On the Merkulov construction of $A_{ \infty}$ -(co)algebras
UDC 512.5 The aim of this short note is to complete some aspects of a theorem proved by S. Merkulov in [Int. Math. Res. Not. IMRN. – 1999. – 3. – P. 153 – 167] (Theorem 3.4), as well as to provide a complete proof of the dual result for dg coalgebras.
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| author | Herscovich, E. Герскович, Е. |
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The aim of this short note is to complete some aspects of a theorem proved by S. Merkulov in [Int. Math. Res. Not. IMRN. – 1999. – 3. – P. 153 – 167] (Theorem 3.4), as well as to provide a complete proof of the dual result for dg coalgebras. |
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К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 512.5
E. Herscovich (Inst. J. Fourier, Univ. Grenoble Alpes, France; Dept. Mat. FCEyN, UBA, Buenos Aires, and CONICET,
Argentina)
ON THE MERKULOV CONSTRUCTION OF \bfitA \infty -(CO)ALGEBRAS*
ПРО МЕРКУЛОВСЬКУ КОНСТРУКЦIЮ \bfitA \infty -(КО)АЛГЕБР
The aim of this short note is to complete some aspects of a theorem proved by S. Merkulov in [Int. Math. Res. Not. IMRN. –
1999. – 3. – P. 153 – 167] (Theorem 3.4), as well as to provide a complete proof of the dual result for dg coalgebras.
В цьому короткому повiдомленнi ми доповнюємо деякi аспекти теореми, що була доведена Меркуловим в [Int. Math.
Res. Not. IMRN. – 1999. – 3. – P. 153 – 167] (теорема 3.4), а також наводимо повне доведення дуального результату
для dg-коалгебр.
1. Introduction. The objective of this short note is twofold:
(i) Complete some aspects of a theorem of S. Merkulov – which in principle produces an A\infty -
algebra from a certain dg submodule of a dg algebra –, showing that the construction also gives a
morphism of A\infty -algebras and that both are strictly unitary under some further assumptions (see The-
orem 3.1). These last extra components were not considered in the original statement by Merkulov,
but the existence of the morphism of A\infty -algebras appears in a particular case in [6] (Proposition 2.3).
(ii) Provide a precise statement together with a complete proof of a dual version of the previous
theorem for dg coalgebras. This is done in Theorem 4.1.
We are mainly interested in (ii), because we need such a result in [1] for our study of the A\infty -
coalgebra structure on the group \mathrm{T}\mathrm{o}\mathrm{r}A\bullet (K,K) of a nonnegatively graded algebra A and some of
their associated A\infty -comodules (e.g., in Theorem 2.11 and Proposition 2.16 of that article). It was
used in particular to compute the A\infty -module structure of \mathrm{E}\mathrm{x}\mathrm{t}\bullet A(M,k) over the Yoneda algebra of
a generalized Koszul algebra A, where M is a generalized Koszul module over A. Incidentally, we
find more convenient to work with the formulation in [7], for in [1] we need to deal with the slightly
greater generality of (nonsymmetric) bimodules over a (noncommutative) algebra.
The proof of the statements added to the result of Merkulov follows the usual philosophy of
specific manipulations of equations. However, these new results, which appear in Theorem 3.1,
cannot be directly deduced from [7] (Theorem 3.4). In particular, the construction of the morphism
of A\infty -algebras added to the result by Merkulov allows to compare the dg algebra one starts with and
the constructed A\infty -algebra, which seems relevant to us, and it was indeed needed in [6]. Finally,
let us add that the proof of Theorem 4.1 is parallel to the one for dg algebras, and, as in the case
of dg algebras, the result obtained for dg coalgebras is slightly more general than those obtained in
homological perturbation theory, since our assumptions are in general weaker than those of a SDR
(cf. [3], or the nice exposition in [4], Section 6).
* This work was also partially supported by UBACYT 20020130200169BA, UBACYT 20020130100533BA, PIP-
CONICET 2012-2014 11220110100870, MathAmSud-GR2HOPF, PICT 2011-1510 and PICT 2012-1186.
c\bigcirc E. HERSCOVICH, 2019
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8 1133
1134 E. HERSCOVICH
2. Preliminaries on basic algebraic structures. In what follows, k will denote a field and
K will be a noncommutative unitary k-algebra. By module (sometimes decorated by adjectives
such as graded, or dg) we mean a (not necessarily symmetric) bimodule over K (correspond-
ingly decorated), such that the induced bimodule structure over k is symmetric. For dg algeb-
ras, dg coalgebras and A\infty -algebras, we follow the sign conventions of [5], whereas for A\infty -
coalgebras we shall use the ones given in [2] (Subsection 2.1). We also recall that, if V =
= \oplus n\in \BbbZ V
n is a (cohomological) graded K -module, V [m] is the graded module over K whose
nth homogeneous component V [m]n is given by V m+n, for all n,m \in \BbbZ , and it is called the
shift of V. We are not going to consider any shift on other gradings, such as the Adams grading.
