On Free Quadratic Modules of Commutative Algebras

Abstract

In this paper, we give a construction of (totally) free quadratic modules of commutative algebras on suitable construction data in terms of free simplicial algebras. Similar freeness results are also explored for reduced quadratic modules and quadratic (chain) complexes of algebras.

1 Introduction

Baues, [11], defined the notion of quadratic module of groups as an algebraic model for homotopy (connected) 3-types and gave a relation between quadratic modules and simplicial groups. This model is also a 2-crossed module structure defined by Conduché (cf. [12]) with additional nilpotent conditions.

In the series of articles [4–6], Arvasi and Porter have started to see how a study of the links between simplicial commutative algebras and classical constructions of homological algebra can be strengthened by interposing crossed algebraic models for the homotopy types of simplicial algebras. In particular, in [6], they gave a construction of a free 2-crossed module of algebras in terms of the 2-skeleton of a free simplicial algebra. In the group case, a closely related structure to that of 2-crossed module is that of a quadratic module (cf. [7]). Furthermore, in [9], a construction of the (totally) free quadratic module of groups was given. It seems clear that the results of papers [6, 7] and [9] can be extended to an algebra version of quadratic modules. In this work, we continue this process using these methods to give a construction of the (totally) free quadratic module of commutative algebras in terms of the 2-skeleton of a free simplicial algebra.

In [10], an alternative description of the top algebra of the free crossed squares of algebras was given on 2-construction data in terms of tensor product and coproduct of crossed modules of commutative algebras, and a link between free crossed squares and Koszul complexes was described. Using the close relationship between crossed squares and quadratic modules, we shall give an alternative construction of the totally free quadratic module. In this context, our result has an advantage; Baues’s theory is also related via a quotient functor $\mathbf q :$(free crossed squares)$\rightarrow$ (free quadratic modules) for the case of commutative algebras in terms of the 2-skeleton of a free simplicial algebra. We also interest on reduced quadratic modules of commutative algebras which are special kind of quadratic modules of algebras, describing the 3-types of simply connected CW-complexes constructed with algebras of nilpotency degree 2. In [20], Muro defined a suspension functor from the category of crossed modules of groups to that of reduced quadratic modules of groups which sends a 2-type to the 3-type of its suspension. Using this association, we give a construction of a totally free reduced quadratic module on commutative algebras.

2 (Free) Simplicial Algebras

In what follows, ‘algebras’ will be commutative algebras over an unspecified commutative ring, k, but for convenience are not required to have a multiplicative identity. The category of commutative algebras will be denoted by $\mathsf {Alg}$.

A simplicial algebra $\mathsf {E}$ consists of a family of algebras $\{E_{n}\}$ together with face and degeneracy maps $d_{i}=d_{i}^{n}:E_{n}\rightarrow E_{n-1},$ $0\leqslant i\leqslant n$, $(n\ne 0)$ and $s_{i}=s_{i}^{n}:E_{n}\rightarrow E_{n+1}$, $0\leqslant i\leqslant n$, satisfying the usual simplicial identities given in [1, 2] or [17]. In fact, it can be completely described as a functor $\mathsf {E}:\Delta ^{op}\rightarrow \mathsf {Alg}$ where $\Delta$ is the category of finite ordinals $[n]=\{0<1<\cdots <n\}$ and increasing maps. We denote the category of simplicial algebras by $\mathsf {SimpAlg}$.

Given a simplicial algebra $\mathsf {E}$, the Moore complex $(\mathsf {NE} ,\partial )$ of $\mathsf {E}$ is the chain complex defined by

$$\begin{aligned} {NE}_{n}=\ker d_{0}^{n}\cap \ker d_{1}^{n}\cap \cdots \cap \ker d_{n-1}^{n} \end{aligned}$$

with $\partial _{n}:NE_{n}\rightarrow NE_{n-1}$ induced from $d_{n}^{n}$ by restriction.

The nth homotopy module $\pi _{n}(\mathsf {E})$ of $\mathsf {E}$ is the nth homology of the Moore complex of $\mathsf {E}$, i.e.

$$\begin{aligned} \pi _{n}\mathsf {(E)}\cong H_{n}(\mathsf {NE},\partial )=\bigcap \limits _{i=0}^{n}\ker d_{i}^{n}/d_{n+1}^{n+1}\left( \bigcap \limits _{i=0}^{n}\ker d_{i}^{n+1}\right) . \end{aligned}$$

We say that the Moore complex $\mathsf {NE}$ of a simplicial algebra is of length k if $NE_{n}=0$ for all $n\geqslant k+1.$

André [1] used simplicial methods to investigate homological properties of commutative algebras and introduced ‘step-by-step’ construction of a resolution of a commutative algebra. The reader is referred to the book of André [1] and to the article of Arvasi and Porter [4] and the references there.

We recall briefly the ‘step-by-step’ construction of a free simplicial algebra from the article [4]. We denote by $\{m,n\}$ the set of increasing surjective maps $[m]\rightarrow [n]$ as given in [1, 4] and [19], where $[n]=\{0<1\cdots <n\}$ is an ordered set.

