I just was thinking about mobius strip, and asked myself simple questions, which I could not convince myself of answers and hope someone can give my a concrete answer.
I know Mobius strip is a line bundle over the circle as real manifolds, but how this transfer to algebraic geometry.
Q1- What algebro-geometric properties of analogous Mobius strip in algebraic geometry? i.e. is it a variety, or some sort of a general scheme?
in case the above answer to Q1 is yes please see Q2 , otherwise thanks.
2-how it's constructed in this respect? i,e. if it's any of the above, any informain about its structure sheaf if it's a scheme, or coordinate ring in case it's a variety?
UPDATE: After the discussion bellow with kenny Wong, I realized that the rightwo question is this: 1- Can we have a mobuis strip analogy as a complex manifold, I mean will it look the same band that twisted around the edges and glued?
I appreciate your help here, thank you.
Yes, the Möbius strip is a perfectly valid algebraic line bundle bundle on the real circle , which is the perfectly valid real algebraic variety $$S=\operatorname {Proj}\mathbb R[X,Y]=\mathbb P^1_{\mathbb R}$$ We can look at the circle also as the subscheme $S^1=V(X^2+Y^2-1)\subset \mathbb A^2_\mathbb R$, i.e. $$S^1=\operatorname {Spec}\frac {\mathbb R[X,Y]}{\langle X^2+Y^2-1\rangle}=:\operatorname {Spec}\mathbb R[x,y]$$ Hence the real circle is both an affine and a projective variety, which is of course only possible because $\mathbb R$ is not algebraically closed!
Over $S$
The Möbius bundle is the total space of the tautological line bundle $\mathcal O_{\mathbb P^1_{\mathbb R}}(-1)$.
Over $S^1$
In the second incarnation $S^1$ of the circle, the Möbius bundle is the line bundle associated to the ideal $\langle y,x-1\rangle\subset \mathbb R[x,y]$, which is a non free projective module of rank one over the ring $\mathbb R[x,y]$.
In other words the Möbius bundle is the line bundle associated to the divisor $1.P$ of the circle, where $P$ is the closed point with coordinates $(1,0)$.