nLab
differential graded vector space

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

Differential (pre-)graded vector spaces. (We continue from graded vector spaces with the terminology of pre-gvs for the general case and gvs for the positively or negatively graded ones.)

A differential (pre-)graded vector space, (dgvs), is a pair (V,)(V,\partial), where VV is a (pre-)graded vector space and Hom 1(V,V)\partial \in Hom_{-1}(V,V) satisfies =0\partial\circ\partial = 0.

This endomorphism, \partial, of degree -1 is called the differential or sometimes the boundary operator of the dgvs.

To link it with differential object, note that using the suspension (as in graded vector space) as the translation (see category with translation) and then the differential is :Vs(V)\partial : V\to s(V). A dgvs is essentially the same as a chain complex of vector spaces. (Some questions of terminology are addressed further down this entry.)

As usual, Ker/ImKer \partial / Im \partial is called the homology of (V,)(V,\partial), denoted H(V,)H(V,\partial).

Let (V,)(V,\partial), (V,)(V',\partial') be two pre-dgvs

Hom(V,V)= pHom p(V,V)Hom(V,V') = \bigoplus_{p\in \mathbb{Z}}Hom_p(V,V')

is a pre-dgvs with differential

Df=f(1) |f|fDf = \partial'\circ f - (-1)^{|f|}f\circ\partial

for ff homogeneous.

A degree rr linear morphism ff is compatible with the differentials if it is a cycle for this differential DD, i.e., Df=0Df = 0 or f=(1) rf.\partial' f = (-1)^{r}f\partial.

A morphism between pre-dgvs is a linear morphism of degree 0 that is compatible with the differentials:

f:(V,)(V,).f: (V,\partial)\to (V',\partial').

This induces H(f):H(V,)H(V,).H(f): H(V,\partial)\to H(V',\partial').

We get a category preDGVSpre - DGVS and HH is a functor H:preDGVSpreGVSH : pre\! -\! DGVS\to pre\!-\! GVS.

Weak equivalences

If f:(V,)(V,)f: (V,\partial)\to (V',\partial') in {\sf pre-DGVS}, then ff is a weak equivalence or quasi-isomorphism if H(f)H(f) is an isomorphism. In this case we write f:(V,)(V,)f: (V,\partial)\stackrel{\simeq}{\to} (V',\partial')

Homotopies

Let f,f:(V,)(V,)f,f' : (V,\partial)\to (V',\partial') be two morphisms in preDGVSpre-DGVS. We say ff and ff' are homotopic denoted fff\sim f' if fff-f' is a boundary in (Hom(V,V),D)(Hom(V,V'),D), i.e., there is some h:VVh : V \to V' of degree +1 such that ff=Dh=h+hf-f' = Dh = \partial' h + h\partial.

Contractible and Acyclic DGVSs

A dgvs (V,)(V,\partial) is contractible if the identity map on VV is homotopic to the zero morphism and is acyclic if H(V,)=0H(V,\partial) = 0.

Remarks

(i) A contracting homotopy h:Id VO Vh : Id_V \sim O_V is a degree 1 map such that h(x)+h(x)=x\partial h(x) + h(\partial x) = x for all xx in V pV_p.

(ii) If (V,)(V,\partial) is contractible, then it is acyclic and conversely. The converse depends strongly on our working with vector spaces. If we worked with modules over a commutative ring, kk, then the correct result would be ‘’If (V,)(V,\partial) is acyclic and projective in each dimension, then it is contractible.’‘ This is important when looking at, for instance, diagrams of dgvs since even if each individual object in the diagram may be contractible, it might be impossible to pick the contracting homotopies to give a map of diagrams, i.e., to be compatible with the structural maps.

Suspension for (pre-) dgvs

If (V,)(V,\partial) is a pre-dgvs, then its rr-suspension (s rV,)(s^rV,\partial), is the rr-suspension, s rVs^rV, of VV together with the differential

s rv=(1) rsr(v).\partial s^rv = (-1)^rs^r(\partial v).

Tensor product of (pre-) dgvs

If (V,)(V,\partial), and (V,)(V',\partial') are pre-dgvs, then we give the tensor product, VVV\otimes V', the differential given on generators by

(vv)=(v)v+(1) |v|v(v), \partial(v\otimes v') = (\partial v) \otimes v' + (-1)^{|v|}v \otimes (\partial' v' ),

and we denote the result by (V,)(V,)(V,\partial)\otimes (V', \partial'). We have (Kunneth theorem)

H((V,)(V,))H(V,)H(V,).H((V,\partial)\otimes (V', \partial')) \cong H(V,\partial)\otimes H(V', \partial').

Duals of (pre-)dgvs

The dual #(V,)=(#V,#)\#(V,\partial) = (\# V,\#\partial) of a pre-dgvs (V,)(V,\partial) is given by #V\# V with differential (#)= t(\# \partial ) = - ^t \partial. This satisfies

(#)f;v+(1) |f|f;v=0.\langle(\# \partial)f; v\rangle + (-1)^{|f|}\langle f; \partial v\rangle = 0.

Chains and cochains: terminology

If (V,)(V,\partial) is a pre-dgvs with ‘lower grading’ that is the summands are written V pV_p, then (V,)(V,\partial) may be called a chain complex and terms such as cycle, boundary, homology are used with the usual meanings.

If (V,)(V,\partial) is presented with the ‘upper grading’, so V pV^p, then the corresponding words will have a ‘co’ as prefix, cochain complex, cocycle, etc.

There is no real distinction between the two cases in the abstract, but in applications there is often a fixed ‘dimensional’ interpretation and then the ‘natural’ and ‘geometric’ aspects determine which is more appropriate or useful.

Revised on January 15, 2011 08:03:51 by Toby Bartels (98.19.56.183)