nLab differential graded algebra

Definition

Abstract definition

A dg-algebra, or differential graded algebra, is equivalently

1. An associative algebra $A$ which is in addition graded algebra and a differential algebra in a compatible way (with the differential derivation being of degree $\pm$ 1);

2. a monoid in the symmetric monoidal category of (possibly unbounded) chain complexes or cochain complexes with its standard structure of a monoidal category by the tensor product of chain complexes;

For the case of chain complexes we also speak of chain algebras.

For the case of cochain complexes we also speak of cochain algebras.

Recall

• that the standard tensor product on (co)chain complexes is given by

  $$(A\otimes B)_n = \sum_{i+j=n} A_i\otimes B_j,$$

  with the differential $d_{A\otimes B} = d_A\otimes B + A\otimes d_B$.

• that a monoid in a monoidal category $C$ is an object $A$ in $C$ together with a morphism

  $$\cdot : A \otimes A \to A$$

  that is unital and associative in the obvious sense.

This implies that a dg-algebra is, more concretely, a graded algebra $A$ equipped with a linear map $d : A \to A$ with the property that

• $d\circ d = 0$

• for $a \in A$ homogeneous of degree $k$, the element $d a$ is of degree $k+1$ (for monoids in cochain complexes) or of degree $k-1$ (for monoids in chain complexes)

• for all $a,b \in A$ with $a$ homogeneous of degree k the graded Leibniz rule holds

  $d (a \cdot b) = (d a) \cdot b + (-1)^k a \cdot (d b)$.

The dg-algebras form a category, dgAlg.

Detailed component definition

Pre-graded algebras

A pre-graded algebra- (pre-ga) or $\mathbb{Z}$-graded algebra is a pre-gvs, $A$, together with an algebra multiplication satisfying $A_p.A_q \subseteq A_{p+q}$ for any $p,q$. The relevant morphisms are pre-gvs morphisms which respect the multiplication. This gives a category $pre GA$.

An augmentation of a pre-ga, $A$, is a homomorphism $\varepsilon : A \to k$. The augmentation ideal of $(A,\varepsilon)$ is $ker \varepsilon$ and will also be denoted $\bar{A}$. The pair $(A,\varepsilon)$ is called an augmented pre-ga.

A morphism $f:(A,\varepsilon)\to (A',\varepsilon')$ of augmented pre-gas is a homomorphism $f : A \to A'$ (thus of degree zero) such that $\varepsilon = \varepsilon ' f$. The resulting category will be written $pre \varepsilon GA$.

Tensor product

If $A$, $A'$ are two pre-gas, then $A\otimes A'$ is a pre-ga with

$$(a\otimes a')(b\otimes b') = (-1)^{|a' | |b|} a b \otimes a' b'$$

for homogeneous $a,b \in A$, $a', b' \in A'$.

If $\varepsilon, \varepsilon$ are augmentations of $A$ and $A'$ respectively, then $\varepsilon\otimes \varepsilon'$ is an augmentation of $A\otimes A'$.

Derivations

Let $A$ be a pre-ga. An (algebra) derivation of degree $p\in \mathbb{Z}$ is a linear map $\theta \in Hom_p(A,A)$ such that

$$\theta(ab) = \theta(a)b + (-1)^{p|a|}a\theta(b)$$

for homogeneous $a,b \in A$.

A derivation $\theta$ of an augmented algebra, $(A, \varepsilon)$, is an algebra derivation which, in addition, satisfies $\varepsilon \theta = 0$.

Let $Der_p(A)$ be the vector space of derivations of degree $p$ of $A$, then $Der(A) = \bigoplus_p Der_p(A)$ is a pre-gvs.

N.B.

In the case of upper gradings, an element of $Der_p(A)$ sends $A^n$ into $A^{n-p}$.

Pre-DGAs

A differential $\partial$ on an (augmented) pre-ga is a derivation of the (augmented) algebra of degree -1 such that $\partial\circ\partial = 0$.

The pair $(A,\partial)$ is called a pre-differential graded algebra (pre-dga). If $A$ is augmented, then $(A,\partial)$ will be called an augmented pre-dga $pre \varepsilon dga$.

If $(A,\partial)$ and $(A',\partial')$ are pre-dgas, then $(A,\partial)\otimes (A',\partial')$, with the conventions already noted, is one as well.

