Tangent complexes and the Diamond Lemma

When studying algebras presented by generators and relations, the central general result is the statement known as the Diamond Lemma. Historically, one would say that it was already implicit in Shirshov’s work [86] who proved an analogous but technically more involved result, the Composition Lemma, in the case of Lie algebras. About a decade after that, this result was proved in the case of associative algebras by Bokut in [17], and independently by Bergman [11], who also proposed the name “Diamond Lemma” in order to emphasize the analogy with the classical result of Newman [68]. Paraphrasing the opening phrase of [11], the main results of our paper are doubly trivial: Bergman’s Diamond Lemma is, in his own words, trivial, and our main goal is to explain a new trivial proof of this trivial statement, as well as some other trivial statements, from the point of view of homotopical algebra. However, we believe that readers of this paper can benefit from it in a number of ways. For a reader whose intuition comes from homotopical algebra, our proof will hopefully feel like a conceptual explanation of useful but seemingly technical criteria of “resolvable ambiguities” for uniqueness of normal forms. For a reader with background in Gröbner bases or term rewriting, our proof will offer intuition behind both the Diamond Lemma and its optimisation, known as the Chain Criterion in the commutative case [21, 27, 48] and as the Triangle Lemma in the case of noncommutative associative algebras [20, 54, 85], as well as precise guidance on how to generalise those for other algebraic structures. Specifically, our work means that computing models of algebras with monomial relations explicitly helps both to state the relevant Diamond Lemmas and to optimise them.

The idea of using Gröbner bases for computing homological invariants of associative algebras is well known. The seemingly earliest instance appears in the work of Priddy [70] who constructed resolutions for the ground field, viewed as the trivial module, for algebras presented by quadratic–linear relations; this was later generalised by Anick [3] to arbitrary presentations. Existence of such resolutions is not at all surprising for the following reason. Defining relations of an associative algebra $A$ form a Gröbner basis if the same monomials give a vector space basis for both algebras $A$ and $A_{\mathrm{mon}}$; the latter is the algebra with monomial relations given by the leading monomials of the relations of $A$ with respect to a suitable ordering. The trivial module for $A_{\mathrm{mon}}$ admits a combinatorially defined free resolution, implicit in the work [4] of Backelin, and explicitly determined by Green, Happel and Zacharia in [32]. Such resolution can be obtained as a contraction of the bar resolution; incorporating the lower terms of relations from a Gröbner basis can obtained from such a contraction by homological perturbation techniques [10, 34, 35, 53]. In fact, the one-sided module resolution of Anick can be generalised to a bimodule resolution, as established by Bardzell [5, 6], leading to a computational method for determining the Hochschild (co)homology of an algebra, and thus for studying deformations of a given algebra. This construction works under a weaker assumption of a convergent rewriting system instead of a Gröbner basis, as shown by Chouhy and Solotar in [23].

Our main wish, motivated by interest in generalising these methods to algebraic structures other than associative algebras, is to go in the opposite direction and re-discover effective criteria for reduction systems with unique normal forms from the deformation theory viewpoint. Since in those situations the multiplication table of an algebra $A$ is obtained from the simple combinatorial multiplication table of $A_{\mathrm{mon}}$ by incorporating appropriate lower terms, we naturally find ourselves in the framework of deformation theory. In fact, very recently the bimodule resolutions mentioned above have been used to study deformations of algebras with monomial relations by Barmeier and Wang in their work on deformation theory of quiver algebras [7] and by Redondo and Rossi Bertone in [75], using the following idea. A free bimodule resolution

