Suppose I have a projective, flat morphism $ pi: X to S $ between smooth projective varieties over $ mathbb {C} $, Through the work of Simpson exists a module space $ M = M (X / S, v) $ of torsion – free (semi – stable) coherent slices deposited on the fibers of $ pi $ with fixed discrete data $ v $, So the sheaves are on twist $ X $provided $ S $ is not a point, but torsion free on the fibers.

I would like to assume that this module space is actually *fine*So there is a universal sheaf $ mathcal {F} $ on $ X times_ {S} M $ that is torsion free. One of the most important features is that for everyone $ m in M $. $ F_ {m} = mathcal {F} | _ {X times_ {S} {m}} $ is the sheaf that represents the point $ m in M $, Basically, I try to understand $ mathcal {F} $ locally:

**(1)** On my first question, I assume $ P = (x, m) in X times_ {S} M $ is such that $ m $ represents a locally free sheaf on the fibers of $ pi $, Will the stalk $ mathcal {F} _ {P} $ be a free one $ mathcal {O} _ {X times_ {S} M, P} $-Module? In words, should universal handle stems be free modules over the local rings at points that represent locally free sheaves?

**(2)** I think my second question is closely related. Due to the universal property, I think it is right for $ P = (x, m) $ We have the isomorphism $ mathcal {F} _ {P} cong (F_ {m}) _ {x} $, But I am confused as to how this isomorphism is given. Suppose the local ring of $ X times_ {S} M $ at the $ P $ is

$$ A = mathbb {C} ((x, y, w, z)) / (xy-wz) $$

from where $ (x, y) $ are coordinates on $ X $ and $ (w, z) $ are coordinates on $ M $, So $ mathcal {F} _ {P} $ should be one $ A $Module while $ (F_ {m}) _ {x} $ should be one $ mathbb {C} ((x, y)) / (xy) $-Module. How is the isomorphism given?

What happens to that $ w, z $ Variables, if this identification is made?