A simple note on the Yoneda (co)algebra of a monomial algebra
UDC 512.7 If $A = TV/\langle R\rangle$ is a monomial $K$-algebra, it is well-known that $\operatorname{Tor}_{p}^{A}(K,K)$ is isomorphic to the space $V^{(p-1)}$ of (Anick) $(p-1)$-chains for $p \geq 1$. The goal of this short note is to show that the next result follows directly from well-establishe...
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| author | Herscovich , E. Herscovich , E. |
| author_facet | Herscovich , E. Herscovich , E. |
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If $A = TV/\langle R\rangle$ is a monomial $K$-algebra, it is well-known that $\operatorname{Tor}_{p}^{A}(K,K)$ is isomorphic to the space $V^{(p-1)}$ of (Anick) $(p-1)$-chains for $p \geq 1$. The goal of this short note is to show that the next result follows directly from well-established theorems on $A_{\infty}$-algebras, without computations: there is an $A_{\infty}$-coalgebra model on $\operatorname{Tor}_{\bullet}^{A}(K,K)$ satisfying that, for $n \geq 3$ and $c \in V^{(p)}$, $\Delta_{n}(c)$ is a linear combination of $c_{1} \otimes \ldots \otimes c_{n}$, where $c_{i} \in V^{(p_{i})}$, $p_{1} + \ldots +p_{n} = p - 1$ and $c_{1} \ldots c_{n} = c$. The proof follows essentially from noticing that the Merkulov procedure is compatible with an extra grading over a suitable category. By a simple argument based on a result by Keller we immediately deduce that some of these coefficients are $\pm 1$. |
| doi_str_mv | 10.37863/umzh.v73i2.6040 |
| first_indexed | 2026-03-24T03:25:43Z |
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DOI: 10.37863/umzh.v73i2.6040
UDC 512.7
E. Herscovich (Inst. Fourier, Univ. Grenoble, Alpes., France)
A SIMPLE NOTE ON THE YONEDA (CO)ALGEBRA
OF A MONOMIAL ALGEBRA
ПРОСТЕ ПОВIДОМЛЕННЯ ПРО (КО)АЛГЕБРУ ЙОНЕДИ
АЛГЕБРИ ОДНОЧЛЕНIВ
If A = TV/\langle R\rangle is a monomial K-algebra, it is well-known that \mathrm{T}\mathrm{o}\mathrm{r}Ap (K,K) is isomorphic to the space V (p - 1) of (Anick)
(p - 1)-chains for p \geq 1. The goal of this short note is to show that the next result follows directly from well-established
theorems on A\infty -algebras, without computations: there is an A\infty -coalgebra model on \mathrm{T}\mathrm{o}\mathrm{r}A\bullet (K,K) satisfying that, for
n \geq 3 and c \in V (p) , \Delta n(c) is a linear combination of c1 \otimes . . . \otimes cn , where ci \in V (pi) , p1 + . . . + pn = p - 1 and
c1 . . . cn = c. The proof follows essentially from noticing that the Merkulov procedure is compatible with an extra grading
over a suitable category. By a simple argument based on a result by Keller we immediately deduce that some of these
coefficients are \pm 1.
Якщо A = TV/\langle R\rangle — K-алгебра одночленiв, то вiдомо, що \mathrm{T}\mathrm{o}\mathrm{r}Ap (K,K) є iзоморфним простору V (p - 1) (p - 1)-
ланцюгiв (Анiка) для p \geq 1. Метою цього повiдомлення є намагання показати, що наступний результат без будь-
яких обчислень безпосередньо випливає з встановлених теорем для A\infty -алгебр: iснує A\infty -коалгебраїчна модель
на \mathrm{T}\mathrm{o}\mathrm{r}A\bullet (K,K) така, що для n \geq 3 i c \in V (p) \Delta n(c) є лiнiйною комбiнацiєю c1 \otimes . . . \otimes cn , де ci \in V (pi) ,
p1 + . . . + pn = p - 1 i c1 . . . cn = c. Доведення, в основному, є наслiдком того, що процедура Меркулова
сумiсна з додатковим градуюванням деякої вiдповiдної категорiї. За допомогою простих аргументiв, що базуються
на результатах Келлера, безпосередньо приходимо до висновку, що деякi з цих коефiцiєнтiв дорiвнюють \pm 1.
