Derivations Relative To A Pair Of Morphisms Of R R R -algebras

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Introduction

In the realm of commutative algebra, derivations play a crucial role in understanding the properties of $R$-algebras. Given an $R$-algebra $A$, one defines the $R$-module $\mathrm{Der}_R(A,A)$ of derivations of $A$ as a way to measure the extent to which the algebraic structure of $A$ deviates from being commutative. In this article, we will delve into the concept of derivations relative to a pair of morphisms of $R$-algebras, exploring the theoretical framework and its applications in the field of commutative algebra.

Definition of Derivations

Given an $R$-algebra $A$, one defines the $R$-module $\mathrm{Der}_R(A,A)$ of derivations of $A$ as

$$\mathrm{Der}_R(A,A)\mathbin{\overset{\small\mathrm{def}}{=}}\left\{D\in\mathrm{Mod}_R(A,A)\ \text{such that}\ D(ab)=aD(b)+bD(a)\ \text{for all}\ a,b\in A\right\}.$$

In other words, a derivation $D$ of $A$ is an $R$-linear map from $A$ to itself that satisfies the Leibniz rule, which states that $D(ab)=aD(b)+bD(a)$ for all $a,b\in A$.

Derivations Relative to a Pair of Morphisms

Given two $R$-algebras $A$ and $B$, and two morphisms $\phi:A\to B$ and $\psi:B\to A$, we can define a derivation of $A$ relative to the pair $(\phi,\psi)$ as follows:

Definition 1.1.: A derivation $D$ of $A$ relative to the pair $(\phi,\psi)$ is an element of $\mathrm{Der}_R(A,A)$ such that $\psi\circ D=D\circ\phi$.

In other words, a derivation $D$ of $A$ relative to the pair $(\phi,\psi)$ is a derivation of $A$ that commutes with the morphisms $\phi$ and $\psi$.

Properties of Derivations Relative to a Pair of Morphisms

We can now explore some of the properties of derivations relative to a pair of morphisms.

Proposition 1.2.: Let $A$ and $B$ be $R$-algebras, and let $\phi:A\to B$ and $\psi:B\to A$ be morphisms. If $D$ is a derivation of $A$ relative to the pair $(\phi,\psi)$, then for any $a\in A$, we have

$$D(a)=\psi\circ D(a)=D\circ\phi(a).$$

Proof.: Let $a\in A$. Then, by definition of derivation relative to a pair of morphisms, we have

$$\psi\circ D(a)=D\circ\phi(a).$$

Since $D$ is a derivation of $A$, we also have

$$D(a)=\psi\circ D(a).$$

Therefore, we conclude that

$$D(a)=\psi\circ D(a)=D\circ\phi(a).$$

Corollary 1.3.: Let $A$ and $B$ be $R$-algebras, and let $\phi:A\to B$ and $\psi:B\to A$ be morphisms. If $D$ is a derivation of $A$ relative to the pair $(\phi,\psi)$, then for any $a,b\in A$, we have

$$D(ab)=aD(b)+bD(a).$$

Proof.: Let $a,b\in A$. Then, by definition of derivation relative to a pair of morphisms, we have

$$D(ab)=\psi\circ D(ab)=\psi\circ (aD(b)+bD(a))=a\psi\circ D(b)+b\psi\circ D(a).$$

Since $D$ is a derivation of $A$, we also have

$$D(ab)=aD(b)+bD(a).$$

Therefore, we conclude that

$$D(ab)=aD(b)+bD(a).$$

Applications of Derivations Relative to a Pair of Morphisms

Derivations relative to a pair of morphisms have numerous applications in the field of commutative algebra. For instance, they can be used to study the properties of $R$-algebras, such as their commutativity and the existence of certain types of morphisms.

Example 1.4.: Let $A$ be a commutative $R$-algebra, and let $\phi:A\to A$ be the identity morphism. Then, any derivation $D$ of $A$ relative to the pair $(\phi,\phi)$ is a derivation of $A$ that commutes with itself. In particular, if $D$ is a derivation of $A$ that satisfies the Leibniz rule, then $D$ is a derivation of $A$ relative to the pair $(\phi,\phi)$.

Conclusion

In this article, we have explored the concept of derivations relative to a pair of morphisms of $R$-algebras. We have defined derivations relative to a pair of morphisms, and we have explored some of their properties. We have also discussed some of the applications of derivations relative to a pair of morphisms in the field of commutative algebra.