A morphism of pre-dgas (or pre-$\varepsilon$-dgas) is a morphism which is both of pre-gdvs and of pre-gas (with $\varepsilon$ as well if used). This gives categories $pre DGA$ and $pre \varepsilon DGA$.

Commutative graded algebras (CGA)

A pre-ga $A$ is said to be graded commutative if $ab = (-1)^{|a||b|}ba$ for each pair, $a, b$, of elements of $A$ of homogeneous degree.

Commutativity is preserved by tensor product.

We get obvious full subcategories $pre CDGA$ and $pre \varepsilon CDGA$ corresponding to the case with differentials.

CDGAs

A cdga is a negatively graded pre-cdga (in upper grading), $A= \bigoplus_{p\geq 0} A^p.$

There is an augmented variant, of course. These definitions give categories $CDGA$, etc.

See at differential graded-commutative algebra.

$n$-connectivity

An $\varepsilon$cdga $(A,d)$ is $n$-connected (resp. cohomologically $n$-connected if $\bar{A}^p = 0$ for $p\leq n$, (resp. $\overline{H(A,d)}^p = 0$ for $p\leq n$). This gives subcategories $CDGA^n$ and $CDGA^{c n}$.

Filtrations

A filtration on a pre-ga, $A$, is a filtration on $A$, so that $F_p A \subseteq F_{p+1}A$, $F_p A.F_n A \subseteq F_{p+n}A$ (and, if $A$ is differential, also $\partial F_p A \subseteq F_p A$).

Example: Word length filtration.

Let $A$ be an augmented pre-ga and denote by

$$\bar{\mu}^p : \bigotimes^{p+1}\bar{A} \to \bar{A},$$

the iterated multiplication. The decreasing word length filtration, $F^p A$ is given by:

$$F^0 A = A, \quad F^p A = Im\bar{\mu}^{(p-1)} if p\geq 1.$$

$Q(A) = \bar{A}/Im\bar{\mu}$ is the space of indecomposables of A.

If $(A,\partial)$ is an augmented pre-dga, $F^p A$ is stable under $\partial$ and we get $Q(\partial)$ is a differential on $Q(A)$ and hence we get a functor $Q: pre \varepsilon DGA\to pre DGVS.$

Free GAs: $T(V)$, the tensor algebra

Given a pre-gvs, $V$, the tensor algebra generated by $V$ is given by $T(V) = \bigotimes_{n\geq 0}V^{\otimes n}$.

The augmentation sends $V$ to 0. $V^{\otimes n}$ is given the tensor product grading, and the multiplication is given by the tensor product.

Lemma (classical: freeness of $T(V)$, $T$ is a left adjoint)

If $A$ is a pre-ga and $f: V\to A$, a morphism to the underlying pre-gvs of $A$, there is a unique extension $\hat{f} :T(V)\to A$, which is a morphism of pre-gvs.

Free CGAs: $\bigwedge V$

This is the tensor product of the exterior algebra on the odd elements and the symmetric algebra on the even ones:

$$\bigwedge V = E(\bigoplus V_{2p+1})\otimes S(\bigoplus V_{2p}).$$

It satisfies $\bigwedge(V \oplus W) \cong (\bigwedge V)\oplus (\bigwedge W)$.

If $A$ is a pre-cga, any morphism, $f : V\to A$, to the underlying pre-gvs of $A$, has a unique extension to a pre-cga morphism $\bar{f} :\bigwedge V \to A$.

If $(e_\alpha)_{\alpha \in I}$ is a homogeneous basis for $V$, $\bigwedge V$ and $T(V)$ may be written $\bigwedge((e_\alpha)_{\alpha \in I})$ and $T((e_\alpha)_{\alpha \in I})$ respectively.

Note:

• $T(V)$ is a non-commutative polynomial algebra,

• $\bigwedge V$ is a commutative polynomial algebra.

Word length filtrations on $\bigwedge V$ and $T(V)$.

On $\bigwedge V$ (resp. $T(V)$) write

$$\bigwedge V = \bigoplus_{k\geq 0}\bigwedge^k V,$$

where $\bigwedge^k V$ is the subspace generated by all $v_1\wedge \ldots \wedge v_k$ with $v_1 \in V$. Then $F^p \bigwedge V = \bigwedge^{\geq p} V = \bigoplus_{k\geq p}\bigwedge^k V$, resp. $T^k (V) = V^{\otimes k}$ and $F^p T(V) = T^{\geq p}(V) = \bigoplus_{k\geq p} T^k (V)$).