Let $\mathsf {E}$ be a simplicial algebra and $k\geqslant 1$ be fixed. Suppose we are given a set $\mathsf {\Omega }$ of elements $\mathsf {\Omega =\{}x_{\lambda }:\lambda \in \Lambda \}$, $x_{\lambda }\in \pi _{k-1}(\mathsf {E})$; then we can choose a corresponding set of elements $w_{\lambda }\in NE_{k-1}$ so that $x_{\lambda }=w_{\lambda }+\partial _{k}(NE_{k})$. We want to ‘kill’ the elements in $\mathsf {\Omega }$. It is formed a new simplicial algebra $\mathsf {F}$ where $F_{n}$ is a free $E_{n}$-algebra, $F_{n}=E_{n}[y_{\lambda ,t}]$ with $\lambda \in \Lambda \text { and }t\in \{n,k\}$, and for $0\leqslant i\leqslant n,$ the algebra homomorphisms $s_{i}^{n}:F_{n}\rightarrow F_{n+1}$ and $d_{i}^{n}:F_{n}\rightarrow F_{n-1}$ are obtained from the homomorphism $s_{i}^{n}:E_{n}\rightarrow E_{n+1}$ and $d_{i}^{n}:E_{n} \rightarrow E_{n-1}$, respectively, together with the relations given in [4].

Thus from the ‘step-by-step’ construction we can give the definition of a free simplicial algebra as follows.

Let $\mathsf {E}$ be a simplicial algebra and $k\geqslant 1$, k-skeletal be fixed. A simplicial algebra $\mathsf {F}$ is called a free if

(i):

$F_{n}=E_{n}$ for $n<k,$

(ii):

$F_{k}=$ a free $E_{k}$-algebra over a set of non- degenerate indeterminates, all of whose faces are zero except the kth, and

(iii):

$F_{n}$ is a free $E_{n}$-algebra over the degenerate elements for $n>k$.

A variant of the ‘step-by-step’ construction gives: if $\mathsf {A}$ is a simplicial algebra, then there exists a free simplicial algebra $\mathsf {E}$ and an epimorphism $\mathsf {E}\rightarrow \mathsf {A}$ which induces isomorphisms on all homotopy modules. The details are omitted as they are well known.

3 Quadratic Modules of Algebras

Baues in [11] defined the quadratic module of groups as an algebraic model of connected 3-types. In [8, Sect. 5], the second author with Arvasi defined the notion of quadratic module for commutative algebras and constructed a functor from 2-crossed modules to quadratic modules of algebras.

First, we give some basic definitions about the structure.

3.1 Nil(n)-Modules for Algebras

For an algebra C,  $C/C^{2}$ is the quotient of the algebra C by its ideal of squares. Then, there is a functor from the category of $\mathsf {k}$ -algebras to the category of $\mathsf {k}$-modules. This functor goes from C to $C/C^{2}$ and plays the role of abelianization in the category of $\mathsf {k}$-algebras. As modules are often called singular algebras (e.g. in the theory of singular extensions), we shall call this functor “singularisation”.

A pre-crossed module of algebras is a homomorphism of algebras $\partial :C\rightarrow R$ together with an action of R on C written $c\cdot r$ for $r\in R$ and $c\in C$ satisfying the condition $\partial (c\cdot r)=\partial (c)r$ for all $r\in R$ and $c\in C$. Let $\partial :C\rightarrow R$ be a pre-crossed module, $P_{1}(\partial )=C$, and $P_{2}(\partial )$ be the Peiffer ideal of C generated by elements of the form

$$\begin{aligned} \left\langle x,y\right\rangle =xy-x\cdot \partial y \end{aligned}$$

which is called the Peiffer element for $x,y\in C.$

The pre-crossed modules together with trivial Peiffer elements precisely are crossed modules. Namely, a crossed module (cf. [21, 22]) is a pre-crossed module $\partial :C\rightarrow R$ satisfying the extra condition

$$\begin{aligned} x\cdot \partial y=xy \text { for all } x,y\in C. \end{aligned}$$

We will denote the category of crossed modules by $\mathsf {XMod}$. Let $\partial :C\rightarrow R$ be a pre-crossed module. Then, the homomorphism

$$\begin{aligned} \partial ^{cr}:C^{cr}=C/P_{2}(\partial )\longrightarrow R \end{aligned}$$

defined by $\partial ^{cr}(xP_2)=\partial (x)$ is the crossed module associated to the pre-crossed module $\partial :C\rightarrow R$. Associating the crossed module $\partial ^{cr}$ to a pre-crossed module $\partial$ gives a functor

$$\begin{aligned} (-)^{cr}:\mathsf {PXMod}\rightarrow \mathsf {XMod} \end{aligned}$$

from the category of pre-crossed modules to that of crossed modules. That is, a morphism (g, f) of pre-crossed modules yields a morphism $(g^{cr},f)$ of crossed modules, and this association satisfies the usual functorial rules.

For any pre-crossed module $\partial :C\rightarrow R$, let $P_n(\partial )$ be the ideal of C generated by Peiffer elements of weight n. We say that $\partial$ is a nil(n)-module of algebras, if $P_{n+1}(\partial )=0$. Thus, by a similar reasoning the related nil(2)-module to the pre-crossed module $\partial$ is $\partial ^{nil}:C^{nil}=C/P_3(\partial )\rightarrow R$ where $\partial ^{nil}$ is given by $\partial ^{nil}(xP_3)=\partial (x)$ for all $x\in C$. The construction of the nil(2)-module $\partial ^{nil}$ from the pre-crossed module of algebras $\partial$ gives a functor from the category of pre-crossed modules to that of nil(2)-modules:

$$\begin{aligned} (-)^{nil}:\mathsf {PXMod}\rightarrow \mathsf {Nil}(2). \end{aligned}$$

Similarly, the related nil(n)-module to a pre-crossed module can be defined. This gives a functor $\mathsf {PXMod}\rightarrow \mathsf {Nil}(n)$ from the category of pre-crossed modules to that of nil(n)-modules of algebras.

