nLab
differential graded algebra

Context

Differential-graded objects

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Definition

Abstract definition

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

  1. An associative algebra AA 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

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

    with the differential d AB=d AB+Ad Bd_{A\otimes B} = d_A\otimes B + A\otimes d_B.

  • that a monoid in a monoidal category CC is an object AA in CC together with a morphism

    :AAA \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 AA equipped with a linear map d:AAd : A \to A with the property that

  • dd=0d\circ d = 0

  • for aAa \in A homogeneous of degree kk, the element dad a is of degree k+1k+1 (for monoids in cochain complexes) or of degree k1k-1 (for monoids in chain complexes)

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

    d(ab)=(da)b+(1) ka(db)d (a \cdot b) = (d a) \cdot b + (-1)^k a \cdot (d b).

Detailed component definition

Pre-graded algebras

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

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

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

Tensor product

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

(aa)(bb)=(1) |a||b|abab(a\otimes a')(b\otimes b') = (-1)^{|a' | |b|} a b \otimes a' b'

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

If ε,ε\varepsilon, \varepsilon are augmentations of AA and AA' respectively, then εε\varepsilon\otimes \varepsilon' is an augmentation of AAA\otimes A'.

Derivations

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

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

for homogeneous a,bAa,b \in A.

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

Let Der p(A)Der_p(A) be the vector space of derivations of degree pp of AA, then Der(A)= pDer p(A)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)Der_p(A) sends A nA^n into A npA^{n-p}.

Pre-DGAs

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

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

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

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

Commutative graded algebras (CGA)

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

Commutativity is preserved by tensor product.

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

CDGAs

A cdga is a negatively graded pre-cdga (in upper grading), A= p0A p.A= \bigoplus_{p\geq 0} A^p.

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

See at differential graded-commutative algebra.

nn-connectivity

An ε\varepsiloncdga (A,d)(A,d) is nn-connected (resp. cohomologically nn-connected if A¯ p=0\bar{A}^p = 0 for pnp\leq n, (resp. H(A,d)¯ p=0\overline{H(A,d)}^p = 0 for pnp\leq n). This gives subcategories CDGA nCDGA^n and CDGA cnCDGA^{c n}.

Filtrations

A filtration on a pre-ga, AA, is a filtration on AA, so that F pAF p+1AF_p A \subseteq F_{p+1}A, F pA.F nAF p+nAF_p A.F_n A \subseteq F_{p+n}A (and, if AA is differential, also F pAF pA\partial F_p A \subseteq F_p A).

Example: Word length filtration.

Let AA be an augmented pre-ga and denote by

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

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

F 0A=A,F pA=Imμ¯ (p1)ifp1.F^0 A = A, \quad F^p A = Im\bar{\mu}^{(p-1)} if p\geq 1.

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

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

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

Given a pre-gvs, VV, the tensor algebra generated by VV is given by T(V)= n0V nT(V) = \bigotimes_{n\geq 0}V^{\otimes n}.

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

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

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

Free CGAs: V\bigwedge V

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

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

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

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

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

Note:

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

  • V\bigwedge V is a commutative polynomial algebra.

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

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

V= k0 kV,\bigwedge V = \bigoplus_{k\geq 0}\bigwedge^k V,

where kV\bigwedge^k V is the subspace generated by all v 1v kv_1\wedge \ldots \wedge v_k with v 1Vv_1 \in V. Then F pV= pV= kp kVF^p \bigwedge V = \bigwedge^{\geq p} V = \bigoplus_{k\geq p}\bigwedge^k V, resp. T k(V)=V kT^k (V) = V^{\otimes k} and F pT(V)=T p(V)= kpT k(V)F^p T(V) = T^{\geq p}(V) = \bigoplus_{k\geq p} T^k (V)).

If (V,d)(\bigwedge V,d) is a pre-cdga, which is free as a pre-cga on a fixed VV, then dd is the sum of derivations d kd_k defined by the condition d k(V) kVd_k (V) \subseteq \bigwedge^k V. There is an isomorphism between VV and Q(V)Q(\bigwedge V), which identifies d 1d_1 with Q(d)Q(d). The derivation d 1d_1 (resp. d 2d_2) is called the linear part (resp. quadratic part) of dd.

Sum and Product of CDGAs.

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

Koszul convention

Given a permutation σ\sigma of a graded object (x 1,,x p)(x_1, \ldots, x_p), the Koszul sign, ε(σ)\varepsilon(\sigma) is defined by

x 1x p=ε(σ)x σ(1)x σ(p)x_1\wedge \ldots \wedge x_p = \varepsilon(\sigma)x_{\sigma(1)} \wedge \ldots \wedge x_{\sigma(p)}

in (x 1,,x p)\bigwedge(x_1, \ldots, x_p ). We note that although we write ε(σ)\varepsilon(\sigma), σ\sigma does not suffice to define it as it depends also on the degrees of the various x ix_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.

Model category structure

There is a standard model category structure on dgAlgdgAlg.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

dg-coalgebra

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

Homological smoothness

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

Formal dg-algebra

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

AH (A) 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

Revised on December 19, 2016 08:54:07 by Urs Schreiber (188.1.230.74)