Definition Of Semilinear Functor On Representation Category And Descent

Mar 13, 2025 by ADMIN 72 views

**Definition of Semilinear Functor on Representation Category and Descent**

Introduction

In the realm of algebraic geometry and representation theory, the concept of semilinear functors plays a crucial role in understanding the behavior of representations of affine group schemes over finite Galois extensions. In this article, we will delve into the definition of semilinear functors on representation categories and explore their connection to descent theory.

Background and Notations

Let $G$ be an affine group scheme defined over a field $k$ of characteristic $0$ . We assume that $G$ is a smooth, affine group scheme, and we denote its coordinate ring by $k[G]$ . Let $K/k$ be a finite Galois extension with Galois group $\Gamma$ . We denote the residue field of $K$ by $\overline{k}$ and the absolute Galois group of $K$ by $\Gamma_{\text{abs}}$ .

Definition of Semilinear Functor

A semilinear functor on the representation category of $G$ over $K$ is a functor $F: \text{Rep}_G(K) \to \text{Rep}_G(K)$ that satisfies the following properties:

Semilinearity: For every $\sigma \in \Gamma$ , there exists an automorphism $\phi_\sigma$ of $F$ such that for every representation $V$ of $G$ over $K$ , we have $F(\sigma \cdot V) = \phi_\sigma(F(V))$ .
Functoriality: For every morphism $f: V \to W$ in $\text{Rep}_G(K)$ , we have $F(f) = \phi_\sigma(f)$ for every $\sigma \in \Gamma$ .

Representation Category and Descent

The representation category of $G$ over $K$ is the category of finite-dimensional representations of $G$ over $K$ . A representation $V$ of $G$ over $K$ is a finite-dimensional vector space over $K$ equipped with a linear action of $G$ .

The descent theory of representations of $G$ over $K$ is a fundamental concept in the study of semilinear functors. It provides a way to construct representations of $G$ over $K$ from representations of $G$ over $k$ .

Descent Data

Let $V$ be a representation of $G$ over $k$ . A descent datum for $V$ is a collection of representations $V_\sigma$ of $G$ over $K$ for every $\sigma \in \Gamma$ , together with a collection of isomorphisms $\alpha_\sigma: V_\sigma \otimes_K K^\sigma \to V$ for every $\sigma \in \Gamma$ , satisfying certain compatibility conditions.

Semilinear Functor and Descent

A semilinear functor $F$ on the representation category of $G$ over $K$ induces a descent datum on every representation $V$ of $G$ over $k$ . This descent datum is defined as follows:

For every $\sigma \in \Gamma$ , we set $V_\sigma = F(V)^\sigma$ .
For every $\sigma \in \Gamma$ , we define the isomorphism $\alpha_\sigma: V_\sigma \otimes_K K^\sigma \to V$ as the composition of the isomorphism $F(V)^\sigma \otimes_K K^\sigma \to F(V)$ with the isomorphism $F(V) \to V$ .

Properties of Semilinear Functor

A semilinear functor $F$ on the representation category of $G$ over $K$ satisfies the following properties:

Semilinearity: For every $\sigma \in \Gamma$ , there exists an automorphism $\phi_\sigma$ of $F$ such that for every representation $V$ of $G$ over $K$ , we have $F(\sigma \cdot V) = \phi_\sigma(F(V))$ .
Functoriality: For every morphism $f: V \to W$ in $\text{Rep}_G(K)$ , we have $F(f) = \phi_\sigma(f)$ for every $\sigma \in \Gamma$ .

Examples of Semilinear Functor

There are several examples of semilinear functors on the representation category of $G$ over $K$ :

Torsion functor: For every representation $V$ of $G$ over $K$ , the torsion functor $T(V)$ is defined as the subspace of $V$ consisting of elements annihilated by some power of the Frobenius endomorphism.
Inertia functor: For every representation $V$ of $G$ over $K$ , the inertia functor $I(V)$ is defined as the subspace of $V$ consisting of elements fixed by the inertia group.

Conclusion

In this article, we have introduced the concept of semilinear functors on the representation category of $G$ over $K$ and explored their connection to descent theory. We have also discussed several properties and examples of semilinear functors. The study of semilinear functors is an active area of research, and we hope that this article will provide a useful introduction to this topic.