3.2 The Definition of Quadratic Modules of Algebras

We can give the definition of a quadratic module of algebras from [8].

Definition 3.1

A quadratic module $(\omega ,\delta ,\partial )$ of algebras is a diagram

of homomorphisms of algebras such that the following axioms are satisfied.

$\mathsf {(QM1)}$ :

The homomorphism $\partial$:$M\rightarrow N$ is a nil(2)-module and the quotient map $M\twoheadrightarrow C=$ $M_{{}}^{cr}/(M_{{}}^{cr})^{2}$ is given by $x\mapsto [x],$ where $[x]\in C$ denotes the class represented by $x\in M$. The map w is defined by Peiffer multiplication, i.e. $w([x]\otimes [y])=xy-x\cdot \partial (y).$

$\mathsf {(QM2)}$ :

The homomorphisms $\delta$ and $\partial$ satisfy $\delta \partial =0$ and the quadratic map $\omega$ is a lift of the map w, that is $\delta \omega =w$ or equivalently

$$\begin{aligned} \delta \omega ([x]\otimes [y])=w([x]\otimes [y])=xy-x\cdot \partial (y) \end{aligned}$$

for $x,y\in M$.

$\mathsf {(QM3)}$ :

L is an N-algebra and all homomorphisms of the diagram are equivariant with respect to the action of N. Moreover, the action of N on L satisfies the following equality:

$$\begin{aligned} a\cdot \partial (x)=\omega ([\delta a]\otimes [x]+[x]\otimes [\delta a]) \end{aligned}$$

for $a\in L,x\in N$.

$\mathsf {(QM4)}$ :

For $a,b\in L$,

$$\begin{aligned} \omega ([\delta a]\otimes [\delta b])=ab. \end{aligned}$$

Example 3.2

Any nil(2)-module $\partial _1 :M\rightarrow N$ gives a quadratic module. Since $(M,N,\partial _1)$ is a nil(2)-module, if we take $L=C\otimes C$ with the natural actions, then the diagram

with the obvious actions is a quadratic module.

4 Free Quadratic Modules of Algebras

In this section, we will define the notion of a totally free quadratic module of algebras and construct it using 2-skeleton of a free simplicial algebra.

Definition 4.1

Let

be a quadratic module and let $\vartheta :Y\rightarrow C_{2}$ be the function. Then $(\omega ,\partial _{2},\partial _{1})$ is said to be free quadratic module with basis $\vartheta$, or alternatively on the function $\partial _{2}\vartheta :Y\rightarrow C_{1}$, if for any quadratic module $(\omega ^{\prime },\partial _{2}^{\prime },\partial _{1})$ and $\vartheta ^{\prime }:Y^{\prime }\rightarrow C_{2}^{\prime }$ such that $\partial _{2}^{\prime }\vartheta ^{\prime }=\partial _{2}\vartheta ,$ there is a unique morphism $\Phi :C_{2}\rightarrow C_{2}^{\prime }$ such that $\partial _{2}^{\prime }\Phi =\partial _{2}.$ We say that a free quadratic module $(\omega ,\partial _{2},\partial _{1})$ is a totally free quadratic module if $\partial _{1}$ is a free nil(2)-module.

Now, we construct the totally free quadratic module of algebras. To give it, we need to recall the 2-skeleton of a free simplicial algebra given by

with the simplicial structure defined as in Sect. 3 of [4].

Analysis of this 2-dimensional construction data shows that it consists of some 1-dimensional data, namely the function $\vartheta :X\rightarrow R,$ that is used to induce $d_{1}:R[X]\rightarrow R$, together with strictly 2-dimensional construction data consisting of the function $\psi :Y\rightarrow R^{+}[X]$, and this function is used to induce $d_{2}:R[s_{0}X,s_{1}X][Y]\rightarrow R[X].$ We will denote this 2-dimensional construction data by $(\vartheta ,\psi ,R)$.

Theorem 4.2

A totally free quadratic module of algebras exists on the 2-dimensional construction data $(\vartheta ,\psi ,R)$.

Proof

Suppose that given a 2-dimensional construction data $(\vartheta ,\psi ,R)$ , i.e. given a function $\vartheta :X\rightarrow R$ and $\psi :Y\rightarrow R^{+}[X]$, with the obvious action of R on $R^{+}[X],$ we have a free pre-crossed module $\partial :R^{+}[X]\rightarrow R$ with basis $\vartheta :X\rightarrow R.$ Here $R^{+}[X]=NE_{1}^{(1)}=\ker d_{0}$ is the positive degree part of R[X]. Now let $P_{3}$ be the ideal of $R^{+}[X]$ generated by the Peiffer elements of length 3 in $R^{+}[X]$. We can define the quotient morphism $q_{1}:R^{+}[X]\rightarrow R^{+}[X]/P_{3}=M$ and then, we have a commutative diagram

where i is the inclusion.