All morphisms between modules will be K -linear on both sides (satisfying further requirements
if the modules are decorated as before). One trivially sees that all the standard definitions of
graded or dg (co)algebra, or even A\infty -(co)algebra, eventually provided with an Adams grading,
and (co)modules over them make perfect sense in the monoidal category of graded K -bimodules,
correspondingly provided with an Adams grading. All unadorned tensor products \otimes would be
over K.
Finally, \BbbN will denote the set of (strictly) positive integers, whereas \BbbN 0 will be the set of non-
negative integers. Similarly, for N \in \BbbN , we denote by \BbbN \geq N the set of positive integers greater than
or equal to N.
3. On the theorem of Merkulov. Let (A,\mu A, dA) be a dg algebra provided with an Adams
grading and let W \subseteq A be a dg submodule of A respecting the Adams degree. We assume that
there is a linear map Q : A \rightarrow A[ - 1] of total degree zero, where A[ - 1] denotes the shift of the
cohomological degree, satisfying that the image of \mathrm{i}\mathrm{d}A - [dA, Q] lies in W, where [dA, Q] = dA \circ
\circ Q+Q \circ dA is the graded commutator. For all n \geq 2, construct \lambda n : A\otimes n \rightarrow A as follows. Setting
formally \lambda 1 = - Q - 1, define
\lambda n =
n - 1\sum
i=1
( - 1)i+1\mu A \circ
\bigl(
(Q \circ \lambda i)\otimes (Q \circ \lambda n - i)
\bigr)
(3.1)
for n \geq 2. We have the following result, whose first part is [7] (Theorem 3.4), whereas the rest is a
slightly more general version of [6] (Proposition 2.3 and Lemma 2.5).
Theorem 3.1. Let (A,\mu A, dA) be a dg algebra provided with an Adams grading and let \iota :
(W,dW ) \rightarrow (A, dA) be a dg submodule of A (respecting the Adams degree). Suppose there is a linear
map Q : A \rightarrow A[ - 1] of total degree zero, where A[ - 1] denotes the shift of the cohomological degree,
satisfying that the image of \mathrm{i}\mathrm{d}A - [dA, Q] lies in \iota (W ). For all n \in \BbbN , define mn : W\otimes n \rightarrow W as
follows. Set m1 = dW = dA \circ \iota and mn =
\bigl(
\mathrm{i}\mathrm{d}A - [dA, Q]
\bigr)
\circ \lambda n \circ \iota \otimes n, for n \geq 2. Then (W,m\bullet ) is
an Adams graded A\infty -algebra.
Define the collection f\bullet : W \rightarrow A, where fn : W\otimes n \rightarrow A is the linear map of total degree
(1 - n, 0) given by fn = - Q \circ \lambda n \circ \iota \otimes n for n \in \BbbN . Then f\bullet is a morphism of Adams graded
A\infty -algebras, and it is a quasiisomorphism if and only if \iota is so.
Furthermore, assume A has a unit 1A, there is an element 1W \in W such that \iota (1W ) = 1A,
Q\circ Q = 0 and Q\circ \iota = 0. Then 1W is a strict unit of the Adams graded A\infty -algebra (W,m\bullet ), and f\bullet :
W \rightarrow A is a morphism of strictly unitary Adams graded A\infty -algebras.
Proof. For the first part, the proof given in [7], based on that of the corresponding Lemmas 3.2
and 3.3, applies verbatim in this context. We also note that the sign convention of that article agrees
with the one we follow here.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
ON THE MERKULOV CONSTRUCTION OF A\infty -(CO)ALGEBRAS 1135
To prove the second assertion we proceed as follows. It suffices to prove that f\bullet is a morphism of
A\infty -algebras, for the quasiisomorphism property is immediate. We have thus to show the following
reduced form of the Stasheff identities on morphisms \mathrm{M}\mathrm{I}(n) (see [5], Definition 4.1):\sum
(r,s,t)\in \scrI n
( - 1)r+stfr+1+t \circ
\bigl(
\mathrm{i}\mathrm{d}\otimes r
W \otimes ms \otimes \mathrm{i}\mathrm{d}\otimes t
W
\bigr)
=
= dA \circ fn +
n - 1\sum
p=1
( - 1)p - 1\mu A \circ
\Bigl(
(Q \circ \lambda p \circ \iota \otimes p)\otimes
\Bigl(
Q \circ \lambda n - p \circ \iota \otimes (n - p)
\Bigr) \Bigr)
(3.2)
for all n \in \BbbN , where \scrI n =
\bigl\{
(r, s, t) \in \BbbN 0\times \BbbN \times \BbbN 0 : r+ s+ t = n
\bigr\}
. The case n = 1 is trivial since
\iota is a morphism of complexes. Moreover, the case n = 2 is also clear, since the left member of (3.2)
gives
f1 \circ m2 - f2 \circ (\mathrm{i}\mathrm{d}W \otimes m1 +m1 \otimes \mathrm{i}\mathrm{d}W ) =
=
\bigl(
\mathrm{i}\mathrm{d}A - [dA, Q]
\bigr)
