The derived category of the dual numbers?

Question

Let $k$ be a field, and let $R = k[x]/x^2$ be the ring of dual numbers over $k$. Let $\mathcal D(R)$ be the derived category of $R$. I'm interested in getting a complete understanding of the structure of $\mathcal D(R)$ as a triangulated category. I'm guessing this appears in the exercises of some standard text, but I don't know where exactly. Where can I learn about the category $\mathcal D(R)$?

Answer 1

Here are a few things I've learned about the ring $k[x]/x^2$, which I'll denote $k[\epsilon]$, and its derived category $\mathcal D(k[\epsilon])$. I'll use cohomological indexing.

1. $k[\epsilon]$ is self-injective. So regardless of whether we take projective or injective resolutions, we find that every chain complex is quasi-isomorphic to one which is levelwise a direct sum of $k[\epsilon]$'s. Call such a chain complex levelwise free.

2. Every levelwise-free chain complex $C_\bullet$ splits as the direct sum $C_\bullet = A_\bullet \oplus B_\bullet$ where $A_\bullet$ is acyclic and $B_\bullet$ has the property that every differential is divisible by $\epsilon$. The proof is an exercise in carefully picking bases in each degree. Call such a complex $B_\bullet$ good (i.e. $B_\bullet$ is "good" if it is levelwise free and each differential is divisible by $\epsilon$). Thus every chain complex is quasi-isomorphic to a good chain complex.

3. If $V$ is a vector space over $k$, then write $V[\epsilon] = V \otimes_k k[\epsilon]$. Let $C_\bullet$ be a good chain complex. Then we may write $C_n = V_n[\epsilon]$ for some vector space $V_n$ over $k$. Moreover, because the differential $\partial_n: V_n[\epsilon] \to V_{n+1}[\epsilon]$ is divisible by $\epsilon$, we have $\partial_n = \epsilon \varphi_n$ for some $k$-linear map $\varphi_n: V_n \to V_{n+1}$, in the sense that the equation $\partial_n(v + \epsilon w) = \epsilon \varphi_n(v)$ for all $v+\epsilon w \in V_n[\epsilon]$. The map $\varphi_n$ may be chosen arbitrarily, since $(\epsilon \varphi_n) (\epsilon \varphi_{n-1}) = 0$ for any $k$-linear maps $\varphi_{n-1},\varphi_n$.

   Thus, modulo the choice of identification $C_n \cong V_n[\epsilon]$, the data of a good chain complex $C_\bullet$ over $k[\epsilon]$ is equivalent to the data $(V_\bullet, \varphi_\bullet)$ which we will call an $A_{\mathbb Z}$-representation which may be described in any of the following equivalent ways:

   • $(V_n)_{n \in \mathbb Z}$ is a $\mathbb Z$-indexed sequence of vector spaces over $k$ and $(\phi_n: V_n \to V_{n+1})_{n \in \mathbb Z}$ is an arbitrary sequence of $k$-linear maps.

   • $(V_n)_{n \in \mathbb Z}$ is a $\mathbb Z$-graded vector space over $k$ and $\varphi_\bullet: V_\bullet \to \Sigma V_\bullet$ is a graded $k$-linear map.

   • $(\cdots \xrightarrow{\varphi_{n-1}} V_n \xrightarrow{\varphi_n} V_{n+1} \xrightarrow{\varphi_{n+1}} \cdots)$ is a functor $\mathbb Z \to Vect_k$ where $\mathbb Z$ is $\mathbb Z$ considered as a linearly ordered set.

   • $(\cdots \xrightarrow{\varphi_{n-1}} V_n \xrightarrow{\varphi_n} V_{n+1} \xrightarrow{\varphi_{n+1}} \cdots)$ is a representation of the quiver $A_{\mathbb Z}$, which has one vertex for each $n \in \mathbb Z$ and one edge $n \to n+1$ for each $n \in \mathbb Z$.

   There is a natural notion of morphism of $A_{\mathbb Z}$-representations $f_\bullet: (V_\bullet, \varphi_\bullet) \to (W_\bullet, \psi_\bullet)$, which can be inferred from any of the above descriptions. It comprises $k$-linear maps $f_n: V_n \to W_n$ such that $f_{n+1} \varphi_{n} = \psi_{n} f_{n}$.

4. Let $C_\bullet, D_\bullet$ be good chain complexes represented by $A_{\mathbb Z}$-representations $(V_\bullet, \varphi_\bullet), (W_\bullet, \psi_\bullet)$. A map of chain complexes comprises $k[\epsilon]$-linear maps $f_n = g_n + \epsilon h_n : V_n[\epsilon] \to W_n[\epsilon]$ (where $f_n, g_n: V_n \to W_n$ are $k$-linear maps) such that $f_\bullet: V_\bullet \to W_\bullet$ is a map of $A_{\mathbb Z}$-representations, but $g_\bullet : V_\bullet \to W_\bullet$ is just a (degree-0) graded $k$-linear map. Composition is the obvious thing, so the category of $A_{\mathbb Z}$-representations is both a subcategory and a quotient category of the category of good chain complexes and chain maps. In fact, this exibits the latter category as a sort of semidirect extension of the former.

5. Finally, $\mathcal D(k[\epsilon])$ is equivalent to the category of good chain complexes and chain maps modulo chain homotopy equivalence. Let $C_\bullet,D_\bullet$ be good chain complexes, identified with $A_{\mathbb Z}$-represntations $(V_\bullet, \varphi_\bullet), (W_\bullet, \psi_\bullet)$. A chain map $f_\bullet + \epsilon g_\bullet: C_\bullet \to D_\bullet$ is nullhomotopic if and only if $f_\bullet = 0$ and there exist $k$-linear maps $h_n: V_n \to W_{n-1}$ such that $g_n = h_{n+1} \varphi_{n} + \psi_{n-1} h_n$. Thus the category of $A_{\mathbb Z}$-representations is both a subcategory and a quotient category of $\mathcal D(k[\epsilon])$. Again, the latter is a semidirect extension of the former.

   In particular $C_\bullet$ and $D_\bullet$ are quasi-isomorphic if and only if $(V_\bullet,\varphi_\bullet)$ and $(W_\bullet, \psi_\bullet)$ are isomorphic $A_{\mathbb Z}$-representations. However, I think an object $C_\bullet$ typically has more automorphisms as an object of $\mathcal D(k[\epsilon])$ than just as an $A_{\mathbb Z}$-representation.