Since $\partial :R^{+}[X]\rightarrow R$ is a pre-crossed module, we have $\partial (P_{3})=0$ and then the map $\partial _{1}: R^{+}[X]/P_3\rightarrow R$ given by $x+P_{3}\mapsto \partial (x)$ for $x\in R^{+}[X]$ is a well-defined homomorphism. Therefore, we have a free nil(2)-module $\partial _{1}:M=R^{+}[X]/P_{3}\rightarrow R$ with basis $\vartheta :X\rightarrow R$, or on the function $\partial _{1}q_{1}i$, since the triple Peiffer elements are trivial in M.

Now, take $D=NE_{2}^{(2)}=R[s_{0}X]^{+}[s_{1}X,Y]\cap ((s_{0}-s_{1})(X))$. Then, as 2-dimensional construction data, the function

$$\begin{aligned} q_{1}\psi :Y\rightarrow M=R^{+}[X]/P_{3} \end{aligned}$$

induces a morphism of algebras

$$\begin{aligned} \theta :R[s_{0}X]^{+}[s_{1}X,Y]\cap ((s_{0}-s_{1})(X))\rightarrow R^{+}[X]/P_{3} \end{aligned}$$

given by $\theta (y)=q_{1}\psi (y)$. Let $P^{\prime }=\partial _{3}(NE_{3}^{(2)})\subset D$ be the Peiffer ideal. Then, $q_{1}\psi (P^{\prime })=0$ as all generator elements of $P^{\prime }$ in $\ker d_{2}.$ Taking the factor algebra $L=D/P^{\prime }$, there is a morphism $\psi ^{\prime }:L\rightarrow M$ such that the diagram

commutes, that is, $\psi ^{\prime }q=q_{1}\psi =\theta$. The Peiffer elements in $R^{+}[X]$ are given by

$$\begin{aligned} \left\langle X_{i},X_{j}\right\rangle =X_{i}X_{j}-\partial (X_{i})X_{j} \end{aligned}$$

for $X_{i},X_{j}\in R^{+}[X]$. Let $P^{\prime }_3$ be the ideal of L generated by elements of the form

$$\begin{aligned} s_1(\langle X_i, X_j\rangle )s_1(X_k)-s_1(\langle X_i, X_j\rangle )s_0(X_k) \end{aligned}$$

and

$$\begin{aligned} s_1(X_i)s_1 (\langle X_j, X_k \rangle )-s_1(X_i)s_0 (\langle X_j, X_k \rangle ) \end{aligned}$$

for all $X_i, X_j, X_k \in R^{+}[X]$. Let $L^{\prime }=L/P_{3}^{\prime }$ and let $q_{2}:L\rightarrow L^{\prime }=L/P_{3}^{\prime }$ be the quotient morphism. We have $\psi (P_{3}^{\prime })=P_{3}.$ Hence, we define a morphism $\psi ^{\prime \prime }$ from $L^{\prime }$ to M such that $\psi ^{\prime \prime }q_{2}=q_{1}\psi$. Then, the diagram

becomes a totally free quadratic module of algebras, where $C=M^{cr}/(M^{cr})^{2}$, and the quadratic map $\omega$ can be given by

$$\begin{aligned} \begin{array}{lll} \omega (\{q_{1}X_{i}\}\otimes \{q_{1}X_{j}\})= & {} q_{2}(s_{1}X_{i}(s_{1}X_{j}-s_{0}X_{j})+P^{\prime }) \end{array} \end{aligned}$$

for $X_{i}\in R^{+}[X],$ $q_{1}(X_{i})\in M=R^{+}[X]/P_{3}$ and $\{q_{1}X_{i}\}\in C=M^{cr}/(M^{cr})^{2}$.

Let

be any quadratic module of algebras and let $\vartheta ^{\prime }:Y\rightarrow A$ be the function. Then there exists a unique morphism $\Phi :L^{\prime }\rightarrow A$ given by $\Phi (q_{2}(y+P^{\prime }))=\vartheta ^{\prime }(y)$ such that $\partial _{2}^{\prime }\Phi =\psi ^{\prime \prime }$. Thus $(\omega ,\psi ^{\prime \prime },\partial _{1})$ is the required totally free quadratic module with basis $q_{1}\psi :Y\rightarrow M.$ $\square$

4.1 From Free Crossed Squares to Free Quadratic Modules

Crossed squares were initially defined by Guin-Waléry and Loday in [16]. The commutative algebra analogue of crossed squares has been studied by Ellis (cf. [13]).

A crossed square of algebras is a commutative diagram of morphisms of algebras

together with the actions of R on L, M and N and a function

$$\begin{aligned} h:M\times N\rightarrow L \end{aligned}$$

satisfying a number of axioms which we do not give in full here.

Ellis, [14], presented the notion of a free crossed square for the case of groups and in the context of CW-complexes gave a neat description of the top term L in a free crossed square of groups. He also constructed a free quadratic module of groups from a free crossed square using the quotient functor q. In [10], a construction of the top algebra L in a free crossed square of algebras was given on the 2-dimensional construction data in terms of tensor and coproduct in the category of crossed modules of algebras.