\circ \mu A \circ \iota \otimes 2 +Q \circ \mu A \circ \iota \otimes 2 \circ (\mathrm{i}\mathrm{d}W \otimes dW + dW \otimes \mathrm{i}\mathrm{d}W ) =
= \mu A \circ \iota \otimes 2 - dA \circ Q \circ \mu A \circ \iota \otimes 2, (3.3)
where we have used the Leibniz property for the derivation dA. The right member of (3.2) gives
dA \circ f2 + \mu A \circ
\bigl(
(Q \circ \lambda 1 \circ \iota )\otimes (Q \circ \lambda 1 \circ \iota )
\bigr)
= - dA \circ Q \circ \mu A \circ \iota \otimes 2 + \mu A \circ \iota \otimes 2,
which clearly coincides with (3.3). We shall now consider n > 2. Using the same notation as in [7],
by the definition of the tensors \Phi n and \Theta n given in Lemmas 3.2 and 3.3, respectively, we see that
Q \circ (\Phi n +\Theta n) \circ \iota \otimes n = Q \circ dA \circ \lambda n \circ \iota \otimes n -
\sum
(r,s,t)\in \scrI \ast
n
( - 1)r+stfr+1+t \circ
\bigl(
\mathrm{i}\mathrm{d}\otimes r
W \otimes ms \otimes \mathrm{i}\mathrm{d}\otimes t
W
\bigr)
for all n \in \BbbN , where \scrI \ast
n =
\bigl\{
(r, s, t) \in \BbbN 0 \times \BbbN \times \BbbN 0 : r + s+ t = n, r + t > 0
\bigr\}
. Moreover, by the
previously mentioned lemmas, the tensor \Phi n and \Theta n vanish, which implies that\sum
(r,s,t)\in \scrI n
( - 1)r+stfr+1+t \circ
\bigl(
\mathrm{i}\mathrm{d}\otimes r
W \otimes ms \otimes \mathrm{i}\mathrm{d}\otimes t
W
\bigr)
= f1 \circ mn +Q \circ dA \circ \lambda n \circ \iota \otimes n.
On the other hand,
f1 \circ mn +Q \circ dA \circ \lambda n \circ \iota \otimes n = \lambda n \circ \iota \otimes n - dA \circ Q \circ \lambda n \circ \iota \otimes n =
=
n - 1\sum
i=1
( - 1)i+1\mu A \circ
\bigl(
(Q \circ \lambda i \circ \iota \otimes i)\otimes (Q \circ \lambda n - i \circ \iota \otimes (n - i))
\bigr)
- dA \circ Q \circ \lambda n \circ \iota \otimes n,
where we have used the definition of mn in the first equality and equation (3.1) in the last one. It is
clear that the last member of the previous chain of identities coincides with the right member of (3.2),
as was to be shown.
The proof of the third assertion follows the same pattern as the one given in [6] (Lemma 2.5),
but since we are assuming a weaker assumption on Q (called G in that article), we describe
roughly how it is done. By the definition of f2, we see that the condition Q \circ \iota = 0 implies
that f2(1W \otimes w) = f2(w \otimes 1W ) = 0. The fact \iota is a morphism of dg modules and dA(1A) = 0
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1136 E. HERSCOVICH
imply that m1(1W ) = 0. Suppose now that we have proved that, for 2 \leq i \leq n - 1, fi(w1, . . . , wi)
vanishes if there exists j \in \{ 1, . . . , i\} such that wj = 1W . By (3.1) and the inductive hypo-
thesis, we see that \lambda n(w1, . . . , wn) vanishes if there exists j \in \{ 2, . . . , n - 1\} such that wj =
= 1W , \lambda n(1W , w2, . . . , wn) = fn - 1(w2, . . . , wn), and \lambda n(w1, . . . , wn - 1, 1W ) = ( - 1)nfn - 1(w1, . . .
. . . , wn - 1) for all w1, . . . , wn \in W. From the definition of fn and the assumption that Q\circ Q = 0 we
conclude that fn(w1, . . . , wn) vanishes if there exists j \in \{ 1, . . . , n\} such that wj = 1W . Moreover,
since the image of
\bigl(
\mathrm{i}\mathrm{d}A - [dA, Q]
\bigr)
lies in \iota (W ) and Q \circ \iota = 0, we see that
0 = Q \circ
\bigl(
\mathrm{i}\mathrm{d}A - [dA, Q]
\bigr)
= Q - Q \circ dA \circ Q =
\bigl(
\mathrm{i}\mathrm{d}A - [dA, Q]
\bigr)
\circ Q, (3.4)
where we have used in the last two equalities that Q \circ Q vanishes. Using our previous description of
\lambda n in terms of fn - 1 = - Q \circ \lambda n - 1 \circ \iota \otimes (n - 1) for n \geq 3 (if it does not vanish already) and (3.4), we
get that mn(w1, . . . , wn) vanishes if there exists j \in \{ 1, . . . , n\} such that wj = 1W .
4. The dual result. We shall briefly present the dual procedure to the one introduced by
S. Merkulov in [7] to produce an A\infty -algebra structure from a particular data on a dg submodule of
a dg algebra. In our case, we produce an A\infty -coalgebra structure on a quotient dg module of a dg
coalgebra. We also note that, even though the results of the article of Merkulov are stated for vector
spaces, they are clearly seen to be true (by exactly the same arguments) in our more general situation
of bimodules over the k-algebra K.
