  We introduce new mathematical aspects of the Bell states using matrix
factorizations, nonnoetherian singularities, and noncommutative blowups. A
matrix factorization of a polynomial $p$ consists of two matrices
$\phi_1,\phi_2$ such that $\phi_1\phi_2 = \phi_2\phi_1 = p \operatorname{id}$.
Using this notion, we show how the Bell states emerge from the separable
product of two mixtures, by defining pure states over complex matrices rather
than just the complex numbers.
  We then show in an idealized algebraic setting that pure states are supported
on nonnoetherian singularities. Moreover, we find that the collapse of a Bell
state is intimately related to the representation theory of the noncommutative
blowup along its singular support. This presents an exchange in geometry: the
nonlocal commutative spacetime of the entangled state emerges from an
underlying local noncommutative spacetime.