Now, we recall from [10] the definition of a free crossed square of algebras on a pair of functions $(f_2, f_3)$:

Let $\mathsf {S_1, S_2}$ and $\mathsf {S_3}$ be sets and $f_2:\mathsf {S_2} \rightarrow R$ is a function from the set $\mathsf {S_2}$ to the free algebra R on $\mathsf {S_1}$. Let $\partial :M\rightarrow R$ be the free pre-crossed module on the function $f_2$. Consider the semi-direct product $M\rtimes R$ and the inclusion $\mu :M\rightarrow M\rtimes R$ and the ideals M and $\overline{M}$ of $M\rtimes R$ where $\overline{M}=\{(m,-\partial m):m\in M\}$ with the inclusion $\mu ^{\prime }:\overline{M}\rightarrow M\rtimes R$. Assume given a function $f_3:\mathsf {S_3}\rightarrow M$, which is to satisfy $\partial f_3=0$. Then there is a corresponding function $\overline{f_3}: \mathsf {S_3}\rightarrow \overline{M}$ given by $y\mapsto (f_3(y),0)$.

We say a crossed square

is totally free on the pair of functions $(f_2, f_3)$ if

(i):

$(M,R,\partial )$ is the free pre-crossed module on $f_2$,

(ii):

$\mathsf {S_3}$ is a subset of L with $f_3$ and $\overline{f_3}$ the restrictions of $\partial _2$ and $\partial ^{\prime }_2$, respectively, and

(iii):

given any crossed square $(L^{\prime }, M, \overline{M}, M\rtimes R)$ and function $\nu :\mathsf {S_3}\rightarrow L^{\prime }$ there is a unique morphism $\phi :L\rightarrow L^{\prime }$ such that $\phi \nu ^{\prime }=\nu$ where $\nu ^{\prime }:S_3\rightarrow L$ is the inclusion.

Proposition 4.3

([10]) Let

be the free crossed square of algebras. Let $\partial :C\rightarrow M\rtimes R$ be the free crossed module on $\mathsf {S_3}\rightarrow M\rtimes R$. Form the crossed module $M\otimes \overline{M}\rightarrow M\rtimes R$. Then

$$\begin{aligned} L\cong \{(M\otimes \overline{M})\sqcup C\}/\sim \end{aligned}$$

where $\sim$ corresponds to the relations

(i):

$i_{M\otimes \overline{M}}(\partial c\otimes \overline{n})\sim j(c)-j( \overline{n}\cdot c)$,

(ii):

$i_{M\otimes \overline{M}}(m\otimes \partial c)\sim j(m\cdot c)-j(c)$

for $c\in C, m\in M$ and $\overline{n}\in \overline{M}$.

The homomorphisms $L\rightarrow M$ and $L\rightarrow \overline{M}$ are given by the homomorphisms $\lambda :M\otimes \overline{M}\rightarrow M$, $(m\otimes \overline{n})\mapsto m\overline{n}-\partial (\overline{n})m$ and $\lambda ^{\prime }:M\otimes \overline{M}\rightarrow \overline{M}$, $(m\otimes \overline{n})\mapsto m\overline{n}-\partial (m)\overline{n}$ and $\partial :C\rightarrow M\cap \overline{M}$. The h-map is given by $h(m, \overline{n})=i(m\otimes \overline{n})$.

Ellis [14] gave a construction of a free quadratic module of groups from a free crossed square. Now, we adapt the Ellis’s construction of a free quadratic module of groups to the algebra context:

Suppose that

is a free crossed square of algebras. We denote by h(M, M, M) the ideal of L generated by elements of the form $h(m,\overline{\langle m^{\prime }, m^{\prime \prime }\rangle })$ and $h(\langle m, m^{\prime }\rangle , \overline{m^{\prime \prime }})$ for $m,m^{\prime },m^{\prime \prime }\in M$. (We are using the notation $\overline{m}=(-m,\delta m)$ for $m\in M$.) Consider the quotient algebra L / h(M, M, M) and the quotient morphism $q_2: L\rightarrow L/h(M,M,M)$.

Let $\langle M, M, M \rangle$ be the ideal of M generated by elements of the form $\langle \langle m,m^{\prime }\rangle , m^{\prime \prime }\rangle$ or $\langle m,\langle m^{\prime },m^{\prime \prime }\rangle \rangle$ for $m,m^{\prime },m^{\prime \prime }\in M$. Consider the quotient algebra $M/\langle M,M,M\rangle$ and the quotient morphism $q_1: M\rightarrow M/\langle M,M,M\rangle$.

From the h-map axioms, we have $\lambda h(M,M,M)=\langle M,M,M\rangle$ and then the map $\overline{\lambda }: L/h(M,M,M)\rightarrow M/\langle M,M,M\rangle$ given by $\overline{\lambda } (l+h(M,M,M))=\lambda (l)+\langle M,M,M\rangle$ is a well-defined homomorphism. Similarly, since $\partial :M\rightarrow R$ is a pre-crossed module, we have $\partial (\langle M,M,M\rangle )=0$ and then the map $\partial ^{\prime }:M/\langle M,M,M\rangle \rightarrow R$ given by $\partial ^{\prime }(m+\langle M,M,M\rangle )=\partial (m)$ is well defined. Let $M^{\prime }=M/\langle M,M,M\rangle$ and $C=(M^{\prime cr}/((M^{\prime cr})^2$. Then

is a free quadratic module of algebras. The h-map $h:M\times \overline{M} \rightarrow L$ induces the quadratic map $\omega : C\otimes C\rightarrow L/h(M,M,M)$.