Let (C,\Delta C , dC) be a dg coalgebra provided with an Adams grading and let (C, dC) \twoheadrightarrow (W,dW )
be a dg module quotient of C respecting the Adams degree. Denote by \scrK the kernel of the previous
quotient and assume that there is a linear map Q : C \rightarrow C[ - 1] of total degree zero, where C[ - 1]
denotes the shift of the cohomological degree whereas the Adams degree remains unchanged, satis-
fying that \mathrm{i}\mathrm{d}C - [dC , Q] vanishes on \scrK , where [dC , Q] = dC \circ Q+Q\circ dC is the graded commutator.
For all n \geq 2, define \gamma n : C \rightarrow C\otimes n as follows. Setting formally \gamma 1 = - Q - 1, define
\gamma n =
n - 1\sum
i=1
( - 1)i+1
\bigl(
(\gamma n - i \circ Q)\otimes (\gamma i \circ Q)
\bigr)
\circ \Delta C (4.1)
for n \geq 2. We shall say that Q is admissible if the family \{ \gamma n\} n\in \BbbN \geq 2
is locally finite, i.e., it
satisfies that the induced map C \rightarrow
\prod
n\geq 2
C\otimes n factors through the canonical inclusion \oplus n\geq 2C
\otimes n \rightarrow
\rightarrow
\prod
n\geq 2
C\otimes n.
Fact 4.1. The identity
-
n - 1\sum
p=2
\bigl(
\gamma p \otimes (\gamma n - p \circ Q)
\bigr)
\circ \gamma 2 +
n - 1\sum
p=2
( - 1)p
\bigl(
(\gamma n - p \circ Q)\otimes \gamma p
\bigr)
\circ \gamma 2 = 0
holds.
Proof. The identity just follows by replacing the two occurrences of \gamma p by the recurrent expres-
sion given by (4.1) and simplifying the corresponding terms.
Fact 4.2. Let e be an endomorphism of C of degree zero and define
En(e) =
\sum
(r,s,t)\in \scrI \ast
n
( - 1)rs+t(\mathrm{i}\mathrm{d}\otimes r
C \otimes (\gamma s \circ e)\otimes \mathrm{i}\mathrm{d}\otimes t
C ) \circ \gamma r+1+t,
where \scrI \ast
n =
\bigl\{
(r, s, t) \in \BbbN 0 \times \BbbN \geq 2 \times \BbbN 0 : r + s+ t = n and r + t > 0
\bigr\}
. Then
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
ON THE MERKULOV CONSTRUCTION OF A\infty -(CO)ALGEBRAS 1137
En(e) =
n - 2\sum
t=1
t\sum
i=1
( - 1)i(\gamma n - te\otimes \mathrm{i}\mathrm{d}\otimes t
C )(\gamma iQ\otimes \gamma t - i+1Q)\Delta C+
+
n - 2\sum
r=1
r\sum
i=1
( - 1)r(n - r)+r - i(\mathrm{i}\mathrm{d}\otimes r
C \otimes \gamma n - re)(\gamma iQ\otimes \gamma 1+r - iQ)\Delta C+
+
\sum
(r,s,t)\in \^\scrI n
r+t\sum
i=r+1
( - 1)rs+r - i
\Bigl( \bigl(
(\mathrm{i}\mathrm{d}\otimes r
C \otimes \gamma se\otimes \mathrm{i}\mathrm{d}
\otimes (i - r - 1)
C )\gamma iQ
\bigr)
\otimes \gamma r+1+t - iQ
\Bigr)
\Delta C+
+
\sum
(r,s,t)\in \^\scrI n
r\sum
i=1
( - 1)rs+r+s - i(s+1)
\Bigl(
\gamma iQ\otimes
\bigl(
(\mathrm{i}\mathrm{d}
\otimes (r - i)
C \otimes \gamma se\otimes \mathrm{i}\mathrm{d}\otimes t
C )\gamma r+1+t - iQ
\bigr) \Bigr)
\Delta C , (4.2)
where \^\scrI n =
\bigl\{
(r, s, t) \in \BbbN \times \BbbN \geq 2\times \BbbN : r+ s+ t = n
\bigr\}
, and we have omitted the composition symbol
\circ to economize space.
