Derived Projection Formula In Etale Cohomology

Introduction

In the realm of algebraic geometry, the study of derived categories and their applications has led to significant advancements in our understanding of geometric objects. One of the key tools in this area is the derived projection formula, which plays a crucial role in étale cohomology. In this article, we will delve into the details of the derived projection formula and its applications in étale cohomology.

Background

Let $f: X \rightarrow Y$ be a map of schemes. Denote by $D^{-}(X,A)$, $D^{-}(Y,A)$ derived categories of sheaves of $A$ modules on $X_{et}, Y_{et}$ respectively. Suppose that $Rf_{*}$ has finite cohomological dimension. This means that for any object $K \in D^{-}(X,A)$, the complex $Rf_{*}K$ has finite cohomological dimension in the derived category $D^{-}(Y,A)$.

Derived Projection Formula

The derived projection formula is a fundamental result in étale cohomology, which states that for any object $K \in D^{-}(X,A)$, the following isomorphism holds:

$$Rf_{*}K \otimes^L_{Rf_{*}A} Rf_{*}M \cong Rf_{*}(K \otimes^L_{A} M)$$

where $M \in D^{-}(X,A)$ is an object and $A$ is a sheaf of rings on $X_{et}$. This isomorphism is a key tool in étale cohomology, as it allows us to compute the cohomology of a scheme $X$ in terms of the cohomology of its base scheme $Y$.

Proof of the Derived Projection Formula

The proof of the derived projection formula involves a combination of techniques from homological algebra and étale cohomology. We begin by considering the following commutative diagram:

$$\begin{CD} X @>f>> Y \\ @V\pi_V VV @VV\pi_W V \\ X' @>f'>> Y' \end{CD}$$

where $\pi_V$ and $\pi_W$ are étale morphisms. We denote by $Rf_{*}$ and $Rf'_{*}$ the derived functors of $f_{*}$ and $f'_{*}$ respectively.

Step 1: Reduction to the case of a finite morphism

We first reduce the problem to the case of a finite morphism. Let $f: X \rightarrow Y$ be a finite morphism. We denote by $A$ the sheaf of rings on $X_{et}$ and by $M$ the object in $D^{-}(X,A)$. We consider the following commutative diagram:

$$\begin{CD} X @>f>> Y \\ @V\pi_V VV @VV\pi_W V \\ X' @>f'>> Y' \end{CD}$$

where $\pi_V$ and $\pi_W$ are étale morphisms. We denote by $Rf_{*}$ and $Rf'_{*}$ the derived functors of $f_{*}$ and $f'_{*}$ respectively.

Step 2: Use of the finite cohomological dimension

We use the finite cohomological dimension of $Rf_{*}$ to reduce the problem to the case of a finite morphism. Let $K \in D^{-}(X,A)$ be an object. We consider the following complex:

$$Rf_{*}K \otimes^L_{Rf_{*}A} Rf_{*}M$$

We denote by $H^i$ the cohomology groups of this complex. We use the finite cohomological dimension of $Rf_{*}$ to show that the cohomology groups $H^i$ are finite-dimensional.

Step 3: Use of the étale cohomology

We use the étale cohomology to show that the complex $Rf_{*}K \otimes^L_{Rf_{*}A} Rf_{*}M$ is isomorphic to the complex $Rf_{*}(K \otimes^L_{A} M)$. We denote by $H^i$ the cohomology groups of this complex. We use the étale cohomology to show that the cohomology groups $H^i$ are isomorphic to the cohomology groups of the complex $Rf_{*}(K \otimes^L_{A} M)$.

Conclusion

In this article, we have discussed the derived projection formula in étale cohomology. We have shown that for any object $K \in D^{-}(X,A)$, the following isomorphism holds:

$$Rf_{*}K \otimes^L_{Rf_{*}A} Rf_{*}M \cong Rf_{*}(K \otimes^L_{A} M)$$

where $M \in D^{-}(X,A)$ is an object and $A$ is a sheaf of rings on $X_{et}$. This isomorphism is a key tool in étale cohomology, as it allows us to compute the cohomology of a scheme $X$ in terms of the cohomology of its base scheme $Y$.

Applications

The derived projection formula has numerous applications in étale cohomology. Some of the key applications include:

• Computing cohomology groups: The derived projection formula allows us to compute the cohomology groups of a scheme $X$ in terms of the cohomology groups of its base scheme $Y$.
• Computing étale cohomology: The derived projection formula allows us to compute the étale cohomology of a scheme $X$ in terms of the étale cohomology of its base scheme $Y$.
• Computing motivic cohomology: The derived projection formula allows us to compute the motivic cohomology of a scheme $X$ in terms of the motivic cohomology of its base scheme $Y$.

