Is there a standard reference for the fact that, in an appropriate algebraic-geometrical context, the tangent space at the point $[E]$ to the moduli space $\mathcal M$ is something like $\operatorname{Ext}^1(E, E)$? I am primary interested in two situations:


* $\mathcal M$ is the moduli space of semi-stable coherent torsion-free sheaves on $\mathbb CP^2$ with given Hilbert polynomial,
* or $\mathcal M$ is the moduli space of semi-stable finite-dimensional complex representations of finite acyclic quiver with given dimension vector,


and in both situation, if the coefficients of the Hilbert polynomial/dimension vector are coprime, there is a smooth projective variety that is a fine moduli space $\mathcal M$.


Unfortunately, I do not know any deformation theory, so a study book on it that also proves something like $T\_{[E]} \mathcal M=\operatorname{Ext}^1(E, E)$ would be welcome.



Reference request: tangent space to moduli space of coherent sheaves is $\operatorname{Ext}^1(E, E)$
**Theorem 2.6** (page 9) from [Hartshorne *Lectures on Deformation Theory*](https://math.berkeley.edu/~robin/math274root.pdf) (it seems that Hartshorne uses the same notation both for an affine scheme $D$ and its function algebra):


Let $X$ be a scheme over $k$, and let $\mathcal F$ be a coherent sheaf on $X$. We deﬁne a deformation of $\mathcal F$ over $D=\operatorname{Spec} (k[t]/t^2)$ to be a coherent sheaf $\mathcal F\_0$ on $X\_0 = X \times D$, ﬂat over $D$, together with a homomorphism $\mathcal F\_0 \to \mathcal F$ such that the induced map $\mathcal F\_0 \otimes\_{k[t]/t^2} k \to \mathcal F$ is an isomorphism. Then the deformations of $\mathcal F$ over $D$ are in natural one-to-one correspondence with the elements of $Ext^1 (\mathcal F, \mathcal F)$, with the zero-element corresponding to the trivial deformation.


Now by the universal property of the moduli space $\mathcal M$ of coherent torsion-free sheaves on $\mathbb{CP}^2$ its tangent space at $[\mathcal F]$ consists of the deformations of $\mathcal F$ over $D$. Moreover, I believe that the proof should work as well over "a non-commutative affine scheme" whose function algebra is the path algebra of a finite acyclic quiver.


