Approximation of subcategories by abelian subcategories

APPR O XIMA TION OF SUBCA TEGORIES BY ABELI AN SUBCA TEGORIES. A. SALCH 1. Intr oduction. The earliest circulated versions of this pap er date fro m 2 010, and stem from ideas I ha d ea rlier, while I was a student. Several versions of this preprint hav e b een made av aila ble, but the main idea in eac h of them is a very simple statement: given an ide a l I of a c ommutative ring R satisfying ve ry mild hyp othe ses, the c ate g ory of L 0 - c omplete R -mo dules is the smal lest ab eli an sub c ate gory of Mod( R ) c ontaining al l the I -adic a l ly c omple te R -mo dules and satisfy ing a f ew r e a sonable c ondi tions 1 . As a s logan, “ the category of L 0 -complete mo dules is the b est ab elian approximation to the categor y of I -adically complete mo dules.” See Theorem 3.3, b elow, for a precise statement. In t his short note, I prov e the main theorem in c lo se to its original level of generality . That level of gener ality is sufficient for every application I hav e ever ha d for the theorem, and it is also sufficien t for every applica tio n I hav e ever seen any one else ha ve fo r the th eor em. In this level of ge ne r ality , the theorem is dramatically easier to pro ve than in the more gener al settings, as you can see from how shor t this note is! T o me it seems that this shor t note is not w orth sending to a jour na l, but I think it is worth ha ving on the arXiv. None of the v ario us v ersions of this pap er were published, but in one v ersion of this pap er (the mo s t gener a l version) that I circula ted some years ago, I b elieve there is a hypo thesis missing from some of the s ta temen ts, so I think it is worthwhile to p ost this short note, to serve as the simple, straightforward, and ea sily-seen-to- be -corre c t “ version of r ecord” for this theorem. 2. The r elev ant defi nitions. Definition 2.1 . Given an ide a l I in a c ommutative ring R , we write Λ for t he I -ad ic c ompletio n functor Λ : Mo d( R ) → Mo d( R ) , i.e., Λ( M ) = lim n M /I n M . We write L 0 Λ for the zer oth lef t-derive d fu nctor L 0 Λ : Mo d( R ) → Mo d( R ) of Λ . Since I -adic completion is, in genera l, not right exact, L 0 Λ( M ) do es not neces- sarily coincide with Λ( M ). Definition 2.2. We say that an R - mo dule M is L 0 -complete if the c anonic al map η M : M → L 0 Λ M is an isomorp hism. We write L 0 Λ Mo d( R ) for the ful l s ub c ate- gory of Mo d( R ) whose obje cts ar e the L 0 -c omplete m o dules. See sections A.2 and A.3 of [3] for an excellent in tro duction to L 0 -completion and L 0 -complete mo dules, including pro ofs of many ba sic prop er ties. 1 Y ou can tak e those conditions to b e “replete,” “exact ,” and “full.” It is not difficult to also prov e v ariants of that result, replacing exactness wi th reflectivity , for example. 1 2 A. SALCH 3. The th eorem. Throughout, let R b e a commut ative ring, and let I b e a we akly pr o -r e gular ideal in R . Many early references on lo ca l homo logy and derived completion, such a s [2] and app endix A of [3], assumed that R is No etherian, or that I is gener a ted by a regular sequence. In [1] and [6], it was established that w eak pro-reg ularity of I is sufficient for the pro ofs of most of the fundamen tal r esults in the are a . Every weakly pro-re g ular ideal is finitely generated. Rather than repr o duce the rather techn ical definition of w eak pro -regular ity her e, I prefer to simply cite the result of [6] whic h states that, in a No e ther ian commutativ e ring, every ideal is weakly pro-reg ular. Conseque ntly , in most practical situa tions, one k nows that the ide a ls one encounters in examples are weakly pro-r egular. The pap ers [5] a nd [4 ] hav e v aluable treatments of prop erties of L 0 Λ when I is weakly pro -regula r. F o r example, Theorem 3.9 (a) of [4] establishes that: Theorem 3 .1. If I is we akly pr o-r e gu lar, then the ful l su b c a te gory L 0 Λ Mo d( R ) of Mo d( R ) is a b