leads to the representative $\operatorname{Hom}(C_{\bullet},A_{\mathrm{mon}})$ of the deformation complex, and one can use a strong deformation retract relating it to Hochschild complex to compute explicitly its $L_{\infty}$-algebra structure using the homotopy transfer theorem. Studying deformations amounts to studying solutions to the Maurer–Cartan equation in that $L_{\infty}$-algebra, and one may obtain various results this way. However, if one adopts this viewpoint, the Diamond Lemma criterion of “resolvable ambiguities” cannot be recovered instantly: it involves a calculation in the free associative algebra which cannot be reproduced directly since the target space of the deformation complex is the monomial algebra $A_{\mathrm{mon}}$. The two deformation complexes are homotopy equivalent, so the necessary result can be proved in principle. However, recovering the classical Diamond Lemma will require some translation, in the spirit of Anick’s slogan “the perturbative construction of the second differential of the resolution is precisely the procedure of resolving ambiguities”, see [3, Sec. 2]. We also refer the reader to a discussion of this phenomenon in the survey of Ufnarovski [85, Sec. 3.8] where one also finds the Triangle Lemma as a way to optimise the algorithm.

It turns out that a natural way to remedy the situation is to move from homology to homotopy, and study deformations of algebras in terms of homotopical algebra. This means working with multiplicative free resolutions, that is resolutions which are free as algebras rather than as bimodules. In this case, the homotopy class of the deformation complex of an algebra can be represented by its tangent complex, that is, the differential graded Lie algebra of derivations of its free resolution. Historically, this approach to deformations first emerged in the case of commutative associative algebras. In this context, it makes sense to note that some of the pioneering works both in the theory of Gröbner bases and in the deformation theory appeared in the algebro-geometric context: the former in the solution of the resolution of singularities problem in characteristic zero by Hironaka [39, 40], and the latter in Palamodov’s work on deformation of analytic spaces [69], where the tangent complex approach is attributed to unpublished work of Tyurina (who tragically died in a kayaking accident at the age of 32). However, the standard bases of Hironaka were just one of many tools in a paper of more than two hundred pages, and the power of this method in algebraic geometry and commutative algebra became apparent only after work of Buchberger [22] who highlighted their algorithmic nature. The tangent complex approach to deformation theory of algebras became widely known from the famous manuscript [78] by Schlessinger and Stasheff. We would like to also remark that a seemingly completely independent path to multiplicative resolutions which however stays away from deformation theory questions emerges in higher-categorical rewriting theory. Original work of Squier [52, 80] on homological finiteness conditions for monoids admitting a convergent presentation first received a higher-categorical flavour in works of Lafont [50] and Citterio [24]; corresponding multiplicative resolutions appear in work of Lafont and Métayer [51] and Guetta [33] relying on theory of resolutions of categories by polygraphs developed by Métayer [67]. This formalism was recently extended to the $\mathbbold{k}$-linear context by Guiraud, Hoffbeck, and Malbos [37], who in particular explained how to construct a polygraphic resolution of an associative algebra by techniques similar to the perturbative construction of $A_{\infty}$-structures [36] and free resolutions of algebras [43].

For us, it is crucial that working with free resolutions means that recovering Bergman’s criterion of resolvable ambiguities stands a chance, at least in principle, since our deformation complexes of algebras with monomial relations have free associative algebras as target spaces. Our approach blends ideas and methods from several different areas, and we tried to offer enough detail when recalling the necessary background information. Our first homotopical re-formulation of uniqueness of normal forms is a general criterion in terms of perturbations of free resolutions (Theorem 4.1). Using that result, we prove another general criterion in terms of approximate solutions to the Maurer–Cartan equation in the tangent complex (Theorem 5.1). Using this latter result, we establish the main result of this paper (Theorem 5.2): every model extending the Shafarevich complex of the given relations gives rise to its own Diamond Lemma, so that the resolvable ambiguities condition of the Diamond Lemma becomes the first non-trivial component of the Maurer–Cartan equation in the tangent complex of the model. Applying that latter result to some specific combinatorial models, we recover the classical Diamond Lemma in the case of the inclusion–exclusion model of the first author and Khoroshkin [25] and the Triangle Lemma in the case of the minimal model of the second author [82]. We discuss some analogues and possible generalisations of our results for other types of algebras towards the end of the article. Already in the case of operads with monomial relations, explicit formulas for minimal models are not known in general, and only a much weaker version of the Triangle Lemma is available [20]; we hope that our work will attract due attention to this question.