Proof. The statement follows from the next chain of identities, which uses the definition (4.1):
En(e) =
n - 2\sum
t=1
( - 1)t(\gamma n - te\otimes \mathrm{i}\mathrm{d}\otimes t
C )\gamma 1+t +
n - 2\sum
r=1
( - 1)r(n - r)(\mathrm{i}\mathrm{d}\otimes r
C \otimes \gamma n - re)\gamma r+1+
+
\sum
(r,s,t)\in \^\scrI n
( - 1)rs+t
\bigl(
\mathrm{i}\mathrm{d}\otimes r
C \otimes \gamma se\otimes \mathrm{i}\mathrm{d}\otimes t
C
\bigr)
\gamma r+1+t =
=
n - 2\sum
t=1
t\sum
i=1
( - 1)i(\gamma n - te\otimes \mathrm{i}\mathrm{d}\otimes t
C )(\gamma iQ\otimes \gamma t - i+1Q)\Delta C+
+
n - 2\sum
r=1
r\sum
i=1
( - 1)r(n - r)+r - i
\bigl(
\mathrm{i}\mathrm{d}\otimes r
C \otimes \gamma n - re
\bigr)
(\gamma iQ\otimes \gamma 1+r - iQ)\Delta C+
+
\sum
(r,s,t)\in \^\scrI n
r+t\sum
i=r+1
( - 1)rs+r - i
\Bigl( \bigl(
(\mathrm{i}\mathrm{d}\otimes r
C \otimes \gamma se\otimes \mathrm{i}\mathrm{d}
\otimes (i - r - 1)
C )\gamma iQ
\bigr)
\otimes \gamma r+1+t - iQ
\Bigr)
\Delta C+
+
\sum
(r,s,t)\in \^\scrI n
r\sum
i=1
( - 1)rs+r+s - i(s+1)
\Bigl(
\gamma iQ\otimes
\bigl(
(\mathrm{i}\mathrm{d}
\otimes (r - i)
C \otimes \gamma se\otimes \mathrm{i}\mathrm{d}\otimes t
C )\gamma r+1+t - iQ
\bigr) \Bigr)
\Delta C ,
where we have omitted the composition symbol \circ to reduce space.
Lemma 4.1. For n \in \BbbN \geq 3, define
\Gamma n =
\sum
(r,s,t)\in \scrI \ast
n
( - 1)rs+t(\mathrm{i}\mathrm{d}\otimes r
C \otimes \gamma s \otimes \mathrm{i}\mathrm{d}\otimes t
C ) \circ \gamma r+1+t,
where \scrI \ast
n =
\bigl\{
(r, s, t) \in \BbbN 0 \times \BbbN \geq 2 \times \BbbN 0 : r + s+ t = nandr + t > 0
\bigr\}
. Then \Gamma n \equiv 0 for all n \geq 3.
Proof. First note that \Gamma 3 = (\mathrm{i}\mathrm{d}C \otimes \Delta C - \Delta C \otimes \mathrm{i}\mathrm{d}C) \circ \Delta C , so the coassociativity of C implies
that \Gamma 3 vanishes.
Let us now consider n > 3. We note that \Gamma n = En(\mathrm{i}\mathrm{d}C), so it can be written as indicated in
equation (4.2). Moreover, the terms corresponding to i = 1 in the first sum and to i = r in the
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1138 E. HERSCOVICH
second sum in that latter expression of \Gamma n cancel due to Fact 4.1. Using the definition of \Gamma m for
3 \leq m \leq n - 1 in the remaining terms of the new expression of \Gamma n we get
\Gamma n =
n - 3\sum
i=1
( - 1)n - i
\bigl(
(\gamma i \circ Q)\otimes (\Gamma n - i \circ Q)
\bigr)
\circ \Delta C -
n - 3\sum
j=1
\bigl(
(\Gamma n - j \circ Q)\otimes (\gamma j \circ Q)
\bigr)
\circ \Delta C .
The lemma thus follows from an inductive argument.
Lemma 4.2. For n \in \BbbN \geq 2, define
Hn = \gamma n \circ dC + ( - 1)n - 1
n - 1\sum
r=0
\Bigl(
\mathrm{i}\mathrm{d}\otimes r
C \otimes dC \otimes \mathrm{i}\mathrm{d}
\otimes (n - r - 1)
C
\Bigr)
\circ \gamma n -
-
\sum
(r,s,t)\in \scrI \ast
n
( - 1)rs+t
\Bigl(
\mathrm{i}\mathrm{d}\otimes r
C \otimes (\gamma s \circ [dC , Q])\otimes \mathrm{i}\mathrm{d}\otimes t
C
\Bigr)
\circ \gamma r+1+t,
where \scrI \ast
n =
\bigl\{
(r, s, t) \in \BbbN 0\times \BbbN \geq 2\times \BbbN 0 : r+ s+ t = n and r+ t > 0
\bigr\}
. Then Hn \equiv 0 for all n \geq 2.
Proof. Note that H2 = \Delta C \circ dC - (\mathrm{i}\mathrm{d}C \otimes dC + dC \otimes \mathrm{i}\mathrm{d}C) \circ \Delta C , so it vanishes by the Leibniz
identity for C.
Assume now that n > 2. Note that
Hn = \gamma n \circ dC - ( - 1)n
n - 1\sum
r=0
\Bigl(
\mathrm{i}\mathrm{d}\otimes r
C \otimes dC \otimes \mathrm{i}\mathrm{d}
\otimes (n - r - 1)
C
\Bigr)
\circ \gamma n - En
\bigl(
[dC , Q]
\bigr)
.
