From submodule categories to preprojective algebras

Abstract

Let $S(n)$ be the category of invariant subspaces of nilpotent operators with nilpotency index at most $n$. Such submodule categories have been studied already in 1934 by Birkhoff, they have attracted a lot of attention in recent years, for example in connection with some weighted projective lines (Kussin, Lenzing, Meltzer). On the other hand, we consider the preprojective algebra $\Pi _n$ of type $\mathbb {A}_n$; the preprojective algebras were introduced by Gelfand and Ponomarev, they are now of great interest, for example they form an important tool to study quantum groups (Lusztig) or cluster algebras (Geiss, Leclerc, Schröer). We are going to discuss the connection between the submodule category $\mathcal {S}(n)$ and the module category $\hbox {mod}\;\Pi _{n-1}$ of the preprojective algebra $\Pi _{n-1}$. Dense functors $\mathcal {S}(n) \rightarrow \hbox {mod}\;\Pi _{n-1}$ are known to exist: one has been constructed quite a long time ago by Auslander and Reiten, recently another one by Li and Zhang. We will show that these two functors are full, dense, objective functors with index $2n$, thus $\hbox {mod}\;\Pi _{n-1}$ is obtained from $\mathcal {S}(n)$ by factoring out an ideal which is generated by $2n$ indecomposable objects. As a byproduct we also obtain new examples of ideals in triangulated categories, namely ideals $\mathcal {I}$ in a triangulated category $\mathcal {T}$ which are generated by an idempotent such that the factor category $\mathcal {T}/\mathcal {I}$ is an abelian category.

Appendix: Objective functors

Let $\mathcal {A}, \mathcal {B}$ be additive categories and let $F{:}~\mathcal {A} \rightarrow \mathcal {B}$ be an (additive) functor. An object $A$ in $\mathcal {A}$ will be called a kernel object for $F$ provided $F(A) = 0.$ The functor $F{:}~\mathcal {A} \rightarrow \mathcal {B}$ will be said to be objective provided any morphism $f{:}~ A \rightarrow A'$ in $\mathcal {A}$ with $F(f) = 0$ factors through a kernel object for $F$. If $F$ is an objective functor, then we will say that the kernel of $F$ is generated by $\mathcal {K},$ provided $\mathcal {K}$ is a class of objects in $\mathcal {A}$ such that $\hbox {add}\;\mathcal {K}$ is the class of all kernel objects for $F$.

Given an additive category $\mathcal {A}$ and an ideal $\mathcal {I}$ in $\mathcal {A}$, we denote by $\mathcal {A}/\mathcal {I}$ the corresponding factor category (it has the same objects, and the homomorphisms in $\mathcal {A}/\mathcal {I}$ are the residue classes of the homomorphisms in $\mathcal {A}$ modulo $\mathcal {I}$).

If $\mathcal {K}$ is a class of objects of the category $\mathcal {A}$, we denote by $\langle \mathcal {K}\rangle$ the ideal of $\mathcal {A}$ given by all maps which factor through a direct sum of objects in $\mathcal {K}.$ Instead of writing $\mathcal {A}/\langle \mathcal {K}\rangle$, we just will write $\mathcal {A}/\mathcal {K}.$

If $F{:}~\mathcal {A} \rightarrow \mathcal {B}$ is a full, dense, objective functor and the kernel of $F$ is generated by $\mathcal {K}$, then $F$ induces an equivalence between the category $\mathcal {A}/\mathcal {K}$ and $\mathcal {B}$. We see that given a full, dense, objective functor $F{:}~\mathcal {A} \rightarrow \mathcal {B}$, the category $\mathcal {B}$ is uniquely determined by $\mathcal {A}$ and a class of indecomposable objects in $\mathcal {A}$ (namely the class of indecomposable kernel objects for $F$); if $F$ is objective, but not necessarily full or dense, then $F$ induces an equivalence between the category $\mathcal {A}/\mathcal {K}$ and the image category of $F$.

