tangent bundle of $\mathbb{P}^1$

Question

Let V be a two dimensional vector space over k. Consider the set of isomorphism classes of all rank 1 $k[x]/x^2$-submodules of $V\otimes k[x]/x^2$. Does this set has a variety structure? Is it tangent bundle of $\mathbb{P}^1$?

Answer 1

It may be classical to consider varieties as sets, but I think it is having counterproductive effects on communication in this case. The set you are considering is now known as the set of $k$-points of the variety.

For any commutative ring $k$, the tangent bundle of $\mathbb{P}^1_k$ can be seen as a functor that sends any $k$-algebra $R$ to the set of rank 1 locally free $R[\epsilon]/(\epsilon^2)$ submodules of $k^2 \otimes_k R[\epsilon]/(\epsilon^2)$ that are direct summands. This follows from the Hom-functor description of tangent bundles $$T_{X/k}(R) = \underline{\operatorname{Hom}}(\operatorname{Spec} k[\epsilon]/(\epsilon^2), X)(\operatorname{Spec} R) = X(\operatorname{Spec} R[\epsilon]/(\epsilon^2)),$$ together with a standard functorial description of $\mathbb{P}^1$. In contrast to the wording in your question, the description of $\mathbb{P}^1$ in terms of submodules does not mention isomorphism classes, and does have a "direct summand" requirement (see for example this Math.SE page). There is an alternative functorial description in terms of invertible quotient modules (for example in the stacks project), and that is where you use an equivalence relation.