elian, and the i nclusion functor ι : L 0 Λ Mo d( R ) → Mo d( R ) is e xact. Theorem 3 .1 a ls o app ears in references that pre-date [4], although g enerally with stronger assumptions: for example, compare Theorem A.6 of [3], whic h is similar but includes the a ssumptions that R is No etherian and that I is r egular. It is also stra ightforw ard (e.g. see Pr op osition 3.7 in [4]) to prov e that: Lemma 3.2. If I is we akly pr o-r e gular, then every I -adic a l ly c o mplete R -mo dule is L 0 -c omplete. With Theorem 3.1 and Lemma 3.2 in hand, w e now hav e little trouble in proving the main theo rem: Theorem 3.3. L et I b e a we akly pr o-r e gular ide al in a c ommu tative ring R . T hen the c ate gory L 0 Λ Mo d( R ) of L 0 -c omplete mo dules is t he unique smal lest r eplete 2 exact 3 ful l sub c ate gory of Mo d( R ) c ontaining al l the I -adic al ly c omplete R -m o dules. Pr o of. Theo rem 3.1 esta blis hes that L 0 Λ Mo d( R ) is a replete exac t full sub catego r y of Mo d( R ), while Lemma 3.2 establishes that L 0 Λ Mo d( R ) contains the I -adically complete R -mo dules. So supp ose that A is a replete exact full sub categ o ry of Mo d( R ) whic h con tains the I -adically complete R -mo dules. W e m ust then pro ve that A contains L 0 Λ Mo d( R ). Supp ose that X is an L 0 -complete R -mo dule. Cho ose an exact sequence P 1 d − → P 0 − → X → 0 in Mo d( R ), with P 0 , P 1 pro jective R - mo dules. Applying Λ, we hav e a short exact sequence (3.1) 0 → im Λ d → Λ P 0 → L 0 Λ X → 0 . W e know that Λ P 0 is in A , since A contains a ll the I -adically complete R - mo dules. If w e c a n show tha t im Λ d is I -a dically complete, then the short exact seq uence (3.1) exhibits X ∼ = L 0 Λ X as the quotient of an I -adically complete R - mo dule by an 2 Recall that a sub category is rep lete if it is closed under i somorphisms, i .e., it con tains ev ery ob ject isomorphic to one of its ob jects. 3 Recall that a nonempty full ab elian subcategory A of an ab elian category C is exact if the inclusion functor A ֒ → C is exact. APPRO XIMA TION OF SUBCA TEGORIES BY ABELIAN SUBCA TEGORIES. 3 I -adically co mplete submo dule, i.e., as the quo tient of an ob ject o f A by a sub ob ject also in A . Exactness and r epleteness of A then gives us that L 0 Λ X is in A as well. So all that is left is to show that the image of an R -mo dule map with I -adically complete domain and I -adically co mplete co domain is a lso I - adically complete. Suppo se f : Y → Z is a morphism of R -mo dules, with Y , Z ea ch I -adically complete. W e have the commut ative diagra m Y π / / ηY ∼ =   im f i / / η im f   Z ηZ ∼ =   Λ Y Λ π / / Λ im f Λ i / / Λ Z where π and i a re the ca no nical pro jection to the image and inclusion of the image, and η : id Mo d( R ) → Λ is the canonical natura l transformatio n sending each mo dule to its I -a dic completion. Since Λ pres e r ves epimorphisms, Λ π is epic, so (Λ π ) ◦ η Y = ( η (im f )) ◦ π is epic, s o η (im f ) is epic. Mean while, ( η Z ) ◦ i = (Λ i ) ◦ η (im f ) is a comp osite of mo nomorphisms, so η (im f ) is monic. So η (im f ) is an isomorphism, i.e., im f is I -adically complete, as des ired.  References [1] Leovigildo A lonso T arr ´ ıo, Ana Jerem ´ ıas L´ opez, and Josep h Lipman. Local homology and cohomology on schemes. Ann. Sci. ´ Ec ole Norm. Sup. (4) , 30(1):1–3 9, 1997. [2] J. P . C. Greenlees and J. P . Ma y . Derived functors of I -adic completion and lo cal homology . J. A lgebr a , 149(2):438–45 3, 1992. [3] Mar k Hov ey and Neil P . Stri c kland. Mora v a K -the ories and lo calisation. M em. A mer. Math. So c. , 139(6 66):viii+100, 1999. [4] Luca Pol and Jordan Williamson. The homoto py theory of complete mo dules. J. Algebr a , 594:74–100, 2022. [5] Mar co Porta , Liran Shaul, and Amnon Y ekutieli. On the homology of completion and torsion. Algebr. R epr esent. The ory , 17(1):31–67, 2014. [6] Peter Sche nzel. Proregular sequences, lo cal cohomology , and completion. Math. Sc and. , 92(2):161– 180, 2003.