ) A o h aeoyo a functors flat of category the for sa bla ru ie ytecedfrua(e 9 o coe for [9] (see formula coend the by given group abelian an is easalctgr.Frayobject any For category. small a be C A C ( F A aeoy hr ucoilproof functorial short A category: easalctgr.Frfunctors For category. small a be eactgr ihdrc iis esythat say We limits. direct with category a be eactgr ihdrc iis nobject An limits. direct with category a be and -mod a , . Ab C C ) (e.g. G n Inj and → C ⊗ ( (( o h ulsbaeoyo ntl rsne bet in objects presented finitely of subcategory full the for A fp A C fp eray2,2017 24, February Mdi ipecrlayo htw ilpoehr,once here, prove will we what of corollary simple a is -Mod A C, F C ( C = A ( A aulDean Samuel − Ab ( = ) ewrite we , -mod a, op A a , Z ) Abstract , Mdfraring a for -Mod − o xml,teeuvlneo aycharacterisa- many of equivalence the example, For . ) Ab C a : ) ⊗ ∈ , A 1 : ) ≃ Ab A A C → ( Flat F c Ga ) → ( 7→ oklk w do). (we like look Ab. A, a = ∼ G ) (( Ab ∈ ⊗ iecategory tive C a F Ab : fp o 3.W rt bfrtectgr of category the for Ab write We [3]. rom Z A ( A C C − ) ( op tive. nition. n n functor any and A a F ) h functor the , o h aeoyo functors of category the for c , l rsne objects. presented ely op r otie ntedsrpinof description the in contained are ) → ) fteeuvlneo aiu char- various of equivalence the of , ) a | fp Ab . C easm oebcgon on background some assume We ral edrmyuestandard use may reader ortable band Ab uu fcoends. of culus C ∈ ) is . C C oal ntl presented finitely locally is h opiain fthe of complications The . ntl presented finitely F : F A nds) : → A → b h tensor the Ab, b hr is there Ab, A C additive → fthe if Ab, if Definition 4. Let C be a locally finitely presented category. For any functor F : fpC → Ab, define −→ F : C → Ab by −→ Fc = C(−,c)|fpC ⊗fpC F for any c ∈ C. −→ Theorem 2. Let C be a locally finitely presented category. For any functor F : fpC → Ab, F −→ preserves direct limits and there is an isomorphism F |fpC =∼ F which is natural in F . If E : C → Ab −→ preserves direct limits and E|fpC =∼ F then E =∼ F .

Proof. Variations of this statement appear in many places, but we will give a proof, for the sake of self-containment, which similar to that at [4, 3.16]. See [2] for a very simple argument when F is ﬁnitely presented. −→ The property that F preserves direct limits and restricts to F on fpC follows directly from the −→ deﬁnition of F and Lemma 1. For such a functor E : C → Ab, let α : F → E|fpC be an in isomorphism and, for each c ∈ C, assemble the morphisms

C(a,c) ⊗Z Fa → Ec : f ⊗ x 7→ ((Ef)αa)x (a ∈ fpC) each of which is natural in c, into a morphism −→ Fc = C(−,c)|fpC ⊗ F → Ec −→ which is natural in c. This morphism is an isomorphism when c ∈ fpC. Since both F and E preserve direct limits, it follows that this morphism is an isomorphism for any c ∈ C.

2 Purity in a locally ﬁnitely presented category

Deﬁnition 5. Let C be a locally ﬁnitely presented category. A sequence

0 → a → b → c → 0 in C is pure exact iff the induced sequence

0 → C(−,a)|fpC → C(−,b)|fpC → C(−,c)|fpC → 0 is pure. Definition 6. For a functor F : A → Ab, we define its dual to be the functor F ∗ : Aop → Ab ∗ defined by F a = HomZ(F a, Q/Z). If the reader is working in a slightly different context, witha k-linear locally finitely presented category, and prefers to replace Z by k and Q/Z by some injective cogenerator in k-Mod, then they may do so. The following theorem will still hold. 1 Definition 7. A functor F : A → Ab is said to be fp-injective if Ext (−, F )|fp(A,Ab) =0. We write Fpinj(A, Ab) for the category of all fp-injective functors A → Ab and Inj(A, Ab) for the category of all injective functors A → Ab. Theorem 3. Let C be a locally finitely presented category. For a sequence of maps

0 → a → b → c → 0 in C, the following are equivalent. 1. It is pure exact. 2. It is a direct limit of split exact sequences.

2 3. For any F ∈ fp(fpC, Ab), the induced sequence −→ −→ −→ 0 → Fa → Fb → Fc → 0

is exact in Ab. 4. For any F ∈ (fpC, Ab), the induced sequence −→ −→ −→ 0 → Fa → Fb → Fc → 0

is exact in Ab. 5. For any F ∈ Fpinj(fpC, Ab), the induced sequence −→ −→ −→ 0 → Fa → Fb → Fc → 0

is exact in Ab. 6. For any F ∈ Inj(fpC, Ab), the induced sequence −→ −→ −→ 0 → Fa → Fb → Fc → 0

is exact in Ab. 7. The induced sequence

∗ ∗ ∗ 0 → C(−,c)|(fpC)op → C(−,b)|(fpC)op → C(−,a)|(fpC)op → 0

is split exact in (fpC, Ab).