The composition of objective functors is not necessarily objective. Here is an example: Let $\mathcal {B}$ be the linearization of the chain of cardinality 3, thus $\mathcal {B}$ has three objects $b_1,b_2,b_3$ with $\mathrm{Hom }(b_i,b_j) = k$ provided $i \le j$ and zero otherwise, such that the composition $\mathrm{Hom }(b_2,b_3)\otimes \mathrm{Hom }(b_1,b_2) \rightarrow \mathrm{Hom }(b_1,b_3)$ is the multiplication map. Let $\mathcal {A}$ be the full subcategory of $\mathcal {B}$ with objects $b_1,b_3$, thus $\mathcal {A}$ is the linearization of a chain of cardinality 2. Let $\mathcal {K} = \{b_2\}$ and $\mathcal {C} = \mathcal {B}/\mathcal {K}$. The inclusion functor $F{:}~\mathcal {A} \rightarrow \mathcal {B}$ and the projection functor $G{:}~\mathcal {B} \rightarrow \mathcal {C}$ both are (full and) objective, however the composition $GF{:}~\mathcal {A} \rightarrow \mathcal {B}$ is not objective (none of the objects $b_1,b_3$ belongs to the kernel of $GF$, we have $\mathrm{Hom }_{\mathcal {A}}(b_1,b_3) = k$ and any non-zero map $b_1 \rightarrow b_3$ is mapped to zero under $GF$). Note that the functor $F$ is not dense.

Let $F{:}~\mathcal {A} \rightarrow \mathcal {B}$ and $G{:}~\mathcal {B} \rightarrow \mathcal {C}$ be objective functors. If $F$ is, in addition, full and dense, then $GF$ is objective (thus, the composition of full, dense, objective functors is full, dense, objective). Proof. Let $a{:}~A_1\rightarrow A_2$ be a morphism with $GF(a) = 0.$ Since $G$ is objective, the morphism $F(a)$ factors through a kernel object $B$ for $G$, say $F(a) = b_2b_1$ where $b_1{:}~F(A_1) \rightarrow B$ and $b_2{:}~B \rightarrow F(A_2)$. By assumption, $F$ is dense, thus there is an isomorphism $b{:}~B \rightarrow F(A)$ for some object $A$ in $\mathcal {A}$. Since $F$ is full, there is a map $a_1{:}~A_1 \rightarrow A$ such that $F(a_1) = bb_1$ and a map $a_2{:}~A \rightarrow A_2$ such that $F(a_2) = b_2b^{-1}$. We have $F(a) = b_2b^{-1}bb_1 = F(a_2)F(a_1) = F(a_2a_1),$ thus $F(a-a_2a_1) = 0.$ Since $F$ is objective, there is a kernel object $A'$ for $F$ such that $a-a_2a_1$ factors through $A'$, say $a-a_2a_1 = a_4a_3$, with $a_3{:}~A_1\rightarrow A', a_4{:}~A' \rightarrow A_2.$ It follows that $a = a_2a_1+a_4a_3 = \begin{bmatrix} a_2&\;a_4\end{bmatrix} \begin{bmatrix} a_1 \\ a_3\end{bmatrix}$, thus this map factors through $A\oplus A'$. But $GF(A\oplus A') = GF(A)\oplus GF(A').$ Now, $F(A)$ is isomorphic to $B$, thus $GF(A)$ is isomorphic to $G(B) = 0$. Also, $F(A') = 0,$ thus $GF(A') = 0$. This shows that $A\oplus A'$ is a kernel object for $GF.$

We recall that an additive category $\mathcal {A}$ is said to be a Krull-Remak-Schmidt category, provided every object in $\mathcal {A}$ is a (finite) direct sum of objects with local endomorphism rings. Assume now that $F{:}~ \mathcal {A} \rightarrow \mathcal {B}$ is an objective functor between Krull-Remak-Schmidt categories $\mathcal {A}$ and $\mathcal {B}$. Then we are interested in the number $i_0(F)$ of isomorphism classes of indecomposable objects in $\mathcal {F}$ which are kernel objects for $F$, as well as in the number $i_1(F)$ of isomorphism classes of indecomposable objects $B$ in $\mathcal {B}$ such that $B$ is not isomorphic to an object of the form $F(A)$ where $A$ is an object in $\mathcal {A}$. If at least one of the numbers $i_0(F), i_1(F)$ is finite, we call $i(F) = i_0(F)-i_1(F)$ the index of $F$.

The objective functors $F$ considered in the paper are also dense, in this case $i(F) = i_0(F)$ is the number of isomorphism classes of indecomposable kernel objects in $\mathcal {A}$.