For a scheme $X$ over $\text{Spec}(K)$, we can consider maps $\text{Sch}(\text{Spec}(K[d] / d^2), X)$, which we can think of as the tangent bundle over $X$. A map $\text{Spec}(K) \rightarrow S$ picks out a point, and a lift of this map along $\text{Spec}(K) \rightarrow \text{Spec}(K[d]/d^2)$ to a map $\text{Spec}(K[d]/d^2) \rightarrow X$ gives an element of the tangent space of $X$ at that point.
On the algebraic side, for a $K$-algebra $A$ with map $A \rightarrow K$, maps $A \rightarrow K[d] / d^2$ lifting $A \rightarrow K$ along $K[d] / d^2 \rightarrow K$ are in correspondence with the $K$-vector space $\text{Der}_K (A ,K)$.
Now consider how a chain complex $C_*$ of $K$-modules can be thought of as a $K[d]/d^2$-module: we form $\bigoplus_{i \in \mathbb{Z}} C_i$, a $K$-module, which gets an action from $d$. If we view this as a graded $K$-module, then the action of $d$ descends a degree in the grading.
I wonder whether these two considerations are related. Could someone explain the relationship here? Perhaps it is just coincidence.
