On the Image of the Totaling Functor

ON THE IMA GE OF THE TOT ALING FUNCTOR KRISTEN A. BECK Abstract. Let A be a D G algebra with a trivial differen tial o ve r a commu- tativ e unital ring. This paper inv estigates the image of the totaling functor, defined from the category of complexes of graded A -mo dules to the category of DG A -mo dules. Sp ecifically , we exhibit a sp ecial class of semif r ee DG A - modules which can alwa ys b e expressed as the totaling of some complex of graded fr ee A -mo dules. As a corollary , we also provide results concerning the image of the totaling functor when A is a p olynomial ring o ver a field. Introduction Let A b e a DG alge br a over a commutativ e unital ring . F urthermor e, let DG ( A ) denote the catego r y of DG A -mo dules and their degree zero chain maps, and let Ch Gr ( A ♮ ) denote the catego r y of (co)c hain complexes of gra ded A ♮ -mo dules and their degree zero ch ain maps. The motiv ation for the work in this paper is to b e tter understand the difference betw een these categories. In the most g eneral case, o ne can easily see that DG ( A ) is a muc h ‘richer’ category than Ch Gr ( A ♮ ) — indeed, its ob jects take into account tw o differentials rather than just one. How ever, when A ha s a triv ial differential (so that A = A ♮ ), the differ ence betw een DG ( A ) a nd Ch Gr ( A ♮ ) is not s o striking . The goa l of this pa per is to address the latter sce na rio by studying the imag e of the so-called totaling functor T ot : Ch Gr ( A ♮ ) → DG ( A ) in the case that A has a trivial differ e n tial. Given a co chain complex X o f graded mo dules ov er such a DG algebra A , one defines T ot X to be the complex whose underly ing graded A ♮ -mo dule structure is given b y (T ot X ) ♮ := M i ∈ Z Σ − i X i and whose differential follows in a natural w ay from tha t of X . Up on defining an A -action o n this co mplex, T ot X admits the structure of a DG A -mo dule. The primary questio n we consider is whether the totaling functor is surjective. In Theorem 2 .4, w e provide a necessary and s ufficien t condition for a DG mo dule ov er a DG alg ebra A with a tr ivial differen tial to b e equal to the totaling of some complex of graded free A ♮ -mo dules. The constructive nature of the pro of of this result further mo re allows us to express a ‘totaling pre-image’ for DG mo dules which lie in the image of the tota ling functor. T o a nswer the ques tion of whether the totaling functor is surjective, we res trict to the case that A is a p olynomial ring ov er a field. In E xample 2.5, we exhibit a DG mo dule ov er a polynomia l ring in Date. No ve mber 21, 2018. 2010 Mathematics Subje ct Classific ation. 13D09, 18E30. Key wor ds and phr ases. T otaling, differen tial graded algebra, deriv ed category . This research was partially supp orted by NSA Grant H98230-07-1-0197 and NSF GK-12 Pro- gram Gran t 0841400. 1 2 K. A. BECK t wo (or more) v a riables which do es not sa tisfy the condition s pec ifie d by Theorem 2.4, and therefor e do es not lie in the imag e of the functor. Mor eov er, in Theorem 2.11, we illustra te tha t every DG module ov er a p olynomial ring in one v ar ia ble is quasiisomor phic to the totaling o f some complex o f graded A ♮ -mo dules. 1. Back ground The results in this pap er will assume a working under standing of differential graded (DG) alg ebras and their mo dules. F or a thor ough treatment of this sub ject, the reader is referred to [1], [2], or [4]. In what follows, A is assumed to b e a DG algebra ov er a commutativ e unital r ing. 1.1. Semifree DG mo dules. W e b egin by in tro ducing a class of DG mo dules which genera lize free mo dules over a r ing . They will for m the basic structures necessary for the construction of (count er- )examples in the sequel, and will also be essential to the statement o f our main re sult. Definition 1 .1.1. Let M be a DG A -mo dule. A subset E ⊆ M ♮ is called a semib asis for M if (1) E is a basis for M ♮ ov er A ♮ , a nd (2) E = F d ∈ N E d (a disjoin t union) such that ∂ ( E d ) ⊆ A G i s ; indeed, these cases represent the tor sion-free part of H( M ). Thus, for s < i ≤ t , one has the following. µ i j : ( 0 → M j if j < r i (Σ r i A ) j → M j if j ≥ r i Now consider the following isomorphism o f graded A -mo dules . ϕ : s M i =1 Σ r i A h i Σ c i A ⊕ t M i = s +1 Σ r i A → H( M ) Fixing 1 ≤ i ≤ t , let 0 6 = z i ∈ M r i be a cycle defined in such a wa y that ϕ ( σ r i 1) = cls( z i ) ∈ H r i ( M ). Since ϕ is A -linear, ϕ  x ℓ σ r i 1  = cls( x ℓ z i ) for ℓ ≥ 0. Notice that, for small enough v alues of ℓ , these classes ar e nonzero. T o b e precise, cls ( x ℓ z i ) = 0 if and o nly if 1 ≤ i ≤ s and ℓ ≥ c i − r i . Therefore, for each 1 ≤ i ≤ s , there exists m i ∈ M c i +1 such that ∂ M c i +1 ( m i ) = x c i − r i z i . W e now pro ceed to define, for each i a nd j , a basis U i j for (T ot G i ) j ov er k . U i j ⊇  { x j − r i σ r i 1 , ( − 1 ) j − c i − 1 x j − c i − 1 σ c i +1 1 } if 1 ≤ i ≤ s a nd j > c i { x j − r i σ r i 1 } otherwise ON THE IMAGE OF THE TOT ALING F UNCTOR 11 Finally , we define µ i j : (T ot G i ) j → M j in the following wa y . µ i j ( u ) =      x j − r i z i if u = x j − r i σ r i 1 x j − c i − 1 m i if u = ( − 1 ) j − c i − 1 x j − c i − 1 σ c i +1 1 0 otherwise By constructio n, µ i = ( µ i j ) is an A - line a r degree zero chain map b etw een T o t G i and M . F urther, one can easily chec k that the Le ibniz rule is satisfied, so that the map µ : T o t F → M given in (2.11.2) is a morphism of DG mo dules . The ab ov e construction also guara nt ees that µ establishes a one-to-one co rresp ondence betw een the g enerator s o f homology of T ot F = L m i =1 T o t G i and that o f M . The result follows.  The following example illustrates the practical use of the co nstruction used in the pr o of of T he o rem 2.1 1. Example 2 .12. Let M be the rank fiv e semifree DG mo dule o ver A = k [ x ] with semibasis giv en b y { e 1 , e 2 , e 3 , e 4 , e 5 } suc h that | e 1 | = 0, | e 2 | = 2, | e 3 | = 4, | e 4 | = 8, | e 5 | = 9, a nd where the differential of M is defined by the following. ∂ ( e 1 ) = 0 ∂ ( e 2 ) = 0 ∂ ( e 3 ) = 0 ∂ ( e 4 ) = x 7 e 1 + x 5 e 2 ∂ ( e 5 ) = x 4 e 3 The homolo gy of M can be decomp osed H( M ) = Ae 1 ⊕ Ae 2 ⊕ Ae 3 A ( x 7 e 1 + x 5 e 2 ) ⊕ Ax 4 e 3 ∼ = A ( x 2 e 1 + e 2 ) A ( x 7 e 1 + x 5 e 2 ) ⊕ Ae 3 Ax 4 e 3 ⊕ Ae 1 (2.12.1) whence one obtains the following deleted minima l free resolution F of H( M ). 0 / / Σ 7 A x 5 / / Σ 2 A / / 0 ⊕ ⊕ F : 0 / / Σ 8 A x 4 / / Σ 4 A / / 0 ⊕ ⊕ 0 / / A / / 0 W e shall now utilize the pro of of Theor em 2.11 to show that T ot F ≃ M . F rom F we obtain the following sub complexes. G 1 : 0 → Σ 7 A x 5 − → Σ 2 A → 0 G 2 : 0 → Σ 8 A x 4 − → Σ 4 A → 0 G 3 : 0 → A → 0 Referring to the decomp ositio n o f homolog y in (2.12.1), one has that the cycles generating H( M ) ov er A are z 1 = x 2 e 1 + e 2 , z 2 = e 3 , and z 3 = e 1 . Then, for 12 K. A. BECK each j ∈ N , a bas is U 1 j of (T ot G 3 ) j ov er k must b e chosen to contain { x j σ 0 1 } . F urther more, the resp ective ba ses of (T ot G 1 ) j and (T o t G 2 ) j ov er k are given a s follows. U 1 j ⊇  { x j − 2 σ 2 1 } if j < 8 { x j − 2 σ 2 1 , ( − 1) j − 8 x j − 8 σ 8 1 } if j ≥ 8 U 2 j ⊇  { x j − 4 σ 4 1 } if j < 9 { x j − 4 σ 4 1 , ( − 1) j − 9 x j − 9 σ 9 1 } if j ≥ 9 Using these bases, one can no w define chain maps µ i = ( µ i j ) : (T ot G i ) j → M j . µ 1 j ( u ) =      x j e 1 + x j − 2 e 2 if u = x j − 2 σ 2 1 x j − 8 e 4 if u = ( − 1 ) j − 8 x j − 8 σ 8 1 0 otherwise µ 2 j ( u ) =      x j − 4 e 3 if u = x j − 4 σ 4 1 x j − 9 e 5 if u = ( − 1 ) j − 9 x j − 9 σ 9 1 0 otherwise µ 3 j ( u ) = ( x j e 1 if u = x j σ 0 1 0 otherwise Finally , µ : T ot F → M is g iven b y µ ( x ) =  µ 1 ( x ) , µ 2 ( x ) , µ 3 ( x )  for each x ∈ T o t F . Ackno wledgments The author would lik e to thank Dav e Jorgensen and Sean Sather- W ag s taff for bo th piquing and cultiv ating her interest in DG algebra. Thanks als o to the referee for ma n y helpful sugges tions which greatly improved the ov era ll quality of this manuscript. References [1] L. L. Avramov, Infinite fr e e r esolutions , Six lectures on comm utative algebra, 201 0, pp. 1– 118. MR2641236 [2] L. L. Avramov, H.-B. F oxb y, and S. Halper in, Differ ential gr ade d homolo gic al algebr a , (in preparation). [3] L. L. Avramov and D. A. Jorgensen, R e alization of c ohomolo gy over a c omplete interse ction , (in preparation). [4] B. Keller, On differ ential g r ade d c ate gories , In ternational Congress of Mathematicians. Vol . II, 2006, pp. 151–190. MR2275593 (2008g:1801 5) [5] C. A. W eibel, An i ntr o duction to homolo gica l algebra , Cambridge Studies in Adv anced Math- ematics, vol. 38, Cam bridge U niv ersity Press, Cambridge, 1994. MR 1269324 Kristen A. Beck, Dep ar tment of M a them a tics, University of Arizona, 617 N. Sant a Rit a A ve., Tucson, AZ, 857 19, U.S.A. E-mail addr ess : kbeck @math.arizon a.edu