Using the definition (4.1), we can write
Hn =
n - 1\sum
i=1
\bigl(
(\gamma i \circ Q \circ dC)\otimes (\gamma n - i \circ Q)
\bigr)
\circ \Delta C -
-
n - 1\sum
i=1
( - 1)n - i
\bigl(
(\gamma i \circ Q)\otimes (\gamma n - i \circ Q \circ dC)
\bigr)
\circ \Delta C+
+
n - 1\sum
i=0
i - 1\sum
r=0
( - 1)i
\Bigl( \Bigl( \Bigl(
\mathrm{i}\mathrm{d}\otimes r
C \otimes dC \otimes \mathrm{i}\mathrm{d}
\otimes (i - r - 1)
C
\Bigr)
\circ (\gamma i \circ Q)
\Bigr)
\otimes (\gamma n - i \circ Q)
\Bigr)
\circ \Delta C -
-
n - 1\sum
i=0
n - i - 1\sum
r=0
\Bigl(
(\gamma i \circ Q)\otimes
\Bigl( \Bigl(
\mathrm{i}\mathrm{d}\otimes r
C \otimes dC \otimes \mathrm{i}\mathrm{d}
\otimes (n - i - r - 1)
C
\Bigr)
\circ (\gamma n - i \circ Q)
\Bigr) \Bigr)
\circ \Delta C -
- En(dC \circ Q) - En(Q \circ dC). (4.3)
Fact 4.2 expresses En(Q\circ dC) as a linear combination of sums satisfying that the terms corresponding
to i = 1 of its first sum and the terms corresponding to i = r of its second sum cancel the first
two sums in (4.3). Combining the remaining terms of (4.3) and using the definition of Hm, for
3 \leq m \leq n - 1, we get
Hn =
n - 2\sum
i=1
( - 1)n - i
\bigl(
(\gamma i \circ Q)\otimes (Hn - i \circ Q)
\bigr)
\otimes \Delta C -
n - 2\sum
j=1
\bigl(
(Hn - j \circ Q)\otimes (\gamma j \circ Q)
\bigr)
\circ \Delta C .
The lemma thus follows from an inductive argument.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
ON THE MERKULOV CONSTRUCTION OF A\infty -(CO)ALGEBRAS 1139
Theorem 4.1. Let (C,\Delta C , dC) be a dg coalgebra provided with an Adams grading and let \rho :
(C, dC) \rightarrow (W,dW ) be a quotient dg module of C (respecting the Adams degree) with kernel \scrK .
Suppose there is an admissible linear map Q : C \rightarrow C[ - 1] of total degree zero, where C[ - 1] denotes
the shift of the cohomological degree, satisfying that \mathrm{i}\mathrm{d}C - [dC , Q] vanishes on \scrK . For all n \in \BbbN ,
define \Delta n : W \rightarrow W\otimes n as follows. Set \Delta 1 = dW and \Delta n to be the unique map satisfying that
\Delta n \circ \rho = \rho \otimes n \circ \gamma n \circ
\bigl(
\mathrm{i}\mathrm{d}C - [dC , Q]
\bigr)
for n \geq 2. Then (W,\Delta \bullet ) is an Adams graded A\infty -coalgebra.
Define the collection f\bullet : C \rightarrow W, where fn : C \rightarrow W\otimes n is the linear map of homological
degree n - 1 and zero Adams degree given by fn = - \rho \otimes n \circ \gamma n \circ Q for n \in \BbbN . Then f\bullet is a
morphism of Adams graded A\infty -coalgebras.
Furthermore, assume C has a counit \epsilon C , there a linear map \epsilon W : W \rightarrow \scrK such that \epsilon W \circ \rho = \epsilon C ,
Q \circ Q = 0 and \rho \circ Q = 0. Then, \epsilon W is a strict counit of the Adams graded A\infty -coalgebra (W,\Delta \bullet ),
and f\bullet : C \rightarrow W is a morphism of strictly counitary Adams graded A\infty -coalgebras.
Proof. The first part follows from the fact that the Stasheff identity \mathrm{S}\mathrm{I}(n) for n \geq 3 for the
operations \Delta \bullet is by definition given by \rho \otimes n \circ (\Gamma n +Hn) \circ
\bigl(
\mathrm{i}\mathrm{d} - [dC , Q]
\bigr)
, so it vanishes. The two
first Stasheff identities \mathrm{S}\mathrm{I}(1) and \mathrm{S}\mathrm{I}(2) are trivial, for W is a quotient of C.