References

[1] Deligne, P. (1977). Cohomologie étale. Lecture Notes in Mathematics, 569.
[2] Grothendieck, A. (1968). Séminaire de géométrie algébrique. Lecture Notes in Mathematics, 151.
[3] SGA 4. (1964). Théorie des topos et cohomologie étale des schémas. Lecture Notes in Mathematics, 269.

Appendix

In this appendix, we provide a brief overview of the necessary background material on affine group schemes, Galois extensions, and representation categories.

Affine Group Schemes

An affine group scheme $G$ over a field $k$ is a smooth, affine group scheme defined by a finitely generated $k$ -algebra $k[G]$ . The coordinate ring of $G$ is denoted by $k[G]$ .

Galois Extensions

A Galois extension $K/k$ is a finite extension of fields with Galois group $\Gamma$ . The residue field of $K$ is denoted by $\overline{k}$ .

Representation Categories

Q: What is a semilinear functor on the representation category of G over K?

A: A semilinear functor on the representation category of G over K is a functor F: Rep_G(K) → Rep_G(K) that satisfies the following properties:

Semilinearity: For every σ ∈ Γ, there exists an automorphism φ_σ of F such that for every representation V of G over K, we have F(σ ⋅ V) = φ_σ(F(V)).
Functoriality: For every morphism f: V → W in Rep_G(K), we have F(f) = φ_σ(f) for every σ ∈ Γ.

Q: What is the connection between semilinear functors and descent theory?

A: The descent theory of representations of G over K is a fundamental concept in the study of semilinear functors. It provides a way to construct representations of G over K from representations of G over k.

Q: What is a descent datum for a representation V of G over k?

A: A descent datum for a representation V of G over k is a collection of representations V_σ of G over K for every σ ∈ Γ, together with a collection of isomorphisms α_σ: V_σ ⊗_K K^σ → V for every σ ∈ Γ, satisfying certain compatibility conditions.

Q: How does a semilinear functor induce a descent datum on a representation V of G over k?

A: A semilinear functor F on the representation category of G over K induces a descent datum on every representation V of G over k. This descent datum is defined as follows:

For every σ ∈ Γ, we set V_σ = F(V)^σ.
For every σ ∈ Γ, we define the isomorphism α_σ: V_σ ⊗_K K^σ → V as the composition of the isomorphism F(V)^σ ⊗_K K^σ → F(V) with the isomorphism F(V) → V.

Q: What are some examples of semilinear functors on the representation category of G over K?

A: There are several examples of semilinear functors on the representation category of G over K:

Torsion functor: For every representation V of G over K, the torsion functor T(V) is defined as the subspace of V consisting of elements annihilated by some power of the Frobenius endomorphism.
Inertia functor: For every representation V of G over K, the inertia functor I(V) is defined as the subspace of V consisting of elements fixed by the inertia group.

Q: What are some applications of semilinear functors in algebraic geometry and representation theory?

A: Semilinear functors have several applications in algebraic geometry and representation theory, including:

The study of Galois representations and their properties.
The study of the cohomology of algebraic groups and their representations.
The study of the geometry of algebraic stacks and their representations.

Q: What are some open problems in the study of semilinear functors?

A: There are several open problems in the study of semilinear functors, including:

The study of the properties of semilinear functors on the representation category of G over K.
The study of the connection between semilinear functors and descent theory.
The study of the applications of semilinear functors in algebraic geometry and representation theory.

Q: What are some resources for learning more about semilinear functors?

A: There are several resources for learning more about semilinear functors, including:

The book "Cohomologie étale" by Pierre Deligne.
The book "Séminaire de géométrie algébrique" by Alexander Grothendieck.
The online notes "Semilinear functors and descent" by [Author].

Q: What are some future directions for research on semilinear functors?

A: There are several future directions for research on semilinear functors, including:

The study of the properties of semilinear functors on the representation category of G over K.
The study of the connection between semilinear functors and descent theory.
The study of the applications of semilinear functors in algebraic geometry and representation theory.

Conclusion

In this Q&A article, we have discussed the definition and properties of semilinear functors on the representation category of G over K, as well as their connection to descent theory. We have also discussed several examples and applications of semilinear functors, and have outlined some open problems and future directions for research in this area.