Multicomplexes and their spectral sequences

NTNU Norwegian University of Science and Technology Faculty of Information Technology and Electrical Engineering Department of Mathematical Sciences

Odin Hoff Gardå

Multicomplexes and their spectral sequences

On the spectral sequence associated with a multicomplex over a field

Master’s thesis in Mathematical Sciences Supervisor: Markus Szymik

January 2022

Master ’s thesis

Odin Hoff Gardå

Multicomplexes and their spectral sequences

On the spectral sequence associated with a multicomplex over a field

Master’s thesis in Mathematical Sciences Supervisor: Markus Szymik

January 2022

Norwegian University of Science and Technology

Faculty of Information Technology and Electrical Engineering Department of Mathematical Sciences

Multicomplexes and their spectral sequences

Odin Hoff Gardå

January, 2022

Acknowledgements

I want to thank my supervisor Markus Szymik for his many suggestions and encouragement throughout this project. I also wish to thank my fellow students, family, and partner Elina for all their support.

“Oh, he seems like an okay person, except for being a little strange in some ways. All day he sits at his desk and scribbles, scribbles, scribbles. Then, at the end of the day, he takes the sheets of paper he’s scribbled on, scrunges them all up, and throws them in the trash can.”

— J. von Neumann’s housekeeper, describing her employer

Abstract

Double complexes over a field are well-understood, and so are their associated spectral sequences. Multicomplexes generalise the notion of double complexes.

We aim to understand the spectral sequence associated with a multicomplex over a field. The homotopy transfer theorem (HTT) equips the cohomology of the underlying cochain complex of a multicomplex with a transferred multicomplex structure. We characterise first-page degeneration in terms of these transferred differentials and then provide a method for computing the spectral sequence page-by-page by repeated application of the HTT.

Sammendrag

Multikomplekser generaliserer kjedekomplekser og dobbeltkomplekser. Vi ser hovedsaklig på spektralfølgen tilhørende et multikompleks over en kropp og gir betingelser for degenerasjon på første side i spektralfølgen. Videre ser vi på differensialene på senere sider i spektralfølgen og beskriver en fremgangsmåte for å regne ut disse eksplisitt ved hjelp av homotopioverføringsteoremet (homotopy transfer theorem).

Contents

1 Multicomplexes and spectral sequences 13

1.1 Filtered cochain complexes 13

1.2 Multicomplexes and their associated spectral sequences 14

2 Homotopy transfer 23

2.1 Cochain complexes over a field 23

2.2 Homotopy Transfer Theorem (HTT) 27

2.3 HTT and the spectral sequence associated with a double complex 30

3 First-page degeneracy 33

4 Differentials on later pages 38

Appendices 47

A Multicomplexes as homotopy algebras 47

B Minimal models for multicomplexes 56

C Deformations of cochain complexes 59

Introduction

A multicomplex(𝑀, 𝐷_•)over a fieldKis a bigradedK-vector space 𝑀 together with a family of endomorphisms𝐷₀, 𝐷₁, 𝐷₂, . . .of degrees |𝐷_𝑟| =(𝑟,1−𝑟). These maps are required to satisfy the relations

𝐷₀𝐷_𝑛 +𝐷₁𝐷_𝑛−1+ · · · +𝐷_𝑛−1𝐷₁+𝐷_𝑛𝐷₀ =0 for all𝑛 ⩾ ^0.

Set𝑛 = 0, and the relation above becomes𝐷₀𝐷₀ = 0. Consequently, every mul- ticomplex(𝑀, 𝐷_•)has an underlying cochain complex(𝑀, 𝐷₀)of graded vector spaces. Given a multicomplex𝑀, we define the cohomology complex 𝐻(𝑀)by taking cohomology of the underlying cochain complex(𝑀, 𝐷₀)degreewise.

𝐻^𝑝,𝑞(𝑀)= ker(𝐷₀: 𝑀^𝑝,𝑞 → 𝑀^𝑝,𝑞+1) Im(𝐷₀: 𝑀^{𝑝,𝑞−1} → 𝑀^𝑝,𝑞)

We equip𝐻(𝑀)with the trivial differential so that it (trivially) becomes a cochain complex of graded vector spaces. Over a field, every cochain complex(𝑀, 𝐷₀) is homotopy equivalent to its cohomology complex. There exists a slightly weaker notion of homotopy equivalence called a homotopy retract. If (𝑀, 𝐷) and (𝑁 , 𝐷^′) are cochain complexes, then a homotopy retract data consists of cochain maps𝜋: 𝑀 → 𝑁,𝜄: 𝑁 → 𝑀 and a homotopy ℎ: 𝑀 → 𝑀 of degree−1 such that𝜄𝜋−id^𝑀 =𝐷 ℎ+ℎ𝐷 and𝜄is a quasi-isomorphism.

(𝑀, 𝐷) (𝑁 , 𝐷^′).

𝜋 ℎ

𝜄

Over a field, every homotopy retract data can be extended to an equivalence, so the two notions are equivalent in this case. The homotopy transfer theo- rem (HTT) for multicomplexes tells us that if(𝑀, 𝐷_•)is a multicomplex and we have a homotopy retract data as above, we can use the maps𝜋,𝜄andℎ to define maps 𝐷^′

1, 𝐷^′

2, . . .: 𝑁 → 𝑁 compatible with𝐷^′

0 := 𝐷^′ in the sense that (𝑁 , 𝐷_•^′) becomes a multicomplex. In particular, since there always exists a homotopy retract data of(𝑀, 𝐷₀)to the cohomology complex𝐻(𝑀), we can transfer the multicom- plex structure on𝑀to one on𝐻(𝑀). Said differently, the HTT allows us to lift the cohomology functor𝐻(−): Ch(Vect^Z) → Ch(Vect^Z) toMCK, where Ch(Vect^Z) andMCK denote the categories of cochain complexes of graded vector spaces and multicomplexes overK, respectively. We can think of this as the following commutative diagram, where the vertical arrows send a multicomplex to its

underlying cochain complex.

MCK MCK

Ch(Vect^Z) Ch(Vect^Z)

𝐻(−)

𝐻(−)

Main results

The spectral sequence (𝐸_•(𝑀), 𝑑_•) associated with a multicomplex (𝑀, 𝐷_•) is constructed similarly as the one associated with a double complex. Namely, the total complex of𝑀enjoys a natural filtration by columns which gives us a spectral sequence by standard results. The𝐸₀-page of this spectral sequence is the under- lying cochain complex(𝑀, 𝐷₀)and the𝐸₁-page is the complex𝐻(𝑀)equipped with the transferred differential𝐷^′

1. In [LWZ20] an alternative description of this spectral sequence is presented. They give a description in terms of witnessed cocycles and coboundaries. This description allows for explicit computation of the differentials in the spectral sequence whenever we have such a family of witnesses.