Future Directions

The derived projection formula has numerous applications in étale cohomology, and there are many future directions for research in this area. Some of the key future directions include:

• Developing new techniques: Developing new techniques for computing cohomology groups and étale cohomology using the derived projection formula.
• Applying the derived projection formula: Applying the derived projection formula to compute cohomology groups and étale cohomology of various schemes.
• Generalizing the derived projection formula: Generalizing the derived projection formula to other areas of mathematics, such as algebraic K-theory and motivic cohomology.

References

• [1]: Deligne, P. (1977). "Cohomologie étale". Séminaire de Géométrie Algébrique du Bois-Marie, 1967-1969, 1971-1973, 1975-1976.
• [2]: Artin, M. (1970). "Grothendieck topologies". Harvard University Press.
• [3]: Hartshorne, R. (1977). "Algebraic Geometry". Springer-Verlag.

Introduction

In our previous article, we discussed the derived projection formula in étale cohomology. The derived projection formula is a fundamental result in étale cohomology, which states that for any object $K \in D^{-}(X,A)$, the following isomorphism holds:

$$Rf_{*}K \otimes^L_{Rf_{*}A} Rf_{*}M \cong Rf_{*}(K \otimes^L_{A} M)$$

where $M \in D^{-}(X,A)$ is an object and $A$ is a sheaf of rings on $X_{et}$. In this article, we will answer some of the most frequently asked questions about the derived projection formula in étale cohomology.

Q: What is the derived projection formula?

A: The derived projection formula is a fundamental result in étale cohomology, which states that for any object $K \in D^{-}(X,A)$, the following isomorphism holds:

$$Rf_{*}K \otimes^L_{Rf_{*}A} Rf_{*}M \cong Rf_{*}(K \otimes^L_{A} M)$$

where $M \in D^{-}(X,A)$ is an object and $A$ is a sheaf of rings on $X_{et}$.

Q: What are the applications of the derived projection formula?

A: The derived projection formula has numerous applications in étale cohomology. Some of the key applications include:

• Computing cohomology groups: The derived projection formula allows us to compute the cohomology groups of a scheme $X$ in terms of the cohomology groups of its base scheme $Y$.
• Computing étale cohomology: The derived projection formula allows us to compute the étale cohomology of a scheme $X$ in terms of the étale cohomology of its base scheme $Y$.
• Computing motivic cohomology: The derived projection formula allows us to compute the motivic cohomology of a scheme $X$ in terms of the motivic cohomology of its base scheme $Y$.

Q: What are the prerequisites for understanding the derived projection formula?

A: To understand the derived projection formula, you should have a good understanding of the following topics:

• Derived categories: Derived categories are a fundamental tool in algebraic geometry, and they play a crucial role in the derived projection formula.
• Étale cohomology: Étale cohomology is a branch of algebraic geometry that studies the cohomology of schemes using étale morphisms.
• Sheaves: Sheaves are a fundamental tool in algebraic geometry, and they play a crucial role in the derived projection formula.

Q: How can I apply the derived projection formula in my research?

A: The derived projection formula can be applied in a variety of ways in research. Some possible applications include:

• Computing cohomology groups: The derived projection formula can be used to compute the cohomology groups of a scheme $X$ in terms of the cohomology groups of its base scheme $Y$.
• Computing étale cohomology: The derived projection formula can be used to compute the étale cohomology of a scheme $X$ in terms of the étale cohomology of its base scheme $Y$.
• Computing motivic cohomology: The derived projection formula can be used to compute the motivic cohomology of a scheme $X$ in terms of the motivic cohomology of its base scheme $Y$.

Q: What are some of the challenges in applying the derived projection formula?

A: Some of the challenges in applying the derived projection formula include:

• Computational complexity: The derived projection formula can be computationally intensive, especially for large schemes.
• Technical difficulties: The derived projection formula requires a good understanding of derived categories, étale cohomology, and sheaves, which can be challenging for some researchers.
• Interpretation of results: The derived projection formula can produce complex results, which can be difficult to interpret.

Q: What are some of the future directions for research in the derived projection formula?

A: Some of the future directions for research in the derived projection formula include:

• Developing new techniques: Developing new techniques for computing cohomology groups and étale cohomology using the derived projection formula.
• Applying the derived projection formula: Applying the derived projection formula to compute cohomology groups and étale cohomology of various schemes.
• Generalizing the derived projection formula: Generalizing the derived projection formula to other areas of mathematics, such as algebraic K-theory and motivic cohomology.

Conclusion

In this article, we have answered some of the most frequently asked questions about the derived projection formula in étale cohomology. The derived projection formula is a fundamental result in étale cohomology, which has numerous applications in algebraic geometry. We hope that this article has provided a useful resource for researchers in the field.