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110.69 Two differential graded categories

In this section we construct two differential graded categories satisfying axioms (A), (B), and (C) as in Differential Graded Algebra, Situation 22.27.2 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 22.26.6.

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.

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