Can you recover the characteristic of a perfect field from the category of smooth projective geometrically connected varieties over it?



Recover the characteristic of $k$ from the category of $k$-varieties
**Correction**. As correctly noted by Remy van Dobben de Bruyn, there is a mistake in Lemma 4. What follows is a corrected argument, with the original (mistaken) post appended below the corrected argument.


Let $k$ be a perfect field. Denote by $\mathbf{V}$ the category of $k$-schemes that are smooth, projective and geometrically connected.


**Proposition**. A non-final object $Y$ of $\mathbf{V}$ is a curve if and only if the natural action of the symmetric group $\mathfrak{S}\_2$ on $Y\times\_{\text{Spec}\ k} Y$ admits a categorical quotient in $\mathbf{V}$.


**Proof**. For a curve, the geometric quotient is smooth, hence it is an object of $\mathbf{V}$.


Let $Y$ be a non-final object that is not a curve. Denote the dimension by $n\geq 2$. In the category of $k$-schemes, there is a categorical quotient $Y\_2$ (even a geometric quotient) of the action of $\mathfrak{S}\_2$ on $Y\times\_{\text{Spec}\ k}Y$, and it is singular along the image of the diagonal, i.e., the embedding dimension is strictly larger than $2n$.


There is a closed immersion of $Y\_2$ into a projective space, and this is an object of $\mathbf{V}$. Thus, if there is a categorical quotient $Z$, it factors this closed immersion. In particular, the morphism from $Y\_2$ to $Z$ is also a closed immersion of $k$-schemes.


Since the embedding dimension of $Y$ is strictly larger than $2n$, also $Z$ has dimension strictly larger than $2n$. Thus, the closed immersion is not surjective on closed points. For any closed point of $Z$ that is not in the image of $Y\_2$, consider the blowing up $\widetilde{Z}$ of the categorical quotient at that closed point. Since the field is perfect, $\widetilde{Z}$ is still a smooth $k$-scheme (this can fail for the blowing up at a closed point with inseparable residue field).


The closed immersion from $Y\_2$ to $Z$ factors through the morphism $\widetilde{Z}\to Z$. Of course the identity map on $Z$ does not factor through $\widetilde{Z}$. This contradicts that $Z$ is a categorical quotient. **QED**


Now we repeat the last part of the argument from the original post.


**Lemma**. A $k$-curve $Y$ in $\mathbf{V}$ is isomorphic to $\mathbb{P}^1\_k$ if and only if every $k$-curve in $\mathbf{V}$ admits a nonconstant morphism to $Y$.


**Proof**. This follows from the fact that $\mathbb{P}^1\_k$ admits no nonconstant morphism to a curve of positive genus, and every $k$-curve of genus $0$ with a $k$-point is isomorphic to $\mathbb{P}^1\_k$. **QED**


**Corollary**. The field $k$ is uniquely determined by the category $\mathbf{V}$.


**Proof**. The proposition and the lemma together establish that there is a characterization of the object $\mathbb{P}^1\_k$ in $\mathbf{V}$ in purely categorical terms. The automorphism group of $(\mathbb{P}^1\_k,0,\infty)$ is $k^\times$. The automorphism group of $(\mathbb{P}^1,\infty)$ is a semidirect product of $k^\times$ and the additive group $k$. With the structure of both the multiplicative group of $k$ and the additive group of $k$, we can recover the field $k$. **QED**


**Original post.** There is a mistake in Lemma 4, as pointed out by Remy van Dobben de Bruyn.


I am writing an answer to address the case of a field $k$ that is not necessarily algebraically closed. Denote by $\mathbf{V}$ the category of $k$-schemes that are smooth, projective and geometrically connected.


**Lemma 1**. The $k$-scheme $\text{Spec}\ k$ is a final object in $\mathbf{V}$.


**Proof**. In fact this is a final object in the larger category of all $k$-schemes. **QED**


**Definition 2**. A morphism in $\mathbf{V}$ is **constant** if it factors through a morphism from the domain to a final object.


**Definition 3**. For an object $Z$ of $\mathbf{V}$, an ordered pair $(f,g)$ of nonconstant morphisms, $f:X\to Z$ and $g:Y\to Z$, is **cofinite** if every ordered pair $(u,v)$ of morphisms $u:W\to X$, $v:W\to Y$ with $f\circ u$ equal to $g\circ v$ is a pair of constant morphisms.


**Lemma 4**. A non-final object of $\mathbf{V}$ is a $k$-curve if and only if there exists no cofinite pair of nonconstant morphisms to the object.


**Proof**. This is a straightforward application of Bertini-type theorems. **QED**


**Lemma 5**. A $k$-curve $Y$ in $\mathbf{V}$ is isomorphic to $\mathbb{P}^1\_k$ if and only if every $k$-curve in $\mathbf{V}$ admits a nonconstant morphism to $Y$.


**Proof**. This follows from the fact that $\mathbb{P}^1\_k$ admits no nonconstant morphism to a curve of positive genus. **QED**


**Proposition 6**. The field $k$ is uniquely determined by the category $\mathbf{V}$.


**Proof**. The lemmas establish that there is a characterization of the object $\mathbb{P}^1\_k$ in $\mathbf{V}$ in purely categorical terms. The automorphism group of $(\mathbb{P}^1\_k,0,\infty)$ is $k^\times$. The automorphism group of $(\mathbb{P}^1,\infty)$ is a semidirect product of $k^\times$ and the additive group $k$. With the structure of both the multiplicative group of $k$ and the additive group of $k$, we can recover the field $k$. **QED**