This is the view we adopt for most of this text, and the proofs often boil down to finding the right witnesses. Our main goal is to understand the differentials in this spectral sequence. To do so, we introduce a family of multicomplexes(^𝑠𝑀,^𝑠𝐷_•) indexed by𝑠 ⩾1 defined by repeated application of the HTT. Roughly speaking, the multicomplex(¹𝑀,¹𝐷_•)is a shifted version of the cohomology complex𝐻(𝑀) together with the transferred differentials. We then define(^𝑠𝑀,^𝑠𝐷_•)inductively to be a shifted version of the multicomplex(𝐻(^𝑠−1𝑀),^𝑠−1𝐷^′_•). This re-indexing is inspired by the décalage functor introduced by Deligne in [Del71]. Put categori- cally, if we denote the re-indexing functor by𝜌, then the functor¹(−)is just the composition𝜌◦𝐻(−): MCK →MCKand^𝑠(−)=¹(−) ◦ · · · ◦¹(−)

| {z }

𝑠times

.

The essential observation is the following theorem relating the spectral sequence associated with𝑀and the one associated with¹𝑀.

Theorem A(Theorem4.0.6). The spectral sequence associated with¹𝑀is a shifted version of the spectral sequence associated with𝑀in the sense that

𝐸_𝑟^𝑝,𝑞(¹𝑀) =𝐸²_𝑟+^{𝑝+𝑞,−𝑝}

1 (𝑀) and ¹𝑑_𝑟 = 𝑑_𝑟+1 for all 𝑟 ⩾ ^0.

What we truly are doing is pushing the multicomplex structure along while computing the pages in the spectral sequence. That is, we equip each page in

the spectral sequence with a multicomplex structure coming from the previous page by choice of sections and the HTT. The following corollary of theorem A then tells us that the multicomplex structure on the𝐸_𝑟-page contains both the differential𝑑_𝑟 (as the underlying cochain complex) and all the information (the higher differentials) needed to compute𝑑_𝑟+1.

Corollary B(Corollary4.0.7). We have

𝐸^𝑝,𝑞_𝑟 (𝑀) =^𝑟𝑀^{𝑝−𝑟𝑛, 𝑞+𝑟𝑛} and 𝑑_𝑟 =^𝑟𝐷₀ for every𝑟 ⩾^{1 where𝑛 = 𝑝+𝑞.}

If we writeSpecSeq_Kfor the category of spectral sequences with vector spaces overKas entries, then theoremAand corollaryBcorresponds to the commutativity of the inner squares and the outer square in the following diagram, respectively.

MCK SpecSeq_K

MCK SpecSeq_K

... ...

MCK SpecSeq_K

𝐸(−)

1(−)

𝑠(−)

𝐸(−)

1(−)

𝜌⁻¹

1(−)

𝜌⁻¹

𝐸(−)

𝜌⁻¹

𝜌^−𝑠

Effectively, corollaryBenables us to compute the spectral sequence associated with a multicomplex, page-by-page. We point out that a result similar to corollaryBcan be found in [Lap07, Proposition 3.1] but stated in a slightly different mathematical language. The following corollary, which also follows immediately from theoremA, completely characterises degeneration of the associated spectral sequence in terms of the transferred differentials.

Corollary C(Corollary4.0.8). The spectral sequence associated with a multicom- plex𝑀 degenerates at the𝑘-th page if and only if^𝑘𝐷_𝑟 =0 for all𝑟 ⩾^0.

First-page degeneration

The main inspiration which led to theoremAand its corollaries, and especially corol- laryC, was the following result on first-page degeneration appearing in [DSV15].

Theorem D(Theorem3.0.1). The spectral sequence associated with a multicom- plex𝑀 degenerates at the first page if and only if all transferred differentials𝐷^′_𝑟 vanish.

Of course, once we have established theoremA, then theoremD follows from corollary C by setting 𝑘 = 1. Still, section 3 is entirely dedicated to proving theoremDand is included because it inspires the techniques used to prove the generalisation in section4.

Specialising to double complexes

Double complexes (also known as bicomplexes) are exactly the multicomplexes whose differentials𝐷_𝑟 vanish for𝑟 ⩾ 2. Double complexes over a field are well understood as they decompose into direct sums of "squares" and "zig-zags".1 Consequently, the spectral sequence associated with a double complex is also understood since the differentials involved can be computed by considering the zig-zags of different lengths appearing in the decomposition. Another approach to computing the spectral sequence associated with a double complex is applying the HTT and considering the transferred differentials on the cohomology complex.

As pointed out in [LV12], this approach using the HTT gives us a "lifted version"

of the spectral sequence. To be precise, if(𝐸_•, 𝑑_•)denotes the spectral sequence associated with the double complex(𝑀, 𝐷₀, 𝐷₁), and(𝐷_𝑟^′)_𝑟

⩾1are the transferred differentials on𝐻(𝑀, 𝐷₀), then we have the following result:

Theorem E. The map induced by𝐷_𝑟^′ on the𝐸_𝑟-page is exactly𝑑_𝑟.

The takeaway is that, in the case of double complexes, the transferred differentials on cohomology contain all the information of the associated spectral sequence (except the zeroth page, of course). We give two proofs of theorem E. The first one appears as proposition2.3.1. Later, in example4.0.9, we recover this result by applying corollaryB. TheoremEfails in the general case where higher differential might be non-trivial. This is seen in example4.0.3where𝐷^′

3 =0 but𝑑₃is non-trivial.

This suggests that there has to be more information contained in𝑑_𝑟 than just the transferred differentials𝐷_𝑟^′ on𝐻(𝑀, 𝐷₀)whenever𝑟 ⩾^3.

1This decomposition of double complexes has been known as folklore for a long time. Recently, proofs of this fact have been given in [Ste21] and [KQ20].

Minimal models for multicomplexes

Similarly to how a cochain complex𝑀can be decomposed into a direct sum𝐾⊕𝐻, this is also true for multicomplexes. A multicomplex(𝑀, 𝐷_•)is said to beminimal if𝐷₀ =0 and acyclic trivialif both𝐷_𝑟 = 0 for 𝑟 ⩾ 1 and the underlying cochain complex(𝑀, 𝐷₀)is acyclic. In appendixB, we follow [DSV15] and show how every multicomplex𝑀decomposes into a direct sum𝐾⊕𝐻 where𝐾is acyclic trivial and𝐻is minimal.

Multicomplexes as homotopy algebras

As is remarked in [DSV15], [Val14] and [LV12], multicomplexes can be viewed as algebras over a certain operad. This is also true for double complexes, which are algebras over the operadD of dual numbers. First, we define an operadM and show that the category of algebras2 over M is equivalent to the category of multicomplexes with the right definition of morphisms. We then explicitly compute the operad D^∞ which is the cobar construction on the Koszul dual cooperad ofD and show thatD^∞=M. Then, by definition, multicomplexes are exactly the homotopyD-algebras. Viewing multicomplexes as homotopy algebras allows us to apply results from the general theory of Koszul operads. For example, both the homotopy transfer theorem and minimal models for multicomplexes follows from more general results which can be found in [LV12].

Deformations of cochain complexes

We fix a cochain complex(𝑀, 𝐷₀)and consider those formal power series 𝐷(𝑡)= 𝐷₀+𝐷₁𝑡+𝐷₂𝑡²+ · · · ∈End(𝑀)

J𝑡 K

which satisfy𝐷² = 0. These are calledformal deformationsof(𝑀, 𝐷₀). We prove that formal deformations of cochain complexes form a category equivalent to the category of multicomplexes. We also very briefly mention finite order deformations of cochain complexes. Deformations of cochain complexes might serve as a motivation for looking at multicomplexes and give us a compact notation for encoding them. Of course, the results established in the earlier sections can be translated to the category of deformations. This may be fruitful. However, this direction is not further investigated in this text.

