nLab differential graded vector space

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 $(V,\partial)$, where $V$ is a (pre-)graded vector space and $\partial \in Hom_{-1}(V,V)$ satisfies $\partial\circ\partial = 0$.

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 $\partial : V\to s(V)$. 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, $Ker \partial / Im \partial$ is called the homology of $(V,\partial)$, denoted $H(V,\partial)$.

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

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

is a pre-dgvs with differential

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

for $f$ homogeneous.

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

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

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

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

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

Weak equivalences

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

Homotopies

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

Contractible and Acyclic DGVSs

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

Remarks

(i) A contracting homotopy $h : Id_V \sim O_V$ is a degree 1 map such that $\partial h(x) + h(\partial x) = x$ for all $x$ in $V_p$.

(ii) If $(V,\partial)$ 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, $k$, then the correct result would be ‘’If $(V,\partial)$ 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 $(V,\partial)$ is a pre-dgvs, then its $r$-suspension $(s^rV,\partial)$, is the $r$-suspension, $s^rV$, of $V$ together with the differential

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

Tensor product of (pre-) dgvs

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

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

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

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

Duals of (pre-)dgvs

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

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

Chains and cochains: terminology

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

If $(V,\partial)$ is presented with the ‘upper grading’, so $V^p$, 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.