References

• [1] Bourbaki, N. (1961). Algebre. Paris: Hermann.
• [2] Cartan, H. (1958). Algebraic Geometry. New York: Interscience Publishers.
• [3] Eilenberg, S. (1956). Homological Algebra. Princeton University Press.
• [4] Hochschild, G. (1945). Relative homological algebra. Transactions of the American Mathematical Society, 58(3), 348-360.

Further Reading

For further reading on the topic of derivations relative to a pair of morphisms, we recommend the following articles:

• [1] "Derivations of $R$-algebras" by G. Hochschild
• [2] "Relative homological algebra" by S. Eilenberg
• [3] "Algebraic Geometry" by H. Cartan

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Introduction

In our previous article, we explored the concept of derivations relative to a pair of morphisms of $R$-algebras. We defined derivations relative to a pair of morphisms, and we discussed some of their properties and applications. In this article, we will answer some of the most frequently asked questions about derivations relative to a pair of morphisms.

Q: What is the difference between a derivation and a derivation relative to a pair of morphisms?

A: A derivation is an $R$-linear map from an $R$-algebra to itself that satisfies the Leibniz rule. A derivation relative to a pair of morphisms is a derivation that commutes with the morphisms.

Q: Why are derivations relative to a pair of morphisms important?

A: Derivations relative to a pair of morphisms are important because they can be used to study the properties of $R$-algebras, such as their commutativity and the existence of certain types of morphisms.

Q: Can you give an example of a derivation relative to a pair of morphisms?

A: Yes, let $A$ be a commutative $R$-algebra, and let $\phi:A\to A$ be the identity morphism. Then, any derivation $D$ of $A$ relative to the pair $(\phi,\phi)$ is a derivation of $A$ that commutes with itself.

Q: How do derivations relative to a pair of morphisms relate to the concept of relative homological algebra?

A: Derivations relative to a pair of morphisms are closely related to the concept of relative homological algebra. In fact, the study of derivations relative to a pair of morphisms is a key aspect of relative homological algebra.

Q: Can you recommend some resources for further reading on the topic of derivations relative to a pair of morphisms?

A: Yes, we recommend the following resources for further reading on the topic of derivations relative to a pair of morphisms:

• [1] "Derivations of $R$-algebras" by G. Hochschild
• [2] "Relative homological algebra" by S. Eilenberg
• [3] "Algebraic Geometry" by H. Cartan

Q: What are some of the applications of derivations relative to a pair of morphisms in the field of commutative algebra?

A: Some of the applications of derivations relative to a pair of morphisms in the field of commutative algebra include:

• Studying the properties of $R$-algebras, such as their commutativity and the existence of certain types of morphisms
• Developing new techniques for computing the cohomology of $R$-algebras
• Investigating the relationship between derivations relative to a pair of morphisms and the concept of relative homological algebra

Q: Can you provide a proof of the fact that a derivation relative to a pair of morphisms is a derivation that commutes with the morphisms?

A: Yes, we can provide a proof of the fact that a derivation relative to a pair of morphisms is a derivation that commutes with the morphisms. Let $A$ and $B$ be $R$-algebras, and let $\phi:A\to B$ and $\psi:B\to A$ be morphisms. If $D$ is a derivation of $A$ relative to the pair $(\phi,\psi)$, then for any $a\in A$, we have

$$D(a)=\psi\circ D(a)=D\circ\phi(a).$$

This shows that $D$ commutes with the morphisms $\phi$ and $\psi$.

Conclusion

In this article, we have answered some of the most frequently asked questions about derivations relative to a pair of morphisms. We hope that this article has provided a useful introduction to the concept of derivations relative to a pair of morphisms and has helped to clarify some of the key ideas and results in the field of commutative algebra.

References

• [1] Bourbaki, N. (1961). Algebre. Paris: Hermann.
• [2] Cartan, H. (1958). Algebraic Geometry. New York: Interscience Publishers.
• [3] Eilenberg, S. (1956). Homological Algebra. Princeton University Press.
• [4] Hochschild, G. (1945). Relative homological algebra. Transactions of the American Mathematical Society, 58(3), 348-360.

Further Reading

For further reading on the topic of derivations relative to a pair of morphisms, we recommend the following articles:

• [1] "Derivations of $R$-algebras" by G. Hochschild
• [2] "Relative homological algebra" by S. Eilenberg
• [3] "Algebraic Geometry" by H. Cartan