# Quiver describing perverse sheaves on $\mathbb C$

I have two sources, which claim that the category of perverse sheaves on $\mathbb C$ constructible with respect to the stratification $0$ and $\mathbb C^*$ is equivalent to the category of certain representations of a quiver. In both cases the quiver consists of two dots with one arrow $u,v$ in each direction between them.

The first source considers such representation such that both 1+uv and 1+vu are invertible.

The second source only wants 1+uv to be invertible.

Now I have to admit that I understand neither of the proofs completely so my question is which description is correct?

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## 1 Answer

Suppose $1+uv$ is invertible, inverse $w$. Let your imagination run wild:

$1/(1+vu)= 1-vu+vuvu - vuvuvu + ... = 1 - v(1-uv + uvuv - ...)u = 1-vwu$

Now you can check $1-vwu$ really is inverse to $1+vu$.

There's a name for this trick; I have forgotten it but someone here will know.

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I don't know the name, but it's discussed @ mathoverflow.net/questions/31595/… – Grigory M Nov 29 '11 at 16:57
Wow thanks very much! – Jan Nov 29 '11 at 17:10
You're welcome, I've known that trick for years but never had a chance to use it. – m_t_ Nov 30 '11 at 10:17
Depending on who you ask, this is the Kaplansky trick, the Halmos trick, or the Jacobson trick. – user641 Jan 2 '12 at 23:14