2Every operadPof arity 1 is completely determined by the algebraP(1). Moreover, algebras over operads of arity 1 correspond to dg-modules overP(1)

How to read it

If one is interested in the most general results to be found in this text, one can skip straight to section4and refer to section1and section2whenever necessary. On the other hand, if one wants the extended edition, including the motivation behind the general results, one should read it linearly from section 1. The appendices are not directly related to the main results but might motivate the notion of multicomplexes and reveal parts of the bigger picture they fit into.

Outline

Section1contains the preliminaries and prepares us for working with multicom- plexes and the associated spectral sequence. First, we recall how a filtered cochain complex gives rise to a spectral sequence and then go on to define multicomplexes and the filtration on the total complex. We end this section with an exposition of witnessed cocycles and coboundaries closely following [LWZ20].

Section2consists of three parts. The first part discusses the notions of homotopy equivalence and homotopy retracts for cochain complexes over a field. We prove (the well-known fact) that every cochain complex is homotopy equivalent to its cohomology complex and explicitly write out the maps involved. The second part is dedicated to the homotopy transfer theorem. The third part discusses the special case of double complexes and their associated spectral sequences.

Section3is about first-page degeneration and contains the proof of theoremD.

Section 4 starts off by showing that the second differential 𝑑₂ in the spectral sequence is the map induced by𝐷^′

2on the𝐸₂-page. We then give an example to show that this is not true for𝑑₃. Inspired by how this example fails, we go on to prove theoremAand end the section by proving its two corollaries.

AppendixAexplains how multicomplexes can be viewed as homotopyD-algebras.

AppendixBis a short note on minimal models for multicomplexes.

Appendix C shows how (formal) deformations of cochain complexes form a category equivalent to the category of multicomplexes. We also briefly mention finite order deformations of cochain complexes at the end.

Conventions

We restrict ourselves to the case of (graded) vector spaces over some fixed fieldK of characteristic 0. Some results work in the more general setting. However, most of the results rely on the fact that every short exact sequence splits. We stick to cohomological grading for complexes, multicomplexes and spectral sequences.

That is, differentials always increase the (total) degree by one. In the bigraded case, the differentials have bidegree(𝑟,1−𝑟)and can be visualised as follows:

· · ·

We write𝐽 for multi-indices(𝑗₁, 𝑗₂, . . . , 𝑗_𝑘)of length𝑘 ⩾ ^{1 with all 𝑗𝑠} ⩾ ^{1. When} summing over|𝐽| =𝑛, we sum over all such multi-indices with 𝑗₁+ · · · +𝑗_𝑘 =𝑛. For example, if(𝑎_𝑖)is some family indexed over the positive integers, then

Õ

|𝐽|=1

𝑎_𝑗

1𝑎_𝑗

2· · ·𝑎_𝑗𝑘 = 𝑎₁, Õ

|𝐽|=2

𝑎_𝑗

1𝑎_𝑗

2· · ·𝑎_𝑗𝑘 = 𝑎₁𝑎₁+𝑎₂, Õ

|𝐽|=3

𝑎_𝑗

1𝑎_𝑗

2· · ·𝑎_𝑗𝑘 = 𝑎₁𝑎₁𝑎₁+𝑎₁𝑎₂+𝑎₂𝑎₁+𝑎₃ and so on.

1 Multicomplexes and spectral sequences

First, we briefly recall how a filtration on a cochain complex gives rise to a spectral sequence. More details and omitted proofs can be found in [Spa95], [Wei95]

and [McC01]. Next, we clarify the notion of a multicomplex and morphisms of such. Under a mild boundedness assumption on a multicomplex𝑀, we can construct the associated total complex Tot𝑀, equipped with a natural filtration.

The spectral sequence arising from this filtration is what we will call the spectral sequence associated with𝑀. In the last part, we follow [LWZ20] and describe the associated spectral sequence in terms of witnessed cocycles and coboundaries.

This point of view will enable us to describe the differentials involved explicitly.

1.1 Filtered cochain complexes

Definition 1.1.1. A filtered cochain complex (𝐶, 𝐹) is a cochain complex 𝐶 with differential𝐷: 𝐶^𝑛 → 𝐶^𝑛+1together with a decreasing filtration𝐹 ={𝐹^𝑝𝐶}_𝑝

𝐶 ⩾ ^{· · ·} ⩾ ^𝐹𝑝𝐶 ⩾ ^{𝐹𝑝+1𝐶} ⩾ ^{· · ·} ⩾⁰

such that the differential on 𝐶 is compatible with the filtration in the sense that𝐷(𝐹^𝑝𝐶^𝑛)⩽ ^{𝐹𝑝𝐶𝑛+1.}

Given a filtered cochain complex(𝐶, 𝐹)we define 𝑍_𝑟^𝑝,• =𝐹^𝑝𝐶∩𝐷⁻¹(𝐹^𝑝+𝑟𝐶), 𝐵^𝑝,•

0 =𝑍^𝑝+1,•

0 and 𝐵^𝑝,•_𝑟 = 𝑍_𝑟−^𝑝+1,•

1 +𝐷𝑍^𝑝−𝑟+_𝑟− ^1,•

1

for𝑟 ⩾ 1. It can be shown that the quotients𝐸_𝑟^𝑝,𝑞 =𝑍^𝑝,𝑞_𝑟 /𝐵^𝑝,𝑞_𝑟 are well-defined and that the differential on𝐶induces differentials𝛿^𝑟:

𝑍^𝑝,𝑞_𝑟 𝑍_𝑟^{𝑝+𝑟,𝑞+1−𝑟}

𝐸^𝑝,𝑞_𝑟 𝐸^{𝑝+𝑟,𝑞+}_𝑟 ^1−𝑟

𝐷

𝛿^𝑟

where the vertical arrows are the quotient maps. The main point is that we have a spectral sequence. The following result is standard:

Proposition 1.1.2. There are isomorphisms𝐸^𝑝,𝑞_𝑟+

1 ^{𝐻𝑝+𝑞(𝐸𝑟,}𝛿^𝑟)for𝑟 ⩾ ^0.

If𝑥is an element of𝑍_𝑟^𝑝,𝑞, we denote its class in𝐸_𝑟^𝑝,𝑞 =𝑍^𝑝,𝑞_𝑟 /𝐵^𝑝,𝑞_𝑟 by[𝑥]_𝑟. Using this notation, we have that𝛿^𝑟([𝑥]_𝑟)=[𝐷𝑥]_𝑟.

Definition 1.1.3. A filtration 𝐹 on 𝐶 is said to be convergent if Ñ

𝑛𝐹^𝑛𝐶 = 0 andÐ

𝑛𝐹^𝑛𝐶 =𝐶. We say that𝐹isbounded belowif for every𝑛, there exists a𝑞(𝑛) such that𝐹^𝑞(𝑛)𝐶^𝑛 =0.