To conclude the introduction, we would like to mention two crucial sources of inspiration for our work. The first of them is the work of Golod who discovered a proof of the Diamond Lemma [29] using the non-commutative analogue of the Koszul complex, the Shafarevich complex associated to a system of elements in the free algebra originally introduced in [31]. That complex is a differential graded algebra whose homotopy type depends on the presentation of the original algebra. Our homotopy invariant free resolutions are obtained from the Shafarevich complex at the cost of adding extra generators of higher degrees; we believe that the benefit of the resulting clarity outweighs the cost. Our second inspiration comes from the work of Latyshev who used normal forms to resolve some particular cases [58, 59] of the celebrated Specht problem on identities of associative algebras [79], and advocated general importance of normal forms and standard bases [55, 56, 57]. Both Golod and Latyshev passed away relatively recently (in July 2018 and April 2020, respectively). We would like to dedicate this work to their memory.

The authors thank Leonid Arkadievich Bokut, Eric Hoffbeck, Anton Khoroshkin and Muriel Livernet for useful discussions. The first author also thanks Dmitri Piontkovski who introduced him to Shafarevich complexes many years ago, and Emil Sköldberg for sharing some computations of minimal models of monomial operads in low homological degrees. The idea of this article emerged during the workshop “Homotopy meets homology” at Hamilton Mathematics Institute (Trinity College Dublin) in May 2019, and was primarily guided by our wish to unveil homotopical algebra behind the work [75] presented at that workshop by María Julia Redondo; we are grateful to the Science Foundation Ireland and the Simons Foundation for allocating funding that made that workshop possible. Part of this work was done during the second author’s visits to Strasbourg funded by IRMA (Institut de Recherche Mathématique Avancée), he is thankful for both the financial support of those visits and the excellent working conditions.

Let us fix once and for all a finite set $X$; we shall denote by $\langle X\rangle$ the free monoid generated by $X$, so that the linear span $\mathbbold{k}\langle X\rangle$ is the free associative $\mathbbold{k}$-algebra generated by $X$. For an element $g\in\mathbbold{k}\langle X\rangle$ we shall denote by $\operatorname{supp}(g)$ the set of all elements of $\langle X\rangle$ that appear in $g$ with a non-zero coefficient.

The right way to think of a rewriting system is as of a collection of rules that allow to rewrite each monomial $w\in W$ into the element $f(w)\in\mathbbold{k}\langle X\rangle$. In this way, each rewriting system $\Sigma$ naturally gives rise to an associative algebra $A_{\Sigma}$ with relations $R=\{w-f(w)\colon w\in W\}$:

Each rewriting rule of $\Sigma$ becomes a way to replace elements of $\mathbbold{k}\langle X\rangle$ by other representatives of the same coset in $A_{\Sigma}$, as follows.

Using reductions is intended to “simplify” representatives of cosets. This is formalised by the notion of a normal form.

In principle, it is possible that one can perform basic reductions infinitely many times. We shall only work with rewriting systems for which it does not happen. To formalise that, we give the following definition.

Note that if an element $g$ is reduction finite, there exists a reduction that sends this element to an irreducible one, for otherwise we could easily find a pseudo-reduction $\rho$ for which $\rho(g)$ is not eventually constant. Thus, for a terminating rewriting system, each element has at least one normal form. All rewriting systems considered in this paper are assumed to be terminating. We shall now recall how to recast termination in terms of partial orderings.

Among rewriting orderings compatible with $\Sigma$ (if they exist at all), there exists the weakest possible one obtained as follows. We first define $<_{\Sigma}$ as the smallest transitive binary relation that is multiplicative and compatible with $\Sigma$. If $<_{\Sigma}$ is a well-ordering, the reflexive closure $\leqslant_{\Sigma}$ is a rewriting ordering compatible with $\Sigma$, and each other rewriting ordering compatible with $\Sigma$ obviously refines the ordering $\leqslant_{\Sigma}$. According to [11, 23], the rewriting system $\Sigma$ is terminating if and only if $<_{\Sigma}$ is a well-ordering.