1. The results. This article arose from discussions with A. Solotar and M. Suárez-Álvarez in 2014,
and more recently with V. Dotsenko and P. Tamaroff, on the A\infty -algebra structure on the Yoneda
algebra of a monomial algebra. I want to thank them for the exchange and in particular the last two
for lately renewing my interest in the problem. My aim is to explain some results describing such
A\infty -algebras that do not seem to be well-known, but follow rather easily from the general theory,
and were meant to be included in the Master thesis of my former student E. Sérandon in 2016. I
would also like to thank the referee for the comments.
In what follows, K will denote a finite product of r copies of a field k . By module we will mean
a (not necessarily symmetric) bimodule over K (see [3], Section 2). All unadorned tensor products
\otimes will be over K, unless otherwise stated. For the conventions on A\infty -(co)algebras we refer the
reader to [5] (Subsection 2.1).
Let M be a small category with a finite set of objects \{ o1, . . . , or\} . As usual, we denote the
set of all arrows of M by M itself, the composition by \star , and the identity of oi by ei . We remark
that m\prime \star m\prime \prime implies that m\prime and m\prime \prime are composable morphisms. Let M \mathrm{M}\mathrm{o}\mathrm{d} be the category of
modules V provided with an M -grading (i.e., a decomposition of modules V = \oplus m\in MVm) and
linear morphisms preserving the degree. This is a monoidal category with the tensor product V \otimes W
whose mth homogeneous component is \oplus m\prime \star m\prime \prime =mVm\prime \otimes Wm\prime \prime , and the unit K = \oplus r
i=1kei , where
ej .kei = kei .ej = \delta i,jkei . Furthermore, it is easy to see that M \mathrm{M}\mathrm{o}\mathrm{d} is a semisimple category. We
say that a strictly unitary A\infty -algebra (A,m\bullet ) has an M -grading if (A,m\bullet ) is a strictly unitary
A\infty -algebra in the monoidal category M \mathrm{M}\mathrm{o}\mathrm{d}. The same applies to M -graded augmented A\infty -
algebras, and to morphisms of M -graded strictly unitary or augmented A\infty -algebras. Moreover, the
c\bigcirc E. HERSCOVICH, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2 275
276 E. HERSCOVICH
definitions of M -graded strictly counitary and coaugmented A\infty -coalgebra as well as the morphisms
between them are also clear.
Proposition 1.1. Let A = TV/\langle R\rangle is a monomial K-algebra, i.e., V is a module of finite
dimension over k and R is a space of relations of monomial type. Then there is a small category
(M, \star ) with r objects such that A is an M -graded unitary algebra with \mathrm{d}\mathrm{i}\mathrm{m}k(Am) \leq 1 for all
m \in M .
Proof. Let \scrB be a basis of the underlying vector space of V such that ej .v.ei vanishes or it is
v, for all v \in \scrB and all i, j \in \{ 1, . . . , r\} , and define M as the free small category generated by \scrB .
Note that TV identifies with the unitary semigroup algebra associated with M . Given m \in M, set
Am as the vector subspace of A generated by the element \=m of A given as the image of m \in TV
under TV \rightarrow A. It is clear that A = \oplus m\in MAm is an M -grading of A and \mathrm{d}\mathrm{i}\mathrm{m}k(Am) \leq 1 for all
m \in M .
The next result follows directly from the definition of the bar construction.
Fact 1.1. If A is an augmented A\infty -algebra over K with an M -grading, then the coaugmented
dg coalgebra B+(A) given by the bar construction is M -graded for the canonically induced grading.
We present now the main result of this short note.
Theorem 1.1. Let A = TV/\langle R\rangle be a monomial K-algebra and let M be the small category
defined in Proposition 1.1. Then there is an M -graded coaugmented A\infty -coalgebra structure on
\mathrm{T}\mathrm{o}\mathrm{r}A\bullet (K,K) together with a quasi-equivalence from it to the M -graded coaugmented dg coalgebra
B+(A).
Proof. We first remark that [4] (Theorem 4.5), holds verbatim if we replace Adams grading
by M -grading, since M \mathrm{M}\mathrm{o}\mathrm{d} is a semisimple category. Using a grading argument based on the
fact that both B+(A) and \mathrm{T}\mathrm{o}\mathrm{r}A\bullet (K,K) are Adams connected modules (see [5], Section 2, for the
definition for vector spaces), we see that the operator Q in [4] (Theorem 4.5), is locally finite (see
[3], Addendum 2.9). Hence, applying [4] (Theorem 4.5), to the coaugmented dg coalgebra B+(A),
which projects onto its homology \mathrm{T}\mathrm{o}\mathrm{r}A\bullet (K,K), we see that the latter has a structure of M -graded
coaugmented A\infty -coalgebra. Moreover, by the same theorem, there is a quasi-isomorphism of
coaugmented A\infty -coalgebras from B+(A) to \mathrm{T}\mathrm{o}\mathrm{r}A\bullet (K,K), which is trivially a quasi-equivalence by
a grading argument.