5 Free Reduced Quadratic Modules

Definition 5.1

A reduced quadratic module $(\omega ,\delta )$ is a following diagram

of algebras such that the following axioms are satisfied.

• 1. The algebra $C_{1}$ is a nil(2)-algebra and $C=C_{1}/(C_{1})^{2}$ is the singularization of algebra $C_{1}.$ The quotient map $q:C_{1}\rightarrow C$ is given by $x\longrightarrow [x]$.

• 2. For $x,y\in C_{1}$, $\delta \omega ([x]\otimes [y])=xy$.

• 3. For $a\in C_{2},x\in C_{1},$ $\omega ([\delta a]\otimes [x]+[x]\otimes [\delta a])=0$.

• 4. For $a,b\in C_{2}$, $\omega ([\delta a]\otimes [\delta b])=ab.$

We denote the category of reduced quadratic modules by $\mathsf {RQM}$.

5.1 From Crossed Modules to Reduced Quadratic Modules of Algebras

Muro, [20], gave suspension functor using central push-out from crossed modules to reduced quadratic modules over groups and showed that this functor preserves the free crossed modules of groups.

Let I have exactly three elements, $I=\{1,2,3\}$, with quasi-order $1<2$ and $1<3$. A direct system with index set I can be given by the following diagram.

Then for this direct system, the following diagram is a push-out:

where $D=(F_{2}\oplus F_{3})/W,$ $W=\{(\varphi _{2}^{1}(a),-\varphi _{3}^{1}(a)):a\in F_{1}\},$ $f:b\longmapsto (0,b)+W$ and $g:c\longmapsto (c,0)+W$.

Let $\partial :L\rightarrow M$ be a crossed module of algebras. We take $F_{1}=L\otimes M$, $F_{2}=L$ and $F_{3}=\left( M/M^{2}\otimes M/M^{2}\right) /K$. We can define

$$\begin{aligned} \begin{array}{crcl} \varphi _{2}^{1}: &{} F_{1} &{} \longrightarrow &{} F_{2} \\ &{} (l\otimes m) &{} \longmapsto &{} l\cdot m. \end{array} \end{aligned}$$

This morphism satisfies the following: $\varphi _{2}^{1}(id\otimes \partial )(l\otimes l^{\prime })=\varphi _{2}^{1}(l\otimes \partial l^{\prime }) = l\cdot \partial l^{\prime }= ll' =w'([l]\otimes [l'])$ and $\partial \varphi _{2}^{1}(l\otimes m) =\partial (l\cdot m) = (\partial l)m=w^{\prime }(\partial \otimes id)(l\otimes m).$ Thus $\partial \varphi _{2}^{1}=w^{\prime }(\partial \otimes id):L\otimes M\rightarrow M.$ We can define the morphism

$$\begin{aligned} \varphi _{3}^{1}:F_{1}=L\otimes M\rightarrow (M/M^{2}\otimes M/M^{2})/K=F_{3} \end{aligned}$$

by composition of the following maps:

where $q:M\rightarrow M/M^{2}$ is the quotient map and K is the image of

$$\begin{aligned} \partial ^{2}\otimes id+id\otimes \partial ^{2}:L/L^{2}\otimes M/M^{2}\longrightarrow M/M^{2}\otimes M/M^{2}. \end{aligned}$$

That is, $\varphi _{3}^{1}(l\otimes m)=\partial ^{2}q(l)\otimes q(m)+K$. Therefore, we have the following push-out diagram:

where

$$\begin{aligned} L^{\sum }=\frac{L\times (M/M^{2}\otimes M/M^{2})/K}{W}, \end{aligned}$$

and

$$\begin{aligned} W=\{(l\cdot m,\partial ^{2}q(l)\otimes qm+K):l\in L,m\in M\}. \end{aligned}$$

There is a morphism w given by

$$\begin{aligned} \begin{array}{rcc} w:(M/M^{2}\otimes M/M^{2})/K &{} \longrightarrow &{} M^{nil} \\ (qm\otimes qm^{\prime })+K &{} \longmapsto &{} mm^{\prime }. \end{array} \end{aligned}$$

Furthermore, there is also a morphism

given by composition of the following maps:

where $\overline{q}:L\rightarrow L^{nil}$ is the quotient map. Obviously, according to above descriptions, we obtain $w(\partial ^{2}\otimes id)(q\otimes q)=\partial ^{nil}\overline{q}\varphi _{2}^{1}.$ Thus, we have the following diagram.

There is a unique morphism $\delta :L^{\sum }\rightarrow M^{nil}$ satisfying $\delta \omega =w$ and $\delta r=\partial ^{nil}\overline{q}$.

5.2 From Free Crossed Modules to Free Reduced Quadratic Modules

Definition 5.2

Let

be a reduced quadratic module, Y be a set and $\nu :Y\rightarrow L$ be a function and M is free nil(2)-algebra, then this reduced quadratic module is called the totally free reduced quadratic module with basis $\nu :Y\rightarrow L,$ or alternatively on the function $\delta \nu :Y\rightarrow M,$ if for any reduced quadratic module $(L^{\prime },M,\delta ^{\prime },\omega ^{\prime })$ and a function $\nu ^{\prime }:Y\rightarrow L^{\prime }$ such that $\delta ^{\prime }\nu ^{\prime }=\delta \nu ,$ there is a unique morphism $\Phi :L\rightarrow L^{\prime }$ such that $\Phi \nu =\nu ^{\prime }$.