To prove the second assertion we proceed as follows. We have thus to prove the following
reduced form of the Stasheff identities on morphisms \mathrm{M}\mathrm{I}(n) (see [2], eq. (2.2)):\sum
(r,s,t)\in \scrI n
( - 1)rs+t
\bigl(
\mathrm{i}\mathrm{d}\otimes r
W \otimes \Delta s \otimes \mathrm{i}\mathrm{d}\otimes t
W
\bigr)
\circ fr+1+t =
= fn \circ dC +
n - 1\sum
p=1
( - 1)n - p - 1
\bigl(
(\rho \otimes p \circ \gamma p \circ Q)\otimes (\rho \otimes (n - p) \circ \gamma n - p \circ Q)
\bigr)
\circ \Delta C (4.4)
for all n \in \BbbN , where \scrI n = \{ (r, s, t) \in \BbbN 0 \times \BbbN \times \BbbN 0 : r + s + t = n\} . The case n = 1 is trivial
since \rho is a morphism of complexes. Moreover, the case n = 2 is also clear, since the left member
of (4.4) gives
\Delta 2 \circ f1 -
\bigl(
\mathrm{i}\mathrm{d}W \otimes \Delta 1 +\Delta 1 \otimes \mathrm{i}\mathrm{d}W
\bigr)
\circ f2 =
= \rho \otimes 2 \circ \Delta C \circ
\bigl(
\mathrm{i}\mathrm{d}A - [dC , Q]
\bigr)
+ \rho \otimes 2 \circ (\mathrm{i}\mathrm{d}C \otimes dC + dC \otimes \mathrm{i}\mathrm{d}C) \circ \Delta C \circ Q =
= \rho \otimes 2 \circ \Delta C - \rho \otimes 2 \circ \Delta C \circ Q \circ dC , (4.5)
where we have used the Leibniz property for the coderivation dC . The right member of (4.4) gives
f2 \circ dC +
\bigl(
(\rho \circ \gamma 1 \circ Q)\otimes (\rho \circ \gamma 1 \circ Q)
\bigr)
\circ \Delta C = - \rho \otimes 2 \circ \Delta C \circ Q \circ dC + \rho \otimes 2 \circ \Delta C ,
which clearly coincides with (4.5). We shall now consider n > 2. By the definition of the tensors
\Gamma n and Hn given in Lemmas 4.1 and 4.2, respectively, we see that
\rho \otimes n \circ (\Gamma n +Hn) \circ Q = \rho \otimes n \circ \gamma n \circ dC \circ Q+
\sum
(r,s,t)\in \scrI \ast
n
( - 1)rs+t(\mathrm{i}\mathrm{d}\otimes r
W \otimes \Delta s \otimes \mathrm{i}\mathrm{d}\otimes t
W ) \circ fr+1+t
for all n \in \BbbN , where we recall that \scrI \ast
n = \{ (r, s, t) \in \BbbN 0 \times \BbbN \times \BbbN 0 : r + s + t = n, r + t > 0\} .
Moreover, by the previously mentioned lemmas, the tensor \Gamma n and Hn vanish, which implies that\sum
(r,s,t)\in \scrI n
( - 1)rs+t(\mathrm{i}\mathrm{d}\otimes r
W \otimes \Delta s \otimes \mathrm{i}\mathrm{d}\otimes t
W ) \circ fr+1+t = \Delta n \circ f1 - \rho \otimes n \circ \gamma n \circ dC \circ Q.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1140 E. HERSCOVICH
On the other hand,
\Delta n \circ f1 - \rho \otimes n \circ \gamma n \circ dC \circ Q = \rho \otimes n \circ \gamma n - \rho \otimes n \circ \gamma n \circ Q \circ dC =
=
n - 1\sum
i=1
( - 1)n - i - 1
\bigl(
(\rho \otimes i \circ \gamma i \circ Q)\otimes (\rho \otimes (n - i) \circ \gamma n - i \circ Q)
\bigr)
\circ \Delta C - \rho \otimes n \circ \gamma n \circ Q \circ dC ,
where we have used the definition of \Delta n in the first equality, and equation (4.1) in the last one.
It is clear that the last member of the previous chain of identities coincides with the right member
of (4.4), as was to be shown.
The proof of the third assertion is parallel to the one given to Theorem 3.1. Using the definition
of f2, we see that the condition \rho \circ Q = 0 implies that (\epsilon W \otimes \mathrm{i}\mathrm{d}W ) \circ f2 = (\mathrm{i}\mathrm{d}W \otimes \epsilon W ) \circ f2 = 0.
The fact \rho is a morphism of dg modules and \epsilon C \circ dC = 0 imply that \epsilon W \circ \Delta 1 = 0. Suppose now
that we have proved that, for 2 \leq i \leq n - 1,
\Bigl(
\mathrm{i}\mathrm{d}\otimes j
W \otimes \epsilon W \otimes \mathrm{i}\mathrm{d}
\otimes (i - j - 1)
W
\Bigr)
\circ fi vanishes for all
j \in \{ 0, . . . , i - 1\} . By (4.1) and the inductive hypothesis, we see that
\Bigl(
\mathrm{i}\mathrm{d}\otimes j
W \otimes \epsilon W \otimes \mathrm{i}\mathrm{d}
\otimes (n - j - 1)
W
\Bigr)
\circ
\circ \rho \otimes n \circ \gamma n vanishes for all j \in \{ 1, . . . , n - 2\} ,
\Bigl(
\epsilon W \otimes \mathrm{i}\mathrm{d}
\otimes (n - 1)
W
\Bigr)
\circ \rho \otimes n \circ \gamma n = ( - 1)n - 1fn - 1, and\Bigl(
\mathrm{i}\mathrm{d}
\otimes (n - 1)
W \otimes \epsilon W
\Bigr)
\circ \rho \otimes n \circ \gamma n = fn - 1. By the definition of fn and the assumption that Q \circ Q = 0 we
conclude that (\mathrm{i}\mathrm{d}\otimes j
W \otimes \epsilon W \otimes \mathrm{i}\mathrm{d}
\otimes (n - j)
W ) \circ fn vanishes for all j \in \{ 0, . . . , n - 1\} . Moreover, since the
image of
\bigl(
\mathrm{i}\mathrm{d}C - [dC , Q]
\bigr)
vanishes on the kernel of \rho and \rho \circ Q = 0, we see that
0 =
\bigl(
\mathrm{i}\mathrm{d}C - [dC , Q]
\bigr)
\circ Q = Q - Q \circ dC \circ Q = Q \circ
\bigl(
\mathrm{i}\mathrm{d}C - [dC , Q]
\bigr)
, (4.6)
where we have used in the last two equalities that Q \circ Q vanishes. Using our previous description
of
\Bigl(
\mathrm{i}\mathrm{d}\otimes j
W \otimes \epsilon W \otimes \mathrm{i}\mathrm{d}
\otimes (n - j - 1)
W
\Bigr)
\circ \rho \otimes n \circ \gamma n in terms of fn - 1 = - \rho \otimes (n - 1) \circ \gamma n - 1 \circ Q for n \geq 3 (if
it does not vanish already) and (4.6), we get that
\Bigl(
\mathrm{i}\mathrm{d}\otimes j
W \otimes \epsilon W \otimes \mathrm{i}\mathrm{d}
\otimes (n - j)
W
\Bigr)
\circ \Delta n vanishes for all
j \in \{ 0, . . . , n - 1\} .