For a terminating rewriting system, the associative algebra

obtained from the rewriting system $\Sigma_{0}=(W,0)$ and called the monomial algebra associated to $\Sigma$ is a relevant object of study. Indeed, an element $g\in\mathbbold{k}\langle X\rangle$ is irreducible if and only if no monomial $m\in\operatorname{supp}(g)$ is divisible by elements of $W$, so irreducible monomials form a basis of the algebra $A_{\Sigma_{0}}$. However, since the same element may have several different normal forms, the cosets of irreducible monomials might not be linearly independent in the algebra $A_{\Sigma}$.

Among the rewriting systems, the best known class is given by Gröbner bases: those are rewriting systems where there is a total order $\leqslant$ on $\langle X\rangle$ satisfying the above three conditions. Let us recall an example [20, 37] showing that sometimes using a partial order is really advantageous. For that, we consider the algebra $A=\mathbbold{k}\langle x,y,z\mid x^{3}+y^{3}+z^{3}-xyz\rangle$. Note that for a total multiplicative order, we always have $xyz<\max(x^{3},y^{3},z^{3})$, and one can check that for such an order the corresponding rewriting system is not convergent, so one needs to extend it by further rewriting rules. However, one can consider the rewriting system $\Sigma=(X,W,f)$ with $X=\{x,y,z\}$, $W=\{xyz\}$ and $f(xyz)=x^{3}+y^{3}+z^{3}$ which does not arise from a total multiplicative order. Let us explain why this rewriting system is terminating. For that, we define, for each monomial $m$, $\Phi(m)=3A(m)+B(m)$, where $A(m)$ is the number of divisors of $m$ equal to $xyz$ and $B(m)$ is the number of divisors of $m$ equal to $y$. Then one can check that any application of our rewriting rule replaces a monomial by a linear combination of terms for which the parameter $\Phi$ is strictly smaller, which guarantees the termination. Later we shall see that this rewriting system is convergent, showing that in this case there is an advantage over the theory of Gröbner bases.

Bergman’s Diamond Lemma [11] establishes a criterion of convergence in terms of the so called “resolvable ambiguities”. If this were a paper on rewriting systems, at this point we would formulate and prove that result. However, our goal is to explain how such criterion can be found, so that one can determine its analogues for other algebraic structures. It is easy to see that a rewriting system $\Sigma$ is convergent if and only if the corresponding irreducible monomials are linearly independent. In other words, $\Sigma$ is convergent if and only if the linear map $A_{{\Sigma_{0}}}\to A_{\Sigma}$ sending each element of the standard monomial basis of $A_{{\Sigma_{0}}}$ to its coset in $A_{\Sigma}$ is an isomorphism of vector spaces. This suggests that convergent rewriting systems may be studied from the viewpoint of deformation theory of associative algebras. In general, the notion of a deformation implies that we deform the structure by adding lower terms with respect to a suitable filtration. For example, for the classical deformation theory set-up [65] defined over an Artinian local ring containing $\mathbbold{k}$, the filtration may be defined by powers of the maximal ideal $\mathfrak{m}$ of that ring. In our case, the filtration is defined out of the partial ordering $<_{\Sigma}$, or, better to say, out of any order that extends it to a total well-ordering. The multiplication table in the algebra $A_{{\Sigma_{0}}}$ is very simple: the product of two basis elements is either a basis element or zero. For a convergent rewriting system, the product of two basis elements in the algebra $A_{{\Sigma}}$ is either a basis element or the linear combination of smaller basis elements obtained by term rewriting. Thus, saying that $\Sigma$ is convergent is equivalent to saying that the algebra $A_{\Sigma}$ is a deformation of the algebra $A_{\Sigma_{0}}$ with respect to the corresponding filtration. This suggests that using deformation theory of algebras might shed light on convergence of rewriting systems. The main part of this paper uses this viewpoint extensively.

