Theorem: Algebraic descent criterion

Let $v:E\to V$ be a surjective linear presentation map, let $Z,Y$ be vector spaces, and let $\lambda :E\otimes Z\to Y$ be linear. There is a unique bilinear map

such that

if and only if

Proof. Necessity follows by evaluating at $v(e)=0$. For sufficiency, define the value using any presentation $e$ of the first argument. Two presentations differ by an element of $\ker v$, so (eq:descentcondition) makes the value independent of the choice. Linearity of $\lambda$ gives bilinearity. Surjectivity of $v$ gives uniqueness.