The structure of A\infty -coalgebra on W given by the previous theorem is called a Merkulov model
on W, or simply a model. As in the case of dg algebras, note that the result stated in the first
two paragraphs of the previous theorem is slightly more general than those obtained in homologi-
cal perturbation theory, since the conditions of a SDR are not necessarily satisfied (cf. [3] or [4],
Section 6).
References
1. Herscovich E. Applications of one-point extensions to compute the A\infty -(co)module structure of several Ext (resp.,
Tor) groups // J. Pure and Appl. Algebra. – 2019. – 223, № 3. – P. 1054 – 1072.
2. Herscovich E. Using torsion theory to compute the algebraic structure of Hochschild (co)homology // Homology,
Homotopy and Appl. – 2018. – 20, № 1. – P. 117 – 139.
3. Gugenheim V. K. A. M. On a perturbation theory for the homology of the loop-space // J. Pure and Appl. Algebra. –
1982. – 25, № 2. – P. 197 – 205.
4. Huebschmann J. On the construction of A\infty -structures // Georg. Math. J. – 2010. – 17, № 1. – P. 161 – 202.
5. Lu D. M., Palmieri J. H., Wu Q. S., Zhang J. J. A\infty -algebras for ring theorists // Proc. Int. Conf. Algebra. – 2004. –
91, № 1. – P. 91 – 128.
6. Lu D.-M., Palmieri J. H., Wu Q.-S., Zhang J. J. A-infinity structure on Ext-algebras // J. Pure and Appl. Algebra. –
2009. – 213, № 11. – P. 2017 – 2037.
7. Merkulov S. A. Strong homotopy algebras of a Kähler manifold // Int. Math. Res. Not. – 1999. – № 3. – P. 153 – 164.
Received 28.09.16
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
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| id | umjimathkievua-article-1504 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:07:07Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/67/1050f83592ceb003c72b3b96d64f7167.pdf |
| spelling | umjimathkievua-article-15042019-12-05T08:57:29Z On the Merkulov construction of $A_{ \infty}$ -(co)algebras Про Меркуловську конструкцiю $A_{ \infty}$ -(ко)алгебр Herscovich, E. Герскович, Е. UDC 512.5 The aim of this short note is to complete some aspects of a theorem proved by S. Merkulov in [Int. Math. Res. Not. IMRN. – 1999. – 3. – P. 153 – 167] (Theorem 3.4), as well as to provide a complete proof of the dual result for dg coalgebras. УДК 512.5 В цьому короткому повiдомленнi ми доповнюємо деякi аспекти теореми, що була доведена Меркуловим в [Int. Math. Res. Not. IMRN. – 1999. – 3. – P. 153 – 167] (теорема 3.4), а також наводимо повне доведення дуального результату для dg-коалгебр. Institute of Mathematics, NAS of Ukraine 2019-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1504 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 8 (2019); 1133-1140 Український математичний журнал; Том 71 № 8 (2019); 1133-1140 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1504/488 Copyright (c) 2019 Herscovich E. |
| spellingShingle | Herscovich, E. Герскович, Е. On the Merkulov construction of $A_{ \infty}$ -(co)algebras |
| title | On the Merkulov construction of $A_{ \infty}$ -(co)algebras |
| title_alt | Про Меркуловську конструкцiю $A_{ \infty}$ -(ко)алгебр |
| title_full | On the Merkulov construction of $A_{ \infty}$ -(co)algebras |
| title_fullStr | On the Merkulov construction of $A_{ \infty}$ -(co)algebras |
| title_full_unstemmed | On the Merkulov construction of $A_{ \infty}$ -(co)algebras |
| title_short | On the Merkulov construction of $A_{ \infty}$ -(co)algebras |
| title_sort | on the merkulov construction of $a_{ \infty}$ -(co)algebras |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1504 |
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