Recall from [4] that given a presentation $P=(R;x_{1},\ldots ,x_{n})$ of an R-algebra B and $\mathsf {E}^{(1)}$ the 1-skeleton of the free simplicial algebra generated by this presentation,

$$\begin{aligned} \delta :NE_{1}^{(1)}/\partial _{2}(NE_{2}^{(1)})\rightarrow NE_{0}^{(1)} \end{aligned}$$

is the free crossed module on $(x_{1},\ldots ,x_{n})\rightarrow R$.

Now we give a totally free reduced quadratic module using the suspension functor. Let R be a free algebra and let Y be a set and $f:Y\rightarrow R$ be a function with codomain R. Let $E=R^{+}[Y]$ be the positively graded part of the polynomial ring on Y so that R acts on E by multiplication. The function f induces a morphism of R-algebras $\theta :R^{+}[Y]\rightarrow R$ given by $\theta (y)=f(y)$. Let $P_{2}$ be Peiffer ideal of $R^{+}[Y]$, then take $C=R^{+}[Y]/P_{2}$. Thus from the 1-skeleton we obtain a totally free crossed module $C\rightarrow R$ (cf. [4]). Then, we have functions: $\varphi :C\otimes R\rightarrow C$ given by $\varphi (y\otimes r)=y\cdot r$ and $\varphi ^{\prime }:C\otimes R\rightarrow (R/R^{2}\otimes R/R^{2})/K$ given by $\varphi ^{\prime }(y\otimes r)=\theta ^{2}q_{1}(y)\otimes q_{2}(r)+K,$ where $q_{1}:C\rightarrow C/C^{2}$ and $q_{2}:R\rightarrow R/R^{2}$ are the quotient maps and K is image of the function

$$\begin{aligned} \theta ^{2}\otimes id+id\otimes \theta ^{2}:C/C^{2}\otimes R/R^{2}\rightarrow R/R^{2}\otimes R/R^{2}. \end{aligned}$$

Thus the diagram

is a push-out, where

$$\begin{aligned} C^{\sum }=\frac{R^{+}[Y]/P_{2}\times (R/R^{2}\otimes R/R^{2})/K}{W} \end{aligned}$$

and

$$\begin{aligned} W=\{(y\cdot r,\theta ^{2}(q_{1}\otimes q_{2})(y,r)):y\in R^{+}[Y]/P_{2},r\in R\}. \end{aligned}$$

Thus, the diagram

is the required totally free reduced quadratic module on the function $f^{nil}:Y\rightarrow R^{nil}$, where $\delta =\theta ^{nil}\overline{q}$ and $\overline{q}:C\rightarrow C^{nil}$ is the quotient map. Of course, this follows immediately from the description of the suspension functor.

6 Free Quadratic Chain Complexes Over Algebras

In this section, we will give a functor from the category of simplicial algebras to that of quadratic chain complexes of algebras. We shall give a totally free quadratic chain complex of algebras from a free simplicial algebra using this functor. First, we give the definition of a quadratic chain complex of algebras analogously to that given by Baues [11] for the group case.

Definition 6.1

A quadratic chain complex of algebras is a diagram of homomorphisms between algebras

in which(i) $C_{0}$ acts on $C_{n}$ for $n\geqslant 1$, the action of $\partial _{1}(C_{1})$ being trivial on $C_{n}$ for $n\geqslant 3$;(ii) each $\partial _{n}$ is a $C_{0}$-module homomorphism and $\partial _{i}\partial _{i+1}=0$ for all $i\geqslant 1$;(iii)

is a quadratic module of algebras.

A quadratic chain map $f:\mathsf {C} \rightarrow \mathsf {C^{\prime }}$ between quadratic chain complexes of algebras is a family of homomorphisms between algebras $(n\geqslant 0)$, $f_n: C_n \rightarrow C^{\prime }_n$ with $f_{n}d_{n+1}=d_{n+1}f_{n+1}$ such that $(f_2, f_1, f_0)$ is a map between quadratic modules of algebras. We will denote the category of quadratic chain complexes of algebras by $\mathsf {Quadchain}$.

A quadratic chain complex of algebras $\mathsf {C}$ will be said to be free if for $n\geqslant 3$, $C_n$ are free algebras and the quadratic module at the base is a free quadratic module of algebras. It will be totally free if in addition the base quadratic module is totally free quadratic module of algebras.

A crossed complex of algebras is a chain complex of algebras

in which

(i):

$\partial _1: \rho _1 \rightarrow \rho _0$ is a crossed module of algebras,

(ii):

for $n>1$, $\rho _n$ is a $\rho _0$-module and $\partial _1 (\rho _1)$ acts trivially, and each $\partial _n$ is a $\rho _0$-module homomorphism,

(iii):

for $n\geqslant 1$ $\partial _{n} \partial _{n+1} =0$.