A sufficiently general recipe to define model categories is to consider simplicial algebras of the given type. However, in the case of associative algebras, it turns out to be possible to do homotopy theory in a slightly more hands-on way, using differential graded algebras.

We denote by $\operatorname{\mathsf{DGA}}$ the category whose objects are differential graded associative algebras and whose morphisms from $(A,d_{A})$ to $(B,d_{B})$ are (degree zero) algebra homomorphisms $f\colon A\to B$ that are chain maps, that is satisfy $f\circ d_{A}=d_{B}\circ f$. Such a morphism $f$ induces a morphism $f_{\bullet}\colon H_{\bullet}(A,d_{A})\to H_{\bullet}(B,d_{B})$ of graded associative algebras obtained by computing homology. We say that $f$ is a quasi-isomorphism if the induced homology morphism $f_{\bullet}$ is an isomorphism.

As indicated above, our general plan is to replace an associative algebra with a better behaved differential graded algebra of the same homotopy type. A mathematically precise meaning of the words “better behaved” is given by the notion of a cofibrant object in a closed model category. A motivated reader is invited to consult [41, 73] for foundations of the general theory of model categories. According to Hinich [38, Th. 4.1.1], the category $\operatorname{\mathsf{DGA}}$ admits a closed model category structure for which weak equivalences are quasi-isomorphisms, and the fibrations are surjections. In fact, Hinich works in the general framework of algebras over operads, and in the only case relevant for this paper, that of differential graded algebras concentrated in non-negative homological degrees, the same result is established by Jardine [45]. To describe cofibrant objects, we need one more definition.

The cofibrant objects for Hinich’s model category structure on $\operatorname{\mathsf{DGA}}$ are exactly the retracts of triangulated quasi-free algebras. We remark that it is useful to think that triangulated algebras are an abstraction of the construction of “killing cycles” of Tate [46, 83]. The relevance of triangulated differential graded commutative algebras in homotopy theory became apparent after the seminal paper of Sullivan [81]; for this reason, the are frequently called Sullivan algebras.

While minimal models of algebras are unique up to isomorphism, their existence is a much more subtle matter; see, for example, the unpublished note of Keller [47]. However, the target algebras $(A,d_{A})$ of interest for us are in fact non-graded and non-differential, in other words, $A_{n}=0$ for $n\neq 0$ and $d_{A}=0$. Since we work over a field, minimal models for such algebras exist under very mild assumptions (see, for example, [8, 43]) which hold in all cases that are of interest to us.

We shall now recall the notion of a Shafarevich complex of a system of elements in an algebra. Those complexes are very rarely models themselves, but they offer a good starting point for constructing a model, which will be one of the guiding principles for our main results.

It is an immediate consequence of the definition that the degree zero homology of $\Sha(S,A)$ is isomorphic to the quotient $\cong A/(S)$. In this paper, we shall only consider the Shafarevich complexes $\Sha(S,\mathbbold{k}\langle X\rangle)$ for the free algebra $A=\mathbbold{k}\langle X\rangle$; it is precisely those that were first defined in work of Golod and Shafarevich [31] and later called the Shafarevich complexes by Golod in [29]. In this particular case, the degree zero homology is the algebra $\mathbbold{k}\langle X\mid S\rangle$. Of course, $\Sha(S,\mathbbold{k}\langle X\rangle)=(T(V),d)$ where $V$ is the graded vector space whose only non-zero components are $V_{0}=\mathbbold{k}X$ and $V_{1}\cong\mathbbold{k}S$, and $d$ is the unique graded derivation satisfying $d(x)=0$ for $x\in X$ and $d(e_{s})=s\in\mathbbold{k}\langle X\rangle\cong T(V_{0})$ for each basis element $s\in V_{1}$. Thus, $\Sha(S,\mathbbold{k}\langle X\rangle)$ is a triangulated quasi-free algebra whose space of generators is concentrated in homological degrees zero and one corresponding to killing certain cycles in $T(V_{0})$. In that capacity, it was rediscovered by Lemaire [60], and used by Anick in his work on Hilbert series of associative algebras [1, 2].