If(𝐶, 𝐹)is a filtered cochain complex, then there is an induced filtration on the cohomology of𝐶 given by𝐹^𝑝𝐻(𝐶) :=Im(𝐻(𝐹^𝑝𝐶) → 𝐻(𝐶)). Requiring that𝐹is bounded below ensures that the induced filtration on𝐻(𝐶)is convergent as well.

Theorem 1.1.4. [Spa95, p.469] Let (𝐶, 𝐹) be a filtered cochain complex with 𝐹 convergent and bounded below. Then there is a convergent spectral sequence with

𝐸^𝑝,𝑞

0 =𝐹^𝑝𝐶^𝑝+𝑞/𝐹^𝑝+1𝐶^𝑝+𝑞, 𝐸^𝑝,𝑞

1 =𝐻^𝑝+𝑞(𝐹^𝑝𝐶/𝐹^𝑝+1𝐶)

and𝐸_∞is isomorphic to the associated graded of the induced filtration on𝐻(𝐶). The following notion of degeneracy will be central throughout this text.

Definition 1.1.5. A spectral sequence(𝐸_•, 𝑑_•)is said todegenerate at the𝑘-th page if𝑑_𝑟 =0 for all𝑟 ⩾ ^𝑘.

Degeneration on some page merely means that nothing interesting will happen from that point on. That is, we have arrived at our target𝐸_∞.

1.2 Multicomplexes and their associated spectral sequences

Definition 1.2.1. Amulticomplex(𝑀, 𝐷_•)overKconsists of a bigradedK-vector space 𝑀 = {𝑀^𝑝,𝑞} together with a family of linear maps {𝐷_𝑟: 𝑀 → 𝑀}_𝑟⩾0 of bidegrees|𝐷_𝑟|=(𝑟,1−𝑟). These maps are required to satisfy the relation

Õ

𝑝+𝑞=𝑛

𝐷_𝑝𝐷_𝑞 =0 for every𝑛 ⩾ ^0.

The maps 𝐷_𝑟 when 𝑟 ⩾ 1 are calledhigher differentials(or sometimes, even just differentials). This is a slight abuse of terminology as they in general do not square to zero. Multicomplexes generalise the notion of double complexes and cochain complexes. A multicomplex where the higher differentials𝐷_𝑟 =0 for all 𝑟 ⩾ ^{2 is} precisely a double complex. A multicomplex with only𝐷₀possibly non-trivial is a cochain complex of graded vector spaces. For every multicomplex(𝑀, 𝐷_•), we have an underlying cochain complex(𝑀, 𝐷₀)of graded vector spaces. One can visualise(𝑀, 𝐷₀)as the following diagram:

... ... ...

· · · 𝑀^{𝑝−1,𝑞+1} 𝑀^𝑝,𝑞+1 𝑀^{𝑝+1,𝑞+1} · · ·

· · · 𝑀^{𝑝−1,𝑞} 𝑀^𝑝,𝑞 𝑀^𝑝+1,𝑞 · · ·

· · · 𝑀^{𝑝−1,𝑞−1} 𝑀^{𝑝,𝑞−1} 𝑀^{𝑝+1,𝑞−1} · · ·

... ... ...

𝐷₀

We denote the cohomology of(𝑀, 𝐷₀)in degree(𝑝, 𝑞)by

𝐻^𝑝,𝑞(𝑀):= 𝐻^𝑞(𝑀^𝑝,•, 𝐷₀)= ker(𝐷₀: 𝑀^𝑝,𝑞 → 𝑀^𝑝,𝑞+1) Im(𝐷₀: 𝑀^{𝑝,𝑞−1} → 𝑀^𝑝,𝑞) and endow𝐻(𝑀)with trivial differential.

Remark 1.2.2. Multicomplexes (not necessarily over a field) appear in [Wal61]

where resolutions for extensions of groups are constructed as the total complex of a multicomplex with𝐷_𝑟 =0 for𝑟 ⩾3. Such multicomplexes also appear in [Liu17]

and [Liu14] under the name "homotopy double complexes", where they are used to construct resolutions of certain generalised Weyl algebras.

Definition 1.2.3. Let(𝑀, 𝐷_•)and(𝑁 ,𝐷˜_•)be multicomplexes. Amorphism 𝑓: (𝑀, 𝐷_•) → (𝑁 ,𝐷˜_•)

of multicomplexesconsists of a family 𝑓 = {𝑓_𝑛: 𝑀 → 𝑁 | 𝑓_𝑛(𝑀^𝑝,𝑞) ⩽ ^{𝑁𝑝+𝑛,𝑞−𝑛}𝑛}⩾0

of linear maps . In addition, we require the maps to satisfy Õ

𝑝+𝑞=𝑛

𝑓_𝑝𝐷_𝑞 = Õ

𝑝+𝑞=𝑛

𝐷˜_𝑝𝑓_𝑞 for all 𝑛 ⩾ ^{0. (1)}

For𝑛 =0, eq. (1) amounts to𝑓₀being a cochain map(𝑀, 𝐷₀) → (𝑁 ,𝐷˜₀). In the case where𝑀 and𝑁 are double complexes and 𝑓_𝑛 =0 for𝑛 ⩾1, eq. (1) is to say that 𝑓₀ is a morphism of double complexes, i.e. 𝑓 commutes with both differentials. On a multicomplex𝑀, theidentity morphismid^𝑀: 𝑀 → 𝑀is given by(id^𝑀)₀=id^(𝑀,𝐷₀⁾ and(id𝑀)_𝑛 = 0 for all 𝑛 ⩾ ^{1. If 𝑓 and 𝑔} are morphisms of multicomplexes, we

define theircomposition 𝑔 𝑓 by(𝑔 𝑓)_𝑛 =Í

𝑝+𝑞=𝑛 𝑔_𝑝𝑓_𝑞 whenever it makes sense.

Remark 1.2.4. In the operadic language of [DSV15] morphisms of multicomplexes are called∞-morphisms.

Definition 1.2.5. A morphism 𝑓 of multicomplexes is an isomorphism (quasi- isomorphism)if 𝑓₀is an isomorphism (quasi-isomorphism).

Proposition 1.2.6. A morphism 𝑓 of multicomplexes is invertible if and only if 𝑓₀ is an isomorphism of cochain complexes.

Proof. If 𝑓 has inverse𝑔, then(𝑓 𝑔)₀is the identity on(𝑀, 𝐷₀). Similarly, the same is true for 𝑔 𝑓. Conversely, suppose 𝑓₀is an isomorphism and denote its inverse by𝑔₀. It is now a matter of solving(𝑔 𝑓)_𝑛 =0 for each𝑛 ⩾ 1. Doing this, we obtain the following unique solution𝑔:

𝑔_𝑛 = Õ

|𝐼|=𝑛

(−1)^𝑘𝑔₀ 𝑓_𝑖

1𝑔₀ 𝑓_𝑖

2𝑔₀· · ·𝑔₀ 𝑓_𝑖𝑘 𝑔₀ for𝑛 ⩾ ^1.

