Why Is The Morphism G M → G M \mathbb{G}_m \to \mathbb{G}_m G M ​ → G M ​ Given By T ↦ T 2 T \mapsto T^2 T ↦ T 2 Etale?

Why is the Morphism $\mathbb{G}_m \to \mathbb{G}_m$ Given by $T \mapsto T^2$ Etale?

In the realm of algebraic geometry, the concept of etaleness is crucial in understanding the properties of morphisms between schemes. A morphism is said to be etale if it is flat and unramified. In this article, we will delve into the specifics of the morphism $\mathbb{G}_m \to \mathbb{G}_m$ given by $T \mapsto T^2$ and explore why it is etale.

To begin with, let's establish some notation and background information. We are working over a field $k$ of characteristic not equal to 2. The multiplicative group $\mathbb{G}_m$ is defined as the scheme $\operatorname{Spec} k[T^{\pm 1}]$. This is a scheme that represents the group of non-zero elements of $k$ under multiplication.

The morphism in question is induced by the ring homomorphism $k[T^{\pm 1}] \to k[T^{\pm 1}], T \mapsto T^2$. This homomorphism is clearly a polynomial map, and it induces a morphism of schemes $\mathbb{G}_m \to \mathbb{G}_m$.

To determine whether this morphism is etale, we need to check two conditions: flatness and unramifiedness.

Flatness

A morphism $f: X \to Y$ is said to be flat if the fiber $X_y$ is a flat $k(y)$-module for every point $y \in Y$. In our case, the fiber $\mathbb{G}_{m,y}$ is the scheme $\operatorname{Spec} k(T^{\pm 1}) \otimes_{k[T^{\pm 1}]} k(T^{\pm 1})$. Since $k(T^{\pm 1})$ is a flat $k[T^{\pm 1}]$-module, the fiber $\mathbb{G}_{m,y}$ is also flat.

Unramifiedness

A morphism $f: X \to Y$ is said to be unramified if the local ring $\mathcal{O}_{X,x}$ is a regular local ring for every point $x \in X$. In our case, the local ring $\mathcal{O}_{\mathbb{G}_m,x}$ is isomorphic to $k[T^{\pm 1}]_{(T)}$, which is a regular local ring.

Now that we have established the conditions for etaleness, let's see why the morphism $\mathbb{G}_m \to \mathbb{G}_m$ given by $T \mapsto T^2$ is etale.

The key insight here is that the morphism is induced by a polynomial map, which is a polynomial in one variable. This means that the morphism is a polynomial map of degree 2, and it is therefore etale.

To see why this is the case, let's consider the fiber $\mathbb{G}_{m,y}$ over a point $y \in \mathbb{G}_m$. The fiber is the scheme $\operatorname{Spec} k(T^{\pm 1}) \otimes_{k[T^{\pm 1}]} k(T^{\pm 1})$. Since the morphism is induced by a polynomial map, the fiber is a flat $k(y)$-module.

Moreover, the local ring $\mathcal{O}_{\mathbb{G}_m,x}$ is isomorphic to $k[T^{\pm 1}]_{(T)}$, which is a regular local ring. This means that the morphism is unramified.

In conclusion, the morphism $\mathbb{G}_m \to \mathbb{G}_m$ given by $T \mapsto T^2$ is etale because it is flat and unramified. The key insight here is that the morphism is induced by a polynomial map, which is a polynomial in one variable. This means that the morphism is a polynomial map of degree 2, and it is therefore etale.

• [1] Hartshorne, R. (1977). Algebraic Geometry. Springer-Verlag.
• [2] Liu, K. (2002). Algebraic Geometry and Arithmetic Curves. Oxford University Press.
• [3] Mumford, D. (1995). The Red Book of Varieties and Schemes. Springer-Verlag.

For further reading on algebraic geometry and etaleness, we recommend the following resources:

• [1] Algebraic Geometry by Robin Hartshorne
• [2] Algebraic Geometry and Arithmetic Curves by Qing Liu
• [3] The Red Book of Varieties and Schemes by David Mumford

These resources provide a comprehensive introduction to algebraic geometry and etaleness, and are highly recommended for anyone interested in the subject.
Q&A: Etaleness of the Morphism $\mathbb{G}_m \to \mathbb{G}_m$ Given by $T \mapsto T^2$

In our previous article, we explored the etaleness of the morphism $\mathbb{G}_m \to \mathbb{G}_m$ given by $T \mapsto T^2$. In this article, we will answer some frequently asked questions about this topic.

A: Etaleness is a property of morphisms between schemes that is defined as follows: a morphism $f: X \to Y$ is etale if it is flat and unramified.

A: Flatness and unramifiedness are two separate properties of morphisms between schemes. Flatness means that the fiber $X_y$ is a flat $k(y)$-module for every point $y \in Y$. Unramifiedness means that the local ring $\mathcal{O}_{X,x}$ is a regular local ring for every point $x \in X$.

A: The morphism $\mathbb{G}_m \to \mathbb{G}_m$ given by $T \mapsto T^2$ is etale because it is induced by a polynomial map, which is a polynomial in one variable. This means that the morphism is a polynomial map of degree 2, and it is therefore etale.

A: The characteristic of the field $k$ is not equal to 2. This is important because it ensures that the polynomial map $T \mapsto T^2$ is well-defined.

A: Yes, consider the morphism $\mathbb{G}_m \to \mathbb{G}_m$ given by $T \mapsto T^3$. This morphism is not etale because it is not flat.

A: Etaleness has many applications in algebraic geometry, including the study of algebraic curves and surfaces, the classification of algebraic varieties, and the construction of moduli spaces.

A: There are many resources available for learning about etaleness and algebraic geometry, including textbooks, research papers, and online courses. Some recommended resources include:

• [1] Algebraic Geometry by Robin Hartshorne
• [2] Algebraic Geometry and Arithmetic Curves by Qing Liu
• [3] The Red Book of Varieties and Schemes by David Mumford

In conclusion, etaleness is a fundamental concept in algebraic geometry that has many important applications. We hope that this Q&A article provided a helpful introduction to the topic and has answered some of the most frequently asked questions.

• [1] Hartshorne, R. (1977). Algebraic Geometry. Springer-Verlag.
• [2] Liu, K. (2002). Algebraic Geometry and Arithmetic Curves. Oxford University Press.
• [3] Mumford, D. (1995). The Red Book of Varieties and Schemes. Springer-Verlag.

For further reading on algebraic geometry and etaleness, we recommend the following resources:

• [1] Algebraic Geometry by Robin Hartshorne
• [2] Algebraic Geometry and Arithmetic Curves by Qing Liu
• [3] The Red Book of Varieties and Schemes by David Mumford