In this section we construct two differential graded categories satisfying axioms (A), (B), and (C) as in Differential Graded Algebra, Situation 22.20.2 whose objects do not come with a $\mathbf{Z}$-grading.

Example I. Let $X$ be a topological space. Denote $\underline{\mathbf{Z}}$ the constant sheaf with value $\mathbf{Z}$. Let $A$ be an $\underline{\mathbf{Z}}$-torsor. In this setting we say a sheaf of abelian groups $\mathcal{F}$ is $A$-graded if given a local section $a \in A(U)$ there is a projector $p_ a : \mathcal{F}|_ U \to \mathcal{F}|_ U$ such that whenever we have a local isomorphism $\underline{\mathbf{Z}}|_ U \to A|_ U$ then $\mathcal{F}|_ U = \bigoplus _{n \in \mathbf{Z}} p_ n(\mathcal{F})$. Another way to say this is that locally on $X$ the abelian sheaf $\mathcal{F}$ has a $\mathbf{Z}$-grading, but on overlaps the different choices of gradings differ by a shift in degree given by the transition functions for the torsor $A$. We say that a pair $(\mathcal{F}, \text{d})$ is an $A$-graded complex of abelian sheaves, if $\mathcal{F}$ is an $A$-graded abelian sheaf and $\text{d} : \mathcal{F} \to \mathcal{F}$ is a differential, i.e., $\text{d}^2 = 0$ such that $p_{a + 1} \circ \text{d} = \text{d} \circ p_ a$ for every local section $a$ of $A$. In other words, $\text{d}(p_ a(\mathcal{F}))$ is contained in $p_{a + 1}(\mathcal{F})$.

Next, consider the category $\mathcal{A}$ with

1. objects are $A$-graded complexes of abelian sheaves, and

2. for objects $(\mathcal{F}, \text{d})$, $(\mathcal{G}, \text{d})$ we set

$\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}((\mathcal{F}, \text{d}), (\mathcal{G}, \text{d})) = \bigoplus \mathop{\mathrm{Hom}}\nolimits ^ n(\mathcal{F}, \mathcal{G})$

where $\mathop{\mathrm{Hom}}\nolimits ^ n(\mathcal{F}, \mathcal{G})$ is the group of maps of abelian sheaves $f$ such that $f(p_ a(\mathcal{F})) \subset p_{a + n}(\mathcal{G})$ for all local sections $a$ of $A$. As differential we take $\text{d}(f) = \text{d} \circ f - (-1)^ n f \circ \text{d}$, see Differential Graded Algebra, Example 22.19.6.

We omit the verification that this is indeed a differential graded category satisfying (A), (B), and (C). All the properties may be verified locally on $X$ where one just recovers the differential graded category of complexes of abelian sheaves. Thus we obtain a triangulated category $K(\mathcal{A})$.

Twisted derived category of $X$. Observe that given an object $(\mathcal{F}, \text{d})$ of $\mathcal{A}$, there is a well defined $A$-graded cohomology sheaf $H(\mathcal{F}, \text{d})$. Hence it is clear what is meant by a quasi-isomorphism in $K(\mathcal{A})$. We can invert quasi-isomorphisms to obtain the derived category $D(\mathcal{A})$ of complexes of $A$-graded sheaves. If $A$ is the trivial torsor, then $D(\mathcal{A})$ is equal to $D(X)$, but for nonzero torsors, one obtains a kind of twisted derived category of $X$.

Example II. Let $C$ be a smooth curve over a perfect field $k$ of characteristic $2$. Then $\Omega _{C/k}$ comes endowed with a canonical square root. Namely, we can write $\Omega _{C/k} = \mathcal{L}^{\otimes 2}$ such that for every local function $f$ on $C$ the section $\text{d}(f)$ is equal to $s^{\otimes 2}$ for some local section $s$ of $\mathcal{L}$. The “reason” is that

$\text{d}(a_0 + a_1t + \ldots +a_ dt^ d) = (\sum \nolimits _{i\text{ odd}} a_ i^{1/2} t^{(i - 1)/2})^2\text{d}t$

(insert future reference here). This in particular determines a canonical connection

$\nabla _{can} : \Omega _{C/k} \longrightarrow \Omega _{C/k} \otimes _{\mathcal{O}_ C} \Omega _{C/k}$

whose $2$-curvature is zero (namely, the unique connection such that the squares have derivative equal to zero). Observe that the category of vector bundles with connections is a tensor category, hence we also obtain canonical connections $\nabla _{can}$ on the invertible sheaves $\Omega _{C/k}^{\otimes n}$ for all $n \in \mathbf{Z}$.

Let $\mathcal{A}$ be the category with

1. objects are pairs $(\mathcal{F}, \nabla )$ consisting of a finite locally free sheaf $\mathcal{F}$ endowed with a connection

$\nabla : \mathcal{F} \longrightarrow \mathcal{F} \otimes _{\mathcal{O}_ C} \Omega _{C/k}$

whose $2$-curvature is zero, and

2. morphisms between $(\mathcal{F}, \nabla _\mathcal {F})$ and $(\mathcal{G}, \nabla _\mathcal {G})$ are given by

$\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}((\mathcal{F}, \nabla _\mathcal {F}), (\mathcal{G}, \nabla _\mathcal {G})) = \bigoplus \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ C}(\mathcal{F}, \mathcal{G} \otimes _{\mathcal{O}_ C} \Omega _{C/k}^{\otimes n})$

For an element $f : \mathcal{F} \to \mathcal{G} \otimes \Omega _{C/k}^{\otimes n}$ of degree $n$ we set

$\text{d}(f) = \nabla _{\mathcal{G} \otimes \Omega _{C/k}^{\otimes n}} \circ f + f \circ \nabla _\mathcal {F}$

with suitable identifications.

We omit the verification that this forms a differential graded category with properties (A), (B), (C). Thus we obtain a triangulated homotopy category $K(\mathcal{A})$.

If $C = \mathbf{P}^1_ k$, then $K(\mathcal{A})$ is the zero category. However, if $C$ is a smooth proper curve of genus $> 1$, then $K(\mathcal{A})$ is not zero. Namely, suppose that $\mathcal{N}$ is an invertible sheaf of degree $0 \leq d < g - 1$ with a nonzero section $\sigma$. Then set $(\mathcal{F}, \nabla _\mathcal {F}) = (\mathcal{O}_ C, \text{d})$ and $(\mathcal{G}, \nabla _\mathcal {G}) = (\mathcal{N}^{\otimes 2}, \nabla _{can})$. We see that

$\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}^ n((\mathcal{F}, \nabla _\mathcal {F}), (\mathcal{G}, \nabla _\mathcal {G})) = \left\{ \begin{matrix} 0 & \text{if} & n < 0 \\ \Gamma (C, \mathcal{N}^{\otimes 2}) & \text{if} & n = 0 \\ \Gamma (C, \mathcal{N}^{\otimes 2} \otimes \Omega _{C/k}) & \text{if} & n = 1 \end{matrix} \right.$

The first $0$ because the degree of $\mathcal{N}^{\otimes 2} \otimes \Omega _{C/k}^{\otimes -1}$ is negative by the condition $d < g - 1$. Now, the section $\sigma ^{\otimes 2}$ has derivative equal zero, hence the homomorphism group

$\mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}((\mathcal{F}, \nabla _\mathcal {F}), (\mathcal{G}, \nabla _\mathcal {G}))$

is nonzero.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).