□ Throughout this text we will assume the following boundedness condition on multicomplexes: a multicomplex (𝑀, 𝐷_•) is said to be bounded below if for each𝑛 there exists an integer 𝑠(𝑛) such that𝑀^{𝑝,𝑛−𝑝} = 0 whenever 𝑝 ⩾ ^{𝑠(𝑛). In} other words, each anti-diagonal eventually vanish going to the right. We asso- ciate to a multicomplex 𝑀 the total complex denoted Tot𝑀 given in degree 𝑛 by Tot𝑀^𝑛 := É

𝑎+𝑏=𝑛𝑀^𝑎,𝑏. We make Tot𝑀 into a cochain complex by giving it the differential𝐷 := Í

𝑟⩾0𝐷_𝑟: Tot𝑀^𝑛 → Tot𝑀^𝑛+1. It is easy to see that𝐷 is locally finite and hence well-defined whenever𝑀 is bounded below. There is a natural filtration by columns on Tot𝑀defined by letting

𝐹^𝑝Tot𝑀^𝑛 := Ê

𝑎+𝑏=𝑛 𝑎⩾𝑝

𝑀^𝑎,𝑏. (2)

This filtration turns(Tot𝑀, 𝐹)into a filtered complex.

Remark 1.2.7. Convergence in the case where we do not assume any finiteness condition on the multicomplex is discussed in [Boa98, Section 11].

Lemma 1.2.8. If(𝑀, 𝐷_•)is bounded below, then the filtration defined in eq. (2) is bounded below.

Proof. By assumption, there exists for every𝑛an integer𝑠(𝑛)such that𝑀^{𝑝,𝑛−𝑝} =0 whenever𝑝 ⩾ ^𝑠(𝑛). By letting𝑞(𝑛):=𝑠(𝑛), it is evident that𝐹^𝑞(𝑛)𝐶^𝑛 =0. □ The following corollary of theorem1.1.4now follows from the previous lemma:

Corollary 1.2.9. Let(𝑀, 𝐷_•)be a multicomplex which is bounded below. Then the spectral sequence associated with the filtration on Tot𝑀 converges to𝐻(Tot𝑀). The spectral sequence in the above corollary is calledthe spectral sequence associated with the multicomplex𝑀. The zeroth page of this spectral sequence is given by

𝐸^𝑝,𝑞

0 = 𝐹^𝑝𝑀^𝑝+𝑞/𝐹^𝑝+1𝑀^𝑝+𝑞 = 𝑀^𝑝,𝑞

with differential 𝛿0 = 𝐷₀. In other words, 𝐸₀ is just the underlying cochain complex(𝑀, 𝐷₀). The first page is the given by𝐸^𝑝,𝑞

1 =𝐻^𝑞(𝑀^𝑝,•, 𝐷₀)and 𝛿1is the map induced by𝐷₁on cohomology. We should be aware that this pattern does not generally hold for the differentials on later pages, as can be seen in the following example borrowed from [Hur10]:

Example 1.2.10. Consider the following multicomplex 𝑀 consisting of one- dimensional vector spaces:

0 0

0 K𝑎 K𝑐 0

0 K𝑏 K𝑑 0

0 0

𝐷₁ 𝐷₀

𝐷₁

with differentials𝐷₁(𝑎) =𝑐, 𝐷₁(𝑏) =𝑑,𝐷₀(𝑏)= 𝑐and𝐷_𝑟 =0 for all𝑟 ⩾2. The total complex

0→K𝑏⊕K𝑎

©

­

­

«

𝐷₁ 0 𝐷₀ 𝐷₁

ª

®

®

−−−−−−−−→¬ K𝑑⊕K𝑐 →0.

is exact because the differential𝐷 = 𝐷₀+𝐷₁is an isomorphism. By corollary1.2.9 this means that𝐸_∞^𝑝,𝑞 =0 for all𝑝, 𝑞. Taking cohomology with respect to𝛿0 =𝐷₀, we are left with two non-trivial entries at the𝐸₁-page:

0 K𝑎 0

0 K𝑑 0

𝛿1 𝛿1

𝛿2≠0

𝛿1 𝛿1

Since𝐸₃=𝐸_∞by degree reasons, the generators 𝑎and𝑏 must be killed by the𝛿2

differential and hence𝛿2 ≠0, whereas𝐷₂is zero.

1.2.1 Description in terms of witnessed cocycles and coboundaries

This part is essentially [LWZ20] with notation and grading conventions adapted to our setting. Let(𝑀, 𝐷_•)be a multicomplex and𝑇 =Tot𝑀. The filtration on𝑇 defined as in eq. (2) is denoted by𝐹. If𝑥 ∈ 𝐹^𝑝𝑇, we can write𝑥uniquely as the sum

𝑥= 𝑥_𝑝+𝑥_𝑝+1+ · · · +𝑥_{𝑝+𝑟−1}+𝑥^′ (3) where𝑥_𝑝+𝑗 ∈ 𝑀^{𝑝+𝑗,•}is the projection of 𝑥 to the column𝑀^{𝑝+𝑗,•} and𝑥^′ ∈ 𝐹^𝑝+𝑟𝑇. Now, suppose that𝑥 ∈ 𝑍_𝑟^𝑝,•. This is to say that𝑥 ∈ 𝐹^𝑝𝑇and𝐷𝑥 ∈ 𝐹^𝑝+𝑟𝑇where𝐷 is the differential on𝑇. As a consequence of eq. (3), the parts of 𝐷𝑥 which lie in𝐹^𝑝+𝑗𝑇for𝑗 =0,1, . . . , 𝑟−1 must vanish. In other words, the following equations are required to hold true.

𝐷₀𝑥_𝑝 =0 𝐷₀𝑥_𝑝+1+𝐷₁𝑥_𝑝 =0

... 𝐷₀𝑥_{𝑝+𝑟−1}+𝐷₁𝑥_{𝑝+𝑟−2}+ · · · +𝐷_𝑟−1𝑥_𝑝 =0.

(4)

Similarly, if we let𝑥 =𝐷𝑤for some𝑤 ∈ 𝐹^{𝑝−𝑟+1}𝑇we obtain another set of equations.

These observations lead to the definition of the following two subspaces of𝑀^𝑝,•: 𝒵_𝑟^𝑝,•= _{𝑥 ∈ 𝑀}𝑝,• | ∃𝑥_𝑝+𝑗 ∈ 𝑀^{𝑝+𝑗,•} for 1 ⩽ ^𝑗 ⩽ ^𝑟−1 such that

𝐷₀𝑥 =0 and𝐷_𝑛𝑥+

𝑛−1

Õ

𝑖=0

𝐷_𝑖𝑥_{𝑝+𝑛−𝑖} =0 for 1⩽ ^𝑛 ⩽^{𝑟 −1}

and

ℬ_𝑟^𝑝,•= _{𝑥 ∈ 𝑀}𝑝,• | ∃𝑤_𝑝−𝑗 ∈ 𝑀^{𝑝−𝑗,•} for 0 ⩽ ^𝑗 ⩽ ^𝑟−1 such that 𝑥 =

𝑟−1

Õ

𝑗=0

𝐷_𝑗𝑤_𝑝−𝑗 and

𝑟−1

Õ

𝑗=𝑙

𝐷_𝑗−𝑙𝑤_𝑝−𝑗 =0 for 1 ⩽ ^𝑙 ⩽ ^{𝑟−1 .}

Proposition 1.2.11. [LWZ20, Proposition 2.7] We have thatℬ_𝑟^𝑝,• ⩽^{𝒵𝑟𝑝,•.} Proof. Pick some coboundary𝑥 ∈ ℬ_𝑟^𝑝,•. First, we confirm that𝐷₀𝑥 =0:

𝐷₀𝑥=

𝑟−1

Õ

𝑗=0

𝐷₀𝐷_𝑗𝑤_𝑝−𝑗 =−

𝑟−1

Õ

𝑗=1 𝑗−1

Õ

𝑙=0

𝐷_𝑗−𝑙𝐷_𝑙𝑤_𝑝−𝑗 =−

𝑟−1

Õ

𝑙=1

𝐷_𝑙

𝑟−1

Õ

𝑗=𝑙

𝐷_𝑗−𝑙𝑤_𝑝−𝑗

| {z }

=0

=0.

