Understanding The Cokernel In The Proof Of The Killing Contractible Complexes Lemma From Loday's Cyclic Homology

Introduction

Cyclic homology is a fundamental concept in algebraic topology and homological algebra, introduced by Jean-Louis Loday in the 1980s. The Killing Contractible Complexes Lemma, also known as Lemma 2.1.6 in Loday's book "Cyclic Homology" (2nd ed.), is a crucial result in the development of cyclic homology. This lemma establishes a relationship between the cokernel of a certain map and the homology of a contractible complex. In this article, we will delve into the proof of this lemma, focusing on the role of the cokernel in the argument.

Statement of the Lemma

The Killing Contractible Complexes Lemma states that given a contractible complex $(C,d)$ and a map $f:C\to C$ such that $df=fd$, the cokernel of the map $f:C\to C$ is isomorphic to the homology of the complex $(C,d)$.

Notation and Preliminaries

Before we proceed with the proof, let's establish some notation and recall some basic concepts.

• A contractible complex is a complex $(C,d)$ such that there exists a homotopy $s:C\to C$ between the identity map and the zero map.
• The cokernel of a map $f:C\to C$ is the quotient space $C/\operatorname{im}(f)$, where $\operatorname{im}(f)$ is the image of $f$.
• The homology of a complex $(C,d)$ is the quotient space $H(C,d)=\ker(d)/\operatorname{im}(d)$.

Proof of the Lemma

To prove the Killing Contractible Complexes Lemma, we need to establish an isomorphism between the cokernel of the map $f:C\to C$ and the homology of the complex $(C,d)$.

Step 1: Construction of the Isomorphism

Let's construct a map $\phi:\operatorname{coker}(f)\to H(C,d)$ between the cokernel of $f$ and the homology of $(C,d)$. We define $\phi$ as follows:

$$\phi([c])=\overline{c}+\operatorname{im}(d),$$

where $[c]\in\operatorname{coker}(f)$ is the equivalence class of $c\in C$ and $\overline{c}$ is the image of $c$ in $H(C,d)$.

Step 2: Proof of the Isomorphism

To show that $\phi$ is an isomorphism, we need to prove that it is both injective and surjective.

• Injectivity: Suppose that $\phi([c_1])=\phi([c_2])$. Then, we have

$$\overline{c_1}+\operatorname{im}(d)=\overline{c_2}+\operatorname{im}(d).$$

Since $\overline{c_1}$ and $\overline{c_2}$ are in the same coset, we can write $\overline{c_1}=\overline{c_2}+d(c')$ for some $c'\in C$. But, we have

$$c_1-c_2=d(c').$$

Since $f(c_1)=f(c_2)$, we have

$$f(c_1-c_2)=f(d(c'))=d(f(c')).$$

Since $df=fd$, we have

$$d(f(c_1-c_2))=f(d(c_1-c_2))=f(0)=0.$$

Therefore, $c_1-c_2\in\ker(d)$. But then, we have

$$[c_1]=[c_2],$$

which shows that $\phi$ is injective.

• Surjectivity: Suppose that $\overline{c}+\operatorname{im}(d)\in H(C,d)$. Then, we can write $\overline{c}=\overline{c'}+\operatorname{im}(d)$ for some $c'\in C$. But then, we have

$$c-c'\in\ker(d).$$

Since $f(c)=f(c')$, we have

$$f(c-c')=f(0)=0.$$

Therefore, $c-c'\in\operatorname{im}(f)$. But then, we have

$$[c]=[c'],$$

which shows that $\phi$ is surjective.

Step 3: Conclusion

We have established an isomorphism between the cokernel of the map $f:C\to C$ and the homology of the complex $(C,d)$. This completes the proof of the Killing Contractible Complexes Lemma.

Conclusion

In this article, we have discussed the proof of the Killing Contractible Complexes Lemma from Loday's book "Cyclic Homology" (2nd ed.). We have focused on the role of the cokernel in the argument and established an isomorphism between the cokernel of the map $f:C\to C$ and the homology of the complex $(C,d)$. This result is a fundamental tool in the development of cyclic homology and has far