Proof. 1 implies 2: This argument is well-known and standard, but we give it for the sake of self- containment. Express as a direct limit of ﬁnitely presented objects, ∈ . Pure exact c c = lim−→λ Λcλ sequences form an exact structure so are closed under pullbacks. Take the pullback of our sequence along the morphisms cλ → c. We obtain a directed system of pure exact sequences

0 → a → bλ → cλ → 0, each of which must be split since cλ is ﬁnitely presented. The direct limit of this sequence is our original sequence. 2 implies 3: Obvious. −→ 3 implies 4: For any object d ∈ C, the functor (fpC, Ab) → Ab : F 7→ F d clearly preserves direct limits because it is a tensor product. By expressing F as a direct limit of ﬁnitely presented functors, F = limF , we obtain the sequence −→ λ −→ −→ −→ 0 → Fa → Fb → Fc → 0 as a direct limit of pure exact sequence −→ −→ −→ 0 → Fλa → Fλb → Fλc → 0, which is exact since since direct limits are exact. 4 implies 5: Obvious. 5 implies 6: Obvious. 6implies7: To showthatoursequenceis split, we needonlyshowthat, forany F ∈ Inj(fpC, Ab), the sequence

∗ ∗ ∗ 0 → (F, C(−,c)|(fpC)op ) → (F, C(−,b)|(fpC)op ) → (F, C(−,a)|(fpC)op ) → 0

3 ∗ is exact. The reason for this is that, since it is thedual of a ﬂat functor, C(−,a)|fpC is injective (there is a standard argument for this – see e.g. [1, 19.14] for something similar), and therefore we may substitute F = C(−,a)∗ to obtain a splitting. Indeed, if F ∈ Inj(fpC, Ab) then, by the hom-tensor duality, this sequence is isomorphic to

∗ ∗ ∗ 0 → (C(−,c)|(fpC)op ⊗fpC F ) → (C(−,b)|fpC ⊗fpC F ) → (C(−,a)|(fpC)op ⊗fpC F ) → 0, which is equal to −→ ∗ −→ ∗ −→ ∗ 0 → Fc → Fb → Fa → 0 which is exact by hypothesis. 7 implies 1: Easy since Q/Z is an injective cogenerator.

Corollary 1 (Well-known). For a ring A and a sequence

0 → L → M → N → 0 in A-Mod = (A, Ab), the following are equivalent. 1. It is pure. 2. It is a direct limit of split exact sequences. 3. For any pp pair ϕ/ψ in the language of left A-modules, the sequence

0 → ϕL/ψL → ϕM/ψM → ϕN/ψN → 0

is exact. 4. For any Y ∈ Mod-A, the induced sequence

0 → Y ⊗A L → Y ⊗A M → Y ⊗A N → 0

is exact. (This condition need only be checked when Y is ﬁnitely presented.) 5. For any pure injective Y ∈ Mod-A, the induced sequence

0 → Y ⊗A L → Y ⊗A M → Y ⊗A N → 0

is exact. 6. The induced sequence 0 → N ∗ → M ∗ → L∗ → 0 is split exact in Mod-A.

Proof. A functor F ∈ (A-mod, Ab) is: – ﬁnitely presented iff it comes from a pp pair [10, Section 10.2.5].

– fp-injective iff it is of the form Y ⊗A − for some Y ∈ Mod-A by [10, Theorem 12.1.6].

– injective iff it is of the form Y ⊗A − for some pure injective Y ∈ Mod-A by [10, Theorem 12.1.6] (uses the fact that 1 is equivalent to 4). ∗ ∗ For each X ∈ A-Mod, there is an isomorphism (−,X) |A-mod =∼ X ⊗A −|A-mod which is natural in X [5, 3.2.11]. Therefore,

∗ ∗ ∗ 0 → (−,N)|A-mod → (−,M)|A-mod → (−,L)|A-mod → is split exact iff

∗ ∗ ∗ 0 → N ⊗A −|A-mod → M ⊗A −|A-mod → L ⊗A −|A-mod → 0 is split exact. Since Mod-A → (A-mod, Ab): Y 7→ Y ⊗ −|A-mod is fully faithful, this is equivalent to 6.

4 Acknowledgements

I thank Sergio Estrada and Pedro Guil Asensio for inviting me to visit the University of Murcia, where I wrote this paper, and for helpful discussions.

References

[1] Frank W. Anderson, Kent R. Fuller, Rings and Categories of Modules, Second Edition. Springer-Verlag, 1992. [2] Maurice Auslander, Large modules over artin algebras. In Algebra, Topology and Category Theory, Academic Press, 1976. [3] William Crawley-Boevey, Locally finitely presented additive categories. Communications in Algebra, 22:5, 1641-1674, 1994. [4] Samuel Dean, Duality and contravariant functors in the representation theory of artin algebras. arXiv:1603.05863v3 (submitted for publication). [5] Edgar E. Enochs, Overtoun M. G. Jenda, Relative Homological Algebra. De Gruyter, 2011. [6] Sergio Estrada, Manuel Saorin, Locally finitely presented categories with no flat objects. De Gruyter, 2011. [7] Janet L. Fisher, The tensor product of functors; Satellites; and derived functors. Journal of Algebra 8, 277-294, 1968. [8] J. Fisher Palmquist, David C. Newell, Bifunctors and adjoint pairs. Transactions of the Amer- ican Mathematical Society, Vol. 155, No. 2, 1971. [9] Saunders Mac Lane, Categories for the Working Mathematician. Springer-Verlag, 1998. [10] Mike Prest, Purity, Spectra and Localisation. Cambridge University Press, 2009.