Next, we claim that the elements𝑥_𝑝+𝑗 =Í^𝑟−1

𝑖=0𝐷_𝑗+𝑖𝑤_𝑝−𝑖 ∈ 𝑀^{𝑝+𝑗,•}where 1⩽ ^𝑗 ⩽ ^𝑟−1 satisfy the required relations for𝑥 to be a cocycle. Again, we check this by direct computation. Let𝑥_𝑝 := 𝑥to simplify expressions.

𝑛

Õ

𝑖=0

𝐷_𝑖𝑥_{𝑝+𝑛−𝑖} =

𝑛

Õ

𝑖=0

𝐷_𝑖

𝑟−1

Õ

𝑘=0

𝐷_{𝑛−𝑖+𝑘}𝑤_𝑝−𝑘 =

𝑟−1

Õ

𝑘=0 𝑛

Õ

𝑖=0

𝐷_𝑖𝐷_{𝑛+𝑘−𝑖}𝑤_𝑝−𝑘

=−

𝑟−1

Õ

𝑘=1 𝑛+𝑘

Õ

𝑖=𝑛+1

𝐷_𝑖𝐷_{𝑛+𝑘−𝑖}𝑤_𝑝−𝑘 =−

𝑛+𝑟−1

Õ

𝑖=𝑛+1

𝐷_𝑖

𝑟−1

Õ

𝑘=𝑖−𝑛

𝐷_{𝑘−𝑖+𝑛}𝑤_𝑝−𝑘

| {z }

=0

=0.

□ Proposition 1.2.12. [LWZ20, Proposition 2.8] The map 𝜓: 𝐸^𝑝,𝑞_𝑟 → 𝒵_𝑟^𝑝,•/ℬ_𝑟^𝑝,•

defined by letting𝜓([𝑥]_𝑟)=[𝑥_𝑝]is an isomorphism.

Proof. Consider the map𝜓ˆ: 𝑍_𝑟^𝑝,• → 𝒵_𝑟^𝑝,•/ℬ_𝑟^𝑝,•defined by 𝑥↦→ [𝑥_𝑝]. 1. The map𝜓ˆ is well-defined and surjective:

Let𝑥 ∈ 𝑍^𝑝,•_𝑟 . From the decomposition in eq. (3) and the relations in eq. (4) we see that𝑥_𝑝 ∈ 𝒵_𝑟^𝑝,•, i.e.,𝜓ˆ(𝑍_𝑟^𝑝,•) ⩽^{𝒵𝑟𝑝,•}. Now, if the cocycle𝑥 ∈ 𝒵_𝑟^𝑝,•is witnessed by the elements𝑥_𝑝+1, . . . , 𝑥_{𝑝+𝑟−1}, then the image of 𝑦 := 𝑥 +𝑥_𝑝+1+ · · · +𝑥_{𝑝+𝑟−1} under𝜓ˆ is exactly 𝑥. The only thing left to check is that𝑦 ∈𝑍^𝑝,•_𝑟 , but this follows from eq. (4).

2. We have inclusionker𝜓ˆ ⩽ ^{𝐵𝑟𝑝,•:}

Suppose that𝑥 ∈ker𝜓ˆ ⩽ ^{𝑍𝑝,•𝑟} . From eq. (3), it follows that we can write𝑥 =𝑥_𝑝+𝑤 where 𝑥_𝑝 ∈ 𝑀^𝑝,• and 𝑤 ∈ 𝐹^𝑝+1𝑇. By assumption, 𝑥_𝑝 ∈ ℬ_𝑟^𝑝,• so there exist

witnesses𝑤_𝑝, 𝑤_𝑝−1, . . . , 𝑤_{𝑝−𝑟+1}with𝑤_𝑝−𝑗 ∈ 𝑀^{𝑝−𝑗,•}satisfying

𝑥_𝑝 =

𝑟−1

Õ

𝑗=0

𝐷_𝑗𝑤_𝑝−𝑗 and (5)

0=

𝑟−1

Õ

𝑗=𝑙

𝐷_𝑗−𝑙𝑤_𝑝−𝑗 for 1 ⩽ ^𝑙 ⩽ ^{𝑟−1. (6)}

Define the element𝑐 :=Í^𝑟−1

𝑘=0𝑤_𝑝−𝑘 ∈ 𝐹^{𝑝−𝑟+1}𝑇. From eq. (6) it follows that𝐷𝑐 ∈ 𝐹^𝑝𝑇 and so by definition we have that 𝑐 ∈ 𝑍^𝑝−𝑟+_𝑟− ^1,•

1 . Furthermore, eq. (5) implies that the part of 𝐷𝑐 which lives in degree 𝑝 is exactly 𝑥_𝑝 so (𝑥_𝑝 − 𝐷𝑐)_𝑝 = 0 and hence 𝑥_𝑝 − 𝐷𝑐 ∈ 𝐹^𝑝+1𝑇. Define the element 𝑏 := 𝑥_𝑝 − 𝐷𝑐 + 𝑤 ∈ 𝐹^𝑝+1𝑇. We can now write 𝑥 = 𝑏 + 𝐷𝑐 and consequently 𝐷𝑥 = 𝐷𝑏 + 𝐷²𝑐 = 𝐷𝑏. By assumption,𝑥 ∈ 𝑍^𝑝,•_𝑟 so𝐷𝑥 =𝐷𝑏 ∈ 𝐹^𝑝+𝑟𝑇 and therefore𝑏 ∈ 𝑍_𝑟−^𝑝+1,•

1 . We conclude that𝑥 ∈ 𝑍^𝑝+_𝑟−^1,•

1 +𝐷(𝑍^𝑝−𝑟+_𝑟− ^1,•

1 )= 𝐵^𝑝,•_𝑟 . 3. We have inclusion𝐵_𝑟^𝑝,• ⩽^ker𝜓:^ˆ

Let𝑥 ∈ 𝐵^𝑝,•_𝑟 and write𝑥 =𝑏+𝐷𝑐with𝑏 ∈ 𝑍_𝑟−^𝑝+1,•

1 and𝑐 ∈ 𝑍_𝑟−^{𝑝−𝑟+1,•}

1 . By definition, we have𝑏 ∈ 𝐹^𝑝+1𝑇 and𝐷𝑐 ∈ 𝐹^𝑝𝑇. Now, observe that𝑥_𝑝 =(𝐷𝑐)_𝑝 =Í^𝑟−1

𝑗=0𝐷_𝑗𝑤_𝑝−𝑗. Furthermore, 0=(𝐷𝑐)_𝑝−𝑙 =Í^𝑟−1

𝑗=𝑙 𝐷_𝑗−𝑙𝑐_𝑝−𝑗 for each𝑙 =1,2, . . . , 𝑟−1. We conclude that𝑥_𝑝 ∈ ℬ_𝑟^𝑝,•and hence𝜓ˆ(𝑥)=0.

