Dice Question Streamline Icon: https://streamlinehq.com

Existence of the pullback map on 1-forms for algebraifold homomorphisms

Establish whether for every algebraifold homomorphism \varphi: A \to B between k-algebraifolds in standard form, there automatically exists an A-module map \Omega_\varphi: \Omega_A \to \varphi^*\Omega_B satisfying \Omega_\varphi(da) = d(\varphi(a)) for all a \in A.

Information Square Streamline Icon: https://streamlinehq.com

Background

An algebraifold homomorphism is defined by requiring the existence of a compatible A-linear pullback map on 1-forms making the diagram with universal derivations commute. In the manifold case, smooth maps always induce such a pullback on 1-forms.

The authors note they have no counterexample and suspect existence may be automatic, but a general proof is lacking because \varphi*\Omega_B need not be finitely generated projective, so their universal property does not immediately apply.

References

In fact, we do not know of any single example of an algebra homomorphism \varphi : A \to B between algebraifolds for which \Omega_A does not exist. Therefore it is conceivable that its existence is automatic. A general proof is not obvious: the universal property of \Omega_A from Theorem 2.7 does not apply because \varphi* \Omega_B need not be fgp.

— Differential geometry and general relativity with algebraifolds (2403.06548 - Fritz, 11 Mar 2024) in Subsection “Algebraifold homomorphisms”, Section “The category of algebraifolds and the problem of products”