Remark 1.1. The previous theorem and its proof hold more generally for any M -graded K-
algebra A that is connected, i.e., Aei = k for all i \in \{ 1, . . . , r\} , and such that A/K has a compatible
(strictly) positive grading. This occurs, e.g., if there is a functor \ell : M \rightarrow \BbbN 0 such that \ell (m) = 0 if
and only if m is an identity of M, where the monoid \BbbN 0 is regarded as a category with one object.
The result in the abstract is obtained from the previous theorem by identifying \mathrm{T}\mathrm{o}\mathrm{r}Ap (K,K) with
the module V (p - 1) generated by the (Anick) (p - 1)-chains for p \geq 1 (see [1], Lemma 3.3, for the
case K is a field, and [2] (Theorem 4.1), for the general case), i.e., given c \in V (p) and n \geq 3,
\Delta n(c) =
\sum
ci \in V (pi), c1 . . . cn = c
pi \in \BbbN 0, p1 + . . . + pn = p - 1
\lambda (c1\otimes ...\otimes cn)c1 \otimes . . .\otimes cn, where \lambda (c1\otimes ...\otimes cn) \in k. (1)
Note that \Delta 2 is given by the usual coproduct of \mathrm{T}\mathrm{o}\mathrm{r}A\bullet (K,K). The (left or right) dual of this
coaugmented A\infty -coalgebra structure on \mathrm{T}\mathrm{o}\mathrm{r}A\bullet (K,K) gives an augmented A\infty -algebra model on
\mathrm{E}\mathrm{x}\mathrm{t}\bullet A(K,K) (see [3], Proposition 2.13).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
A SIMPLE NOTE ON THE YONEDA (CO)ALGEBRA OF A MONOMIAL ALGEBRA 277
With no extra effort we can say a little more about the coefficients in (1)1.
Theorem 1.2. Assume the same hypotheses as in the previous theorem. Given c \in V (p), n \geq 3,
and ci \in V (pi) pi \in \BbbN 0 such that c1 . . . cn = c, p1 + . . . + pn = p - 1 and p = pj + 1 for some
j \in \{ 1, . . . , n\} , then \lambda (c1\otimes ...\otimes cn) = \pm 1.
Proof. By [5] (Theorem 4.2) (or [3], Theorem 4.1) the twisted tensor product Ae \otimes \tau C is iso-
morphic to the minimal projective resolution of the regular A-bimodule A, where C = \mathrm{T}\mathrm{o}\mathrm{r}A\bullet (K,K)
is the previous coaugmented A\infty -algebra and \tau is the twisting cochain given in that theorem. Com-
paring the differential of Ae\otimes \tau C given in [5], (4.1), with the one in [2] (Theorem 4.1) (see also [6],
Section 3), it follows that the mentioned coefficient is \pm 1.
Remark 1.2. In the examples, the computation of the remaining coefficients in (1) is in general
rather simple to carry out, by imposing that the Stasheff identities are fulfilled.
References
1. D. J. Anick, On the homology of associative algebras, Trans. Amer. Math. Soc., 296, № 2, 641 – 659 (1986).
2. M. J. Bardzell, The alternating syzygy behavior of monomial algebras, J. Algebra, 188, № 1, 69 – 89 (1997).
3. E. Herscovich, Applications of one-point extensions to compute the A\infty -(co)module structure of several Ext (resp.,
Tor) groups, J. Pure and Appl. Algebra, 223, № 3, 1054 – 1072 (2019).
4. E. Herscovich, On the Merkulov construction of A\infty -(co)algebras, Ukr. Mat. Zh., 71, № 8, 1133 – 1140 (2019).
5. E. Herscovich, Using torsion theory to compute the algebraic structure of Hochschild (co)homology, Homology,
Homotopy and Appl., 20, № 1, 117 – 139 (2018).
6. E. Sköldberg, A contracting homotopy for Bardzell’s resolution, Math. Proc. Roy. Irish Acad., 108, № 2, 111 – 117
(2008).