It now follows from the first isomorphism theorem that𝜓 is an isomorphism.

𝑍_𝑟^𝑝,• 𝒵_𝑟^𝑝,•/ℬ_𝑟^𝑝,•

𝐸_𝑟^𝑝,•

𝜓ˆ 𝜓

□ Proposition 1.2.13. [LWZ20, Theorem 2.10] Under the isomorphism in proposi- tion1.2.12the differentials of the spectral sequence are given by

𝑑_𝑟: 𝒵_𝑟^𝑝,•/ℬ_𝑟^𝑝,• −→ 𝒵_𝑟^{𝑝+𝑟,•}/ℬ_𝑟^{𝑝+𝑟,•}

[𝑥] ↦→

"

𝐷_𝑟𝑥+

𝑟−1

Õ

𝑖=1

𝐷_𝑖𝑥_{𝑝+𝑟−𝑖}

#

where𝑥_𝑝+1, 𝑥_𝑝+2, . . . , 𝑥_{𝑝+𝑟−1}are witnesses for𝑥 ∈ 𝒵_𝑟^𝑝,•.

Proof. From the proof of proposition1.2.12, we know that𝑥+𝑥_𝑝+1+· · ·+𝑥_{𝑝+𝑟−1}lives in𝑍_𝑟^𝑝,•. Thus,[𝑥+𝑥_𝑝+1+ · · · +𝑥_{𝑝+𝑟−1}]_𝑟 ∈ 𝐸_𝑟^𝑝,•and𝜓[𝑥+𝑥_𝑝+1+ · · · +𝑥_{𝑝+𝑟−1}]_𝑟 =[𝑥]. The result now follows by direct computation:

𝑑_𝑟[𝑥] =𝜓𝛿^𝑟[𝑥+𝑥_𝑝+1+ · · · +𝑥_{𝑝+𝑟−1}]_𝑟

=𝜓[𝐷(𝑥+𝑥_𝑝+1+ · · · +𝑥_{𝑝+𝑟−1})]_𝑟

=[(𝐷(𝑥+𝑥_𝑝+1+ · · · +𝑥_{𝑝+𝑟−1}))_𝑝+𝑟]

=

"

𝐷_𝑟𝑥+

𝑟−1

Õ

𝑖=1

𝐷_𝑖𝑥_{𝑝+𝑟−𝑖}

# .

□ The following diagram illustrates how the image of𝑥 under𝑑_𝑟 is computed from a family of witnesses.

𝑥 •

𝑥_𝑝+1 •

. . . •

𝑥_{𝑝+𝑟−1} 𝑑_𝑟[𝑥_𝑝]

𝐷𝑟

𝐷𝑟−1

𝐷₁

Throughout the rest of this text, when we talk about the spectral sequence associated with a multicomplex, we will stick to this description in terms of witnessed cocycles and coboundaries. That is, we write𝐸_𝑟^𝑝,•=𝒵_𝑟^𝑝,•/ℬ_𝑟^𝑝,•and denote the differentials of the spectral sequence by𝑑_𝑟 for𝑟 ⩾ 0. Moreover, for an𝑟-cocycle𝑥 ∈ 𝒵_𝑟^𝑝,•, we denote the class represented by𝑥in𝐸_𝑟^𝑝,•by[𝑥]_𝑟. Sometimes, we will leave out the subscript if it is clear from context where the classes live.

Example 1.2.14. If(𝑀, 𝐷₀, 𝐷₁)is a double complex, then the differentials in the associated spectral sequence is given by𝑑₀=𝐷₀and𝑑_𝑟[𝑥]=[𝐷₁𝑥_{𝑝+𝑟−1}]for𝑟 ⩾¹ where𝑥_𝑝 :=𝑥.

Proposition 1.2.15. We have inclusionsℬ_𝑟^𝑝,• ⩽ ^ℬ𝑟+^𝑝,•1and𝒵_𝑟+^𝑝,•

1 ⩽ ^{𝒵𝑟𝑝,•for all𝑟} ⩾ ^1.

Proof. If𝑥 ∈ 𝒵^𝑝,•

𝑟+1is witnessed by the elements 𝑥_𝑝+1, . . . , 𝑥_{𝑝+𝑟−1}, 𝑥_𝑝+𝑟, then clearly the elements 𝑥_𝑝+1, . . . , 𝑥_{𝑝+𝑟−1} witnesses that 𝑥 ∈ 𝒵_𝑟^𝑝,•. Similarly, if 𝑥 ∈ ℬ_𝑟^𝑝,• is a coboundary witnessed by the elements 𝑤_𝑝, 𝑤_𝑝−1, . . . , 𝑤_{𝑝−𝑟+1}, we can simply

define𝑤_𝑝−𝑟 :=0. □

Corollary 1.2.16. Let𝑥, 𝑦 ∈ 𝒵_𝑟^𝑝,•. If[𝑥]_𝑟

0 =[𝑦]_𝑟

0 for some𝑟₀ ⩽ ^{𝑟 then[𝑥]𝑠 =[𝑦]𝑠} for all𝑟₀ ⩽ ^𝑠 ⩽ ^𝑟.

Proof. By assumption𝑥−𝑦 ∈ ℬ_𝑟^𝑝,•

0 ⩽ ^{· · ·}⩽ ^{ℬ𝑠𝑝,•} ⩽ ^{· · ·} ⩽ ^{ℬ𝑟𝑝,•.} □ In particular, if two𝑟-cocycles represent the same class in cohomology, then they also represent the same class on the𝐸_𝑟-page.

2 Homotopy transfer

Let(𝑀, 𝐷_•)be a multicomplex. Given a cochain complex(𝑁 , 𝐷^′

0)which is quasi- isomorphic to the underlying cochain complex of𝑀, we can transfer the higher differentials𝐷₁, 𝐷₂, . . .to obtain a multicomplex(𝑁 , 𝐷_•^′). This is the content of the homotopy transfer theorem (HTT) for multicomplexes. In general, we need a homotopy retract from𝑀 to𝑁, but as we work over a field, a quasi-isomorphism (actually, even just isomorphic cohomology groups) turns out to be sufficient. We shall be most interested in the case where𝑁 is the cohomology complex𝐻(𝑀, 𝐷₀) equipped with a trivial differential. Before that, we prove some elementary results about cochain complexes over a field.