If $(\bigwedge V,d)$ is a pre-cdga, which is free as a pre-cga on a fixed $V$, then $d$ is the sum of derivations $d_k$ defined by the condition $d_k (V) \subseteq \bigwedge^k V$. There is an isomorphism between $V$ and $Q(\bigwedge V)$, which identifies $d_1$ with $Q(d)$. The derivation $d_1$ (resp. $d_2$) is called the linear part (resp. quadratic part) of $d$.

Sum and Product of CDGAs.

If $(A,d)$ and $(A',d')$ are two cdgas, their (categorical) sum (i.e. coproduct) is their tensor product, $(A,d)\otimes(A',d' )$, whilst their product is the ‘direct sum’, $(A,d)\oplus (A',d' )$.

Koszul convention

Given a permutation $\sigma$ of a graded object $(x_1, \ldots, x_p)$, the Koszul sign, $\varepsilon(\sigma)$ is defined by

$$x_1\wedge \ldots \wedge x_p = \varepsilon(\sigma)x_{\sigma(1)} \wedge \ldots \wedge x_{\sigma(p)}$$

in $\bigwedge(x_1, \ldots, x_p )$. We note that although we write $\varepsilon(\sigma)$, $\sigma$ does not suffice to define it as it depends also on the degrees of the various $x_i$.

Terminology

Baues (in his book on Algebraic Homotopy) has suggested using the terminology chain algebra for positively graded differential algebras and cochain algebras for the negatively graded ones. This seems to be a very useful convention.

Enrichment of dg-algebras over dg-coalgebras

Given a dg-coalgebra $C$ and a dg-algebra $B$, the vector space of linear maps $C\to B$ admits a dg-algebra structure using the coproduct on $C$ and product on $B$. This dg-algebra is known as the convolution algebra of $C$ and $B$ and is denoted by $[C,B]$.

The resulting functor

$$[-,-]: dgCoalg^op \times dgAlg \to dgAlg$$

is a part of an adjunction in two variables?, whose other adjoints are

$$\triangleright: dgCoalg\times dgAlg\to dgAlg$$

(the Sweedler product) and

$$\{-,-\}:dgAlg^op\times dgAlg\to dgCoalg$$

(the Sweedler hom).

Taken together, these three functors turn the symmetric monoidal category of dg-algebras into an enriched category over the closed symmetric monoidal category of dg-coalgebras.

See Anel and Joyal for more information.

Related concepts

• differential algebra

Model category structure

There is a standard model category structure on $dgAlg$.See model structure on dg-algebras.

Cosimplicial algebras

The monoidal Dold-Kan correspondence effectively identifies non-negatively graded chain complex algebras with simplicial algebras, and non-negatively graded cochain complex algebras with cosimplicial algebras.

Since cosimplicial algebras have a fundamental interpretation dual to ∞-space, as described at ∞-quantity, this can be understood as explaining the great role differential graded algebras are playing in various context, suchh as notably in

• rational homotopy theory.

dg-coalgebra

Dually, a comonoid in chain complexes is a dg-coalgebra.

Homological smoothness

A dga $A$ is homologically smooth if as a dg-bimodule $_A A_A$ over itself it has a bounded resolution by finitely generated projective dg-bimodules.

Formal dg-algebra

A dg-algebra $A$ is a formal dg-algebra if there exists a morphism

$$A \to H^\bullet(A)$$

to its chain (co)homology (regarded as a dg-algebra with trivial differential) that is a quasi-isomorphism.

Curved dg-algebra

• curved dg-algebra

References

• Mathieu Anel, André Joyal, Sweedler theory of (co)algebras and the bar-cobar constructions, pdf

Twisted tensor products of smooth dg algebras:

• Dmitri Orlov. Smooth DG algebras and twisted tensor product, Russian Math. Surveys, 78 (2023), 5, 853–880. (arXiv:2305.19799).

• Dmitri Orlov. Twisted tensor product, smooth DG algebras, and noncommutative resolutions of singular curves (2024). (arXiv:2405.04738).