6.1 From Simplicial Algebras to Quadratic Chain Complexes

For a simplicial algebra $\mathsf {E}$,let

$$\begin{aligned} \mathsf {C}^{(1)}(\mathsf {E})_n=\rho _n=\frac{NE_{n}}{(NE_{n}\cap D_{n})+d_{n+1}(NE_{n+1}\cap D_{n+1})}. \end{aligned}$$

This gives a crossed complex of algebras $\mathsf {C}^{(1)}(\mathsf {E})$ starting from the Moore complex $\mathsf {(NE,\partial _n)}$ of $\mathsf {E}$. The map $\partial _{n}:\mathsf {C}^{(1)}(\mathsf {E})_n \rightarrow \mathsf {C}^{(1)}(\mathsf {E})_{n-1}$ will be that induced by $d_{n}^{n}$. If $\mathsf {E}$ is a free simplicial algebra, then $\mathsf {C}^{(1)}(\mathsf {E})$ is a free crossed complex of algebras.

Now, we give a quadratic chain complex of algebras from the Moore complex of a simplicial algebra. Given a simplicial algebra $\mathsf {E}$ with Moore complex $\mathsf {NE}$, define $C_n= \mathsf {C}^{(2)}(\mathsf {E})_n$ by $C_0=NE_0$, $C_1=NE_1/P_3$, $C_2=(NE_2/\partial _3(NE_3\cap D_3))/P^{\prime }_3$, and for $n\geqslant 3$, $C_n=NE_n/(NE_n \cap D_n)+d_{n+1}(NE_{n+1} \cap D_{n+1})$ with $\partial _n$ induced by the differential of $\mathsf {NE}$ and $C=(C^{cr}_1)/(C^{cr}_1)^2$ and where $P_{3}$ is the ideal of $NE_{1}$ generated by triple brackets $\left\langle x,\left\langle y,z\right\rangle \right\rangle \text { and }\left\langle \left\langle x,y\right\rangle ,z\right\rangle$ and $P_{3}^{\prime }$ is the ideal of $NE_{2}/\partial _{3}(NE_{3}\cap D_3)$ generated by elements of the form:

$$\begin{aligned} s_1\langle x,y\rangle [s_1z-s_0z]+\partial _3(NE_3\cap D_3) \end{aligned}$$

and

$$\begin{aligned} s_1x[s_1\langle y,z \rangle -s_0\langle y,z \rangle ]+\partial _3(NE_3\cap D_3) \end{aligned}$$

for $x,y,z\in NE_{1}$.

Note that this structure for $n=0,1,2$, forms a quadratic module and for $n\geqslant 3$ the algebras $C_n$ are the same as the corresponding crossed complex term $\mathsf {C}^{(1)}(\mathsf {E})_n$ as defined above.

We thus have the following result.

Proposition 6.2

For a simplicial algebra $\mathsf {E}$ with the above structure, $\mathsf {C}= \mathsf {C}^{(2)}(\mathsf {E})$ is a quadratic chain complex of algebras.

Proof

The structure $(C_2, C_1, C_0, \omega , w)$ forms a quadratic module. The only thing remaining is to check that $\partial _2 \partial _3$ is trivial which is straightforward. $\square$

Thus, we can define a functor from the category of simplicial algebras to that of quadratic chain complexes of algebras. We denote it by

$$\begin{aligned} \mathsf {C}^{(2)}:\mathsf {SimpAlg}\rightarrow \mathsf {Quadchain}. \end{aligned}$$

6.2 From Quadratic Chain Complexes to Crossed Complexes of Algebras

Baues [11] defined a functor from quadratic chain complexes to crossed complexes denoting it by $\lambda$. Denoting it $\lambda$ we will give its algebra version. That is, we define a functor from the category of quadratic chain complexes to that of crossed chain complexes of algebras.

This functor carries a quadratic chain complex of algebras $\mathsf {C}$ to a crossed complex of algebras $\lambda (\mathsf {C} )=(\rho _n,d_n)_{n\geqslant 0}$. The extra structures are $\rho _0=\lambda ( \mathsf {C})_0=C_0$, $\rho _1=\lambda (\mathsf {C})_1=C_1/w(C\otimes C)=C_1^{cr}$, $\rho _2=\lambda (\mathsf {C})_2=C_2/(\omega (C\otimes C))$ and $\rho _n=C_n$ for $n\geqslant 3$. Thus we have the following complex of algebras:

in which $d_1:\rho _1\rightarrow C_0$ is a crossed module of algebras since the Peiffer elements are zero in the factor algebra $\rho _1$.

As a corollary, for a simplicial algebra $\mathsf {E}$ there is the following isomorphism

$$\begin{aligned} \lambda \mathsf {C}^{(2)}(\mathsf {E})\cong \mathsf {C}^{(1)}(\mathsf {E}). \end{aligned}$$

Proposition 6.3

If $\mathsf {E}$ is a free simplicial algebra, then $\mathsf {C}^{(2)}(\mathsf {E} )$ is a totally free quadratic chain complex.

Proof

We have already seen in Theorem 4.2 that the base quadratic module of $\mathsf {C}^{(2)}(\mathsf {E})$ is totally free on 2-dimensional construction data. It remains to show that $C_n$ for $n\geqslant 3$ are free on the corresponding data, but here we can use the case of crossed complexes, and it was proved in [4] that these terms in higher dimension are free algebras on the corresponding data. $\square$

It is clearly seen that for a free simplicial algebra $\mathsf {E}$, from the isomorphism $\lambda \mathsf {C}^{(2)}(\mathsf {E})\cong \mathsf {C}^{(1)}(\mathsf {E})$, $\lambda \mathsf {C}^{(2)}(\mathsf {E})$ is a totally free crossed complex of algebras.