# Questions tagged [characteristic-p]

Fields of characteristic $p$, i.e., fields for which there is a prime $p$ such that $px=0$ for each $x$. Do not use this tag for questions on characteristic polynomials of a matrix.

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### What direction does the derivation of an inseparable algebraic variable point in?

I've been thinking about the geometry of inseparable field extensions lately, since I'm studying smoothness in commutative rings in an advanced topics course this semester. I've generally come to the ...
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### Degree-2 étale covers of curves in characteristic 2 vs torsion points on the Jacobian

It can be found, at Hartshorne exercise 4.2.7 for example, that in the case where $\operatorname{char}(k) \neq 2$ we have a nice correspondence between etale degree 2 covers of a curve $C$ and 2-...
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### Finite, normal subgroups of reductive groups in positive characteristic

Consider the following statement about a connected, reductive group $G$ over a field $k$: Every finite, normal subgroup $N$ of $G$ is central. In characteristic $0$, this is true, and the proof is ...
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### Frobenius endomorphism is not flat

I am actually going through "Twenty four hours of local cohomology". They discuss the Frobenius endomorphism in Chapter 21. Here is the Exercise 21.6 which I am finding hard to solve: Find a ...
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### How big are small inverse powers of 2 mod powers of 3?

Let $a \bmod b$ take value as an integer in $[0, b)$. For any $T \ge 1$, for what $R \in [0, 3^n)$ is $$\min \{2^{-t}\bmod 3^n: t =1, \dotsc, T\} := \min A_T > R?$$ When $T$ is fixed as $n$ ...
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### Lifting of families of curves to characteristic 0

Let $k$ be a finite field, $X_0$ be a smooth affine variety over $k$ and $C\rightarrow X$ a smooth projective family of curves of genus $\geq 2$. By a result of Elkik we can always lift $X_0$ to a ...
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