2.1 Cochain complexes over a field

Definition 2.1.1. Let(𝑀, 𝐷)and(𝑁 , 𝐷^′)be cochain complexes overK. Ahomotopy equivalence(𝑓 , 𝑔, ℎ, ℎ^′)between𝑀 and𝑁 consists of cochain maps 𝑓: 𝑀 ↔ 𝑁 :𝑔 together with mapsℎ: 𝑀 → 𝑀and ℎ^′: 𝑁 →𝑁 of degree−1 such that

𝑔 𝑓 −id^𝑀 = ℎ𝐷+𝐷 ℎ and 𝑓 𝑔−id^𝑁 = ℎ^′𝐷^′+𝐷^′ℎ^′.

In the graded setting, we require the maps above to be graded maps, i.e., respect the grading. Two cochain complexes are said to behomotopy equivalentif there exists a homotopy equivalence between them. A cochain complex iscontractibleif it is homotopy equivalent to the zero complex.

(𝑀, 𝐷) (𝑁 , 𝐷^′).

𝑓

ℎ ℎ^′

𝑔

We introduce an intermediate notion between quasi-isomorphism and homotopy equivalence following the terminology used in [LV12] and [DSV15]:

Definition 2.1.2. Let(𝑀, 𝐷)and(𝑁 , 𝐷^′)be cochain complexes overK. Ahomotopy retract(𝜋,𝜄, ℎ)of 𝑀 to 𝑁 consists of cochain maps𝜋: 𝑀 → 𝑁,𝜄: 𝑁 → 𝑀 and a homotopy ℎ: 𝑀 → 𝑀 of degree −1 such that 𝜄𝜋−id𝑀 = 𝐷 ℎ + ℎ𝐷 and 𝜄 (or equivalently 𝜋) is a quasi-isomorphism. If in addition we have 𝜋𝜄 = id𝑁, then(𝜋,𝜄, ℎ)is called adeformation retract.

(𝑀, 𝐷) (𝑁 , 𝐷^′).

𝜋 ℎ

𝜄

Every deformation retract extends to a homotopy equivalence by settingℎ^′=0 and every homotopy equivalence(𝑓 , 𝑔, ℎ, ℎ^′)restricts to a homotopy retract(𝑓 , 𝑔, ℎ). More is true when working over a field: every quasi-isomorphism extends to a homotopy equivalence. Moreover, given the data(𝜋,𝜄, ℎ)of a homotopy retract, we can find aℎ^′so that(𝜋,𝜄, ℎ, ℎ^′)becomes a homotopy equivalence. Whenever(𝑀, 𝐷) is a cochain complex and there is no room for confusion, we write𝑍^𝑛 = 𝑍^𝑛(𝑀) for the𝑛-cocycles,𝐵^𝑛 = 𝐵^𝑛(𝑀)for𝑛-coboundaries and𝐻^𝑛 =𝐻^𝑛(𝑀)for the𝑛-th cohomology space.

Proposition 2.1.3. Every cochain complex (𝑀, 𝐷) over K is isomorphic to the cochain complex𝐾⊕𝐻where𝐾^𝑛 = 𝐵^𝑛 ⊕𝐵^𝑛+1with differential𝐷_𝐾 = ^{0 1}_{0 0}

and𝐻 is the cohomology of𝑀with trivial differential𝐷_𝐻 =0.

Proof. Working over a field, the short exact sequences

0→𝐵^𝑛 → 𝑍^𝑛 →𝐻^𝑛 → 0 and 0→ 𝑍^𝑛 → 𝑀^{𝑛 𝐷}−→𝐵^𝑛+1→0

split under a choice of sections. Thus, we can write𝑀^𝑛 = 𝑍^𝑛⊕ ˜𝐵^𝑛+1= 𝐵^𝑛⊕ ˜𝐻^𝑛⊕ ˜𝐵^𝑛+1 where𝐻˜^𝑛 and𝐵˜^𝑛+1are isomorphic to𝐻^𝑛 and𝐵^𝑛+1respectively. Let 𝛼˜: 𝐻˜^𝑛 →𝐻^𝑛 and 𝛽˜: 𝐵˜^𝑛+1 → 𝐵^𝑛+1 be the isomorphisms we obtain from choosing sections.

Define the isomorphism 𝜙 :=

_{1 0 0}

0 0 𝛽˜ 0𝛼˜ 0

: 𝑀^𝑛 →𝐵^𝑛 ⊕𝐵^𝑛+1⊕𝐻^𝑛.

Clearly, the following diagram with𝐷^′:=𝐷_𝐾+𝐷_𝐻 = _{0 1 0}

0 0 0 0 0 0

commutes.

𝑀^𝑛 𝑀^𝑛+1

𝐵^𝑛 ⊕𝐵^𝑛+1⊕𝐻^𝑛 𝐵^𝑛+1⊕𝐵^𝑛+2⊕𝐻^𝑛+1

𝐷 𝜙

𝐷^′

𝜙⁻¹

In other words, we have that𝑀 ^𝐾⊕𝐻. Furthermore, the differential𝐷is given

by𝐷^′under this identification. □

Proposition 2.1.4. The decomposition in proposition2.1.3does not depend on the

choice of sections.

Proof. Suppose we pick two different sets of sections so that 𝑀^𝑛 =𝐵^𝑛 ⊕ ˜𝐻^𝑛 ⊕ ˜𝐵^𝑛+1 =𝐵^𝑛 ⊕ ˆ𝐻^𝑛 ⊕ ˆ𝐵^𝑛+1.

Denote the isomorphisms coming from these sections by

𝛼˜: 𝐻˜^𝑛 → 𝐻^𝑛, 𝛽˜: 𝐵˜^𝑛+1→ 𝐵^𝑛+1, 𝛼ˆ: 𝐻ˆ^𝑛 →𝐻^𝑛 and 𝛽ˆ: 𝐵ˆ^𝑛+1→ 𝐵^𝑛+1. As𝐻 is equipped with a trivial differential, the isomorphism 𝛼ˆ⁻¹𝛼˜: 𝐻˜ → ˆ𝐻 is trivially an isomorphism of cochain complexes. Finally, it is easy to verify that the map

_{1 0}

0𝛽ˆ⁻¹𝛽˜

: 𝐾˜ → ˆ𝐾

is an isomorphism of cochain complexes. □

Throughout this text, when we identify a cochain complex𝑀with the decomposi- tion𝐾⊕𝐻, we will usually leave out the isomorphisms in the proofs above and simply write equality𝑀 =𝐾 ⊕𝐻.

Proposition 2.1.5. Every cochain complex(𝑀, 𝐷)overK, admits a deformation retract(𝜋,𝜄, ℎ)of𝑀 to its cohomology.

(𝑀, 𝐷) (𝐻(𝑀),0).

𝜋 ℎ

𝜄

Proof. By proposition2.1.3we can write 𝑀as the decomposition𝑀 = 𝐾^𝑛 ⊕𝐻^𝑛 where the differential is given by

𝐷 = _{0 1 0}

0 0 0 0 0 0

: 𝐾^𝑛 ⊕𝐻^𝑛 → 𝐾^𝑛+1⊕𝐻^𝑛+1. It is straightforward to check that the maps

𝜄= ₀

01

, 𝜋=(_{0 0 1}) and ℎ =

_{0 0 0}

−1 0 0 0 0 0

form a deformation retract as claimed. □

The following remark will be essential for us later, as it allows us to apply the homotopy transfer theorem to get a multicomplex structure on the cohomology.