a noncommutative topology 2

A qurve is an affine algebra such that $~\Omega^1~A$ is a projective $~A~$-bimodule. Alternatively, it is an affine algebra allowing lifts of algebra morphisms through nilpotent ideals and as such it is the ‘right’ noncommutative generalization of Grothendieck’s smoothness criterium. Examples of qurves include : semi-simple algebras, coordinate rings of affine smooth curves, hereditary orders over curves, group algebras of virtually free groups, path algebras of quivers etc.

Hence, qurves behave a lot like curves and as such one might hope to obtain one day a ‘birational’ classification of them, if we only knew what we mean by this. Whereas the etale classification of them is understood (see for example One quiver to rule them all or Qurves and quivers ) we don’t know what the Zariski topology of a qurve might be.

Usually, one assigns to a qurve $~A~$ the Abelian category of all its finite dimensional representations $\wis{rep}~A$ and we would like to equip this set with a topology of sorts. Because $~A~$ is a qurve, its scheme of n-dimensional representations $\wis{rep}n~A$ is a smooth affine variety for each n, so clearly $\wis{rep}~A$ being the disjoint union of these acquires a trivial but nice commutative topology. However, we would like open sets to hit several of the components $\wis{rep}n~A$ thereby ‘connecting’ them to form a noncommutative topological space associated to $~A~$.

In a noncommutative topology on rep A I proposed a way to do this and though the main idea remains a good one, I’ll ammend the construction next time. Whereas we don’t know of a topology on the whole of $\wis{rep}~A$, there is an obvious ordinary topology on the subset $\wis{simp}~A$ of all simple finite dimensional representations, namely the induced topology of the Zariski topology on $~\wis{spec}~A$, the prime spectrum of twosided prime ideals of $~A~$. As in commutative algebraic geometry the closed subsets of the prime spectrum consist of all prime ideals containing a given twosided ideal. A typical open subset of the induced topology on $\wis{simp}~A$ hits many of the components $\wis{rep}n~A$, but how can we extend it to a topology on the whole of the category $\wis{rep}~A$ ?

Every finite dimensional representation has (usually several) Jordan-Holder filtrations with simple successive quotients, so a natural idea is to use these filtrations to extend the topology on the simples to all representations by restricting the top (or bottom) of the Jordan-Holder sequence. Let W be the set of all words w such as $U1U2 \hdots Uk$ where each $Ui$ is an open subset of $\wis{simp}~A$. We can now define the left basic open set $\mathcal{O}w^l$ consisting of all finite dimensional representations M having a Jordan-Holder sequence such that the i-th simple factor (counted from the bottom) belongs to $Ui$. (Similarly, we can define a right basic open set by counting from the top or a symmetric basic open set by merely requiring that the simples appear in order in the sequence). One final technical (but important) detail is that we should really consider equivalence classes of left basic opens. If w and w’ are two words we will denote by $\wis{rep}(w \cup w’)$ the set of all finite dimensional representations having a Jordan-Holder filtration with enough simple factors to have one for each letter in w and w’. We then define $\mathcal{O}^lw \equiv \mathcal{O}^l{w’}$ iff $\mathcal{O}^lw \cap \wis{rep}(w \cup w’) = \mathcal{O}^l{w’} \cap \wis{rep}(w \cup w’)$. Equivalence classes of these left basic opens form a partially ordered set (induced by set-theoretic inclusion) with a unique minimal element 0 (the empty set corresponding to the empty word) and a uunique maximal element 1 (the set $\wis{rep}~A$ corresponding to the letter $w=\wis{simp}~A$). Set-theoretic union induces an operation $\vee$ and the operation $~\wedge$ is induced by concatenation of words, that is, $\mathcal{O}^lw \wedge \mathcal{O}^l{w’} \equiv \mathcal{O}^l{ww’}$. This then defines a left noncommutative topology on $\wis{rep}~A$ in the sense of Van Oystaeyen (see part 1 ). To be precise, it satisfies the axioms in the left and middle column of the following picture and similarly, the right basic opens give a right noncommutative topology (satisfying the axioms of the middle and right columns) whereas the symmetric opens satisfy all axioms giving the basis of a noncommutative topology. Even for very simple finite dimensional qurves such as $\begin{bmatrix} \mathbb{C} & \mathbb{C} \ 0 & \mathbb{C} \end{bmatrix}$ this defines a properly noncommutative topology on the Abelian category of all finite dimensional representations which obviously respect isomorphisms so is really a noncommutative topology on the orbits. Still, while this may give a satisfactory local definition, in gluing qurves together one would like to relax simple representations to Schurian representations. This can be done but one has to replace the topology coming from the Zariski topology on the prime spectrum by the partial ordering on the bricks of the qurve, but that will have to wait until next time…

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