It is well known that a scheme over a perfect field is smooth at $x$ if and only if it is regular at $x$, and that these two properties are not equivalent over non-perfect fields. What is an example of a finite-type scheme over the algebraic closure of $\mathbb{F}_p(T)$ with a regular, singular point? And is there a simple additional condition that can can be added to regularity to ensure smoothness?
2026-03-25 11:00:50.1774436450
Regularity vs smoothness in positive characteristic
206 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in ALGEBRAIC-GEOMETRY
- How to see line bundle on $\mathbb P^1$ intuitively?
- Jacobson radical = nilradical iff every open set of $\text{Spec}A$ contains a closed point.
- Is $ X \to \mathrm{CH}^i (X) $ covariant or contravariant?
- An irreducible $k$-scheme of finite type is "geometrically equidimensional".
- Global section of line bundle of degree 0
- Is there a variant of the implicit function theorem covering a branch of a curve around a singular point?
- Singular points of a curve
- Find Canonical equation of a Hyperbola
- Picard group of a fibration
- Finding a quartic with some prescribed multiplicities
Related Questions in SCHEMES
- Is $ X \to \mathrm{CH}^i (X) $ covariant or contravariant?
- Do torsion-free $\mathcal{O}_X$-modules on curves have dimension one?
- $\mathbb{C}[x,y]$ is the sections of Spec $\mathbb{C}[x,y]$ minus the origin?
- Finitely generated $k-$algebras of regular functions on an algebraic variety
- Is every open affine subscheme of an algebraic $k-$variety an affine $k-$variety?
- Scheme Theoretic Image (Hartshorne Ex.II.3.11.d)
- Is this a closed embedding of schemes?
- Adjunction isomorphism in algebraic geometry
- Closed connected subset of $\mathbb{P}_k^1$
- Why can't closed subschemes be defined in an easier way?
Related Questions in POSITIVE-CHARACTERISTIC
- Characteristic of an integral domain: doubt in the proof
- Cyclic Artin-Schreier-Witt extension of order $p^2$
- Characteristic of a finite field - equivalent formula
- A field with an irreducible, separable polynomial with roots $\alpha$ and $\alpha + 1$ must have positive characteristic.
- Characteristic of residue field in a Dedekind domain.
- Galois extension of exponent $mp^r$ in characteristic $p$
- Peirce decomposition of a ring: must the ideal generators be idempotent in characteristic 2?
- Indecomposable modules of a $p$-group in characteristic $p$
- Example of a characteristic zero local ring with a quotient of positive characteristic
- "Purely inseparable, algebraic extension if and only if trivial automorphism group"
Related Questions in REGULAR-RINGS
- $k[[x,y]]/I$ is a Gorenstein ring implies that $I$ is generated by 2 elements
- Polynomial rings are regular
- Is a complete intersection ring, which is a quotient of a maximal $A$-sequence, Artinian?
- Projective Nullstellensatz and regular rings
- How to find a regular parameter system of the local ring $ \mathbb{C} [X_0, \dots, X_n]_{(P \ )} $?
- On cancelling $\mathfrak m$-primary ideal of regular local ring $(R,\mathfrak m)$
- reflexive ideal in regular local ring
- Exercise 4.4.1 in Weibel's 'An Introduction to Homological Algebra'.
- Is the trivial ring regular?
- Prove polynomial ring over a discrete valuation ring quotient by powers of maximal ideal is regular?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?