In general, the positive degree homology of $\Sha(S,\mathbbold{k}\langle X\rangle)$ does not vanish, so that differential graded algebra is not a model of $\mathbbold{k}\langle X\mid S\rangle$. In fact, it is known that the homology of $\Sha(S,\mathbbold{k}\langle X\rangle)$ is generated by elements of homological degrees zero and one [30], but this remarkable result will not be immediately useful for us. We however will be mainly interested in models that extend Shafarevich complexes.

We begin with a brief recollection of differential graded Lie algebras and their Maurer–Cartan elements.

From the homotopical algebra point of view, one of the protagonist in the homotopy theory for differential graded Lie algebras is the set of Maurer–Cartan elements of such algebra.

Let $L$ be a graded Lie algebra. Note that for an element $x\in L$ of degree $-1$ and any $a\in L$, the Jacobi identity implies

and therefore

Thus, if we suppose that $x$ satisfies the condition $[x,x]=0$ (note that for elements of odd degree the graded antisymmetry property is actually symmetry, so this condition is non-empty), the degree $-1$ endomorphism $[x,-]$ squares to zero; the Jacobi identity implies that it also satisfies the derivation property. Thus, $(L,[x,-])$ is a differential graded Lie algebra.

Our Lie algebras of interest arise as Lie algebras of derivations of models of associative algebras. Suppose that $A$ is an associative algebra, and that $(B,d)$ is a model of $A$. The graded vector space $\operatorname{Der}(B)$ of all derivations of the graded associative algebra $B$ has a natural structure of a graded Lie algebra: the graded bracket

of two derivations is again a derivation. Clearly, $d\in\operatorname{Der}(B)$ and

as we saw above, the map $[d,-]\colon\operatorname{Der}(B)\to\operatorname{Der}(B)$ makes $\operatorname{Der}(B)$ a differential graded Lie algebra. We note that in the previous section, $d$ denoted the differential of the Lie algebra, and now our context forces us to consider differential graded Lie algebras whose differentials are of the form $[d,-]$ with $d$ being an element of the algebra, the differential of the model; we hope that this does not lead to a confusion. This algebra is called the tangent complex associated to the model $B$. Using the work [77] of Schlessinger and Stasheff as an inspiration, Hinich showed in [38] that the homotopy type of this Lie algebra does not depend on the choice of a model.

In our context, the Maurer–Cartan heuristics of deformation theory acquires very precise meaning. Since the differential of the tangent complex of a model is of the form $[d,-]$, the Maurer–Cartan equation in that differential graded Lie algebra is equivalent to the equation $[d+x,d+x]=0$, which in turn is equivalent to the square-zero condition for the derivation $d+x$. Thus, the Maurer–Cartan equation for the differential graded Lie algebra $(\operatorname{Der}(B),[d,-])$ describes perturbations of the differential of $B$.

In this section, we shall prove what is essentially the Diamond Lemma in disguise. It claims that having the Maurer–Cartan equation satisfied on generators of homological degree at most two of a model is sufficient to ensure existence of a perturbation. As above, we consider a rewriting system $\Sigma=(X,W,f)$.

In this section, we are finally rewarded for going through all the previous arguments involving models and the Maurer–Cartan equation; the reward is a general result stating that each word-homogeneous acyclic extension of $\Sha(W,\mathbbold{k}\langle X\rangle)$ leads to its own Diamond Lemma criterion for convergence of $\Sigma$. At the core of such result is the following definition of an S-polynomial associated to a generator of homological degree two of such model.

We remark that our usage of the term “obstruction” is different from that of Anick in [3]: for us, as it will become apparent below, $S_{c}$ is an actual obstruction to convergence of $\Sigma$.

We are now ready to state and prove our model-specific Diamond Lemma. As above, we consider a rewriting system $\Sigma=(X,W,f)$.