An improper arithmetically closed Borel subalgebra of P(ω) mod FIN

We show the existence of a subalgebra A⊆P(ω) that satisfies the following three conditions:
• A is Borel (when P(ω) is identified with ω2).
• A is arithmetically closed (i.e., A is closed under the Turing jump, and Turing reducibility).
• The forcing notion (A,⊆) modulo the ideal FIN of finite sets collapses the continuum to ℵ0.