7. P. Tamaroff, Minimal models for monomial algebras (2018), 28 p., available at https://arxiv.org/abs/1804.01435.
Received 12.04.18
1 P. Tamaroff has told me that, by carefully choosing the SDR data for B+(A) and following all the steps in the
recursive Merkulov procedure, he can even prove that all nonzero coefficients are \pm 1, at least if K is a field (see [7]). Our
results are not so general but they are immediate, since we did not need to look at the interior of the Merkulov construction.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
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| id | umjimathkievua-article-6040 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| last_indexed | 2026-03-24T03:25:43Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-60402025-03-31T08:48:28Z A simple note on the Yoneda (co)algebra of a monomial algebra A simple note on the Yoneda (co)algebra of a monomial algebra Herscovich , E. Herscovich , E. homological algebra dg (co)algebras $A__{∞}$-(co)algebras homological algebra dg (co)algebras $A__{∞}$-(co)algebras UDC 512.7 If $A = TV/\langle R\rangle$ is a monomial $K$-algebra, it is well-known that $\operatorname{Tor}_{p}^{A}(K,K)$ is isomorphic to the space $V^{(p-1)}$ of (Anick) $(p-1)$-chains for $p \geq 1$. The goal of this short note is to show that the next result follows directly from well-established theorems on $A_{\infty}$-algebras, without computations: there is an $A_{\infty}$-coalgebra model on $\operatorname{Tor}_{\bullet}^{A}(K,K)$ satisfying that, for $n \geq 3$ and $c \in V^{(p)}$, $\Delta_{n}(c)$ is a linear combination of $c_{1} \otimes \ldots \otimes c_{n}$, where $c_{i} \in V^{(p_{i})}$, $p_{1} + \ldots +p_{n} = p - 1$ and $c_{1} \ldots c_{n} = c$. The proof follows essentially from noticing that the Merkulov procedure is compatible with an extra grading over a suitable category. By a simple argument based on a result by Keller we immediately deduce that some of these coefficients are $\pm 1$. УДК 512.7 Просте повiдомлення про (ко)алгебру Йонеди алгебри одночленiв   Якщо $A = TV/\langle R\rangle$ — $K$\!-алгебра одночленів, то відомо, що $\operatorname{Tor}_{p}^{A}(K,K)$ є ізоморфним простору $V^{(p-1)}$ $(p-1)$-ланцюгів (Аніка) для $p \geq 1$. Метою цього повідомлення є намагання показати, що наступний результат без будь-яких обчислень безпосередньо випливає з встановлених теорем для $A_{\infty}$-алгебр: існує $A_{\infty}$-коалгебраїчна модель на $\operatorname{Tor}_{\bullet}^{A}(K,K)$ така, що для $n \geq 3$ і $c \in V^{(p)}$ \ $\Delta_{n}(c)$ є лінійною комбінацією $c_{1} \otimes \ldots \otimes c_{n}$, де $c_{i} \in V^{(p_{i})}$, $p_{1} + \ldots +p_{n} = p - 1$ і $c_{1} \ldots c_{n} = c$. Доведення, в основному, є наслідком того, що процедура Меркулова сумісна з додатковим градуюванням деякої відповідної категорії. За допомогою простих аргументів, що базуються на результатах Келлера, безпосередньо приходимо до висновку, що деякі з цих коефіцієнтів дорівнюють $\pm 1$. Institute of Mathematics, NAS of Ukraine 2021-02-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6040 10.37863/umzh.v73i2.6040 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 2 (2021); 275 - 277 Український математичний журнал; Том 73 № 2 (2021); 275 - 277 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6040/8971 |
| spellingShingle | Herscovich , E. Herscovich , E. A simple note on the Yoneda (co)algebra of a monomial algebra |
| title | A simple note on the Yoneda (co)algebra of a monomial algebra |
| title_alt | A simple note on the Yoneda (co)algebra of a monomial algebra |
| title_full | A simple note on the Yoneda (co)algebra of a monomial algebra |
| title_fullStr | A simple note on the Yoneda (co)algebra of a monomial algebra |
| title_full_unstemmed | A simple note on the Yoneda (co)algebra of a monomial algebra |
| title_short | A simple note on the Yoneda (co)algebra of a monomial algebra |
| title_sort | simple note on the yoneda (co)algebra of a monomial algebra |
| topic_facet | homological algebra dg (co)algebras $A__{∞}$-(co)algebras homological algebra dg (co)algebras $A__{∞}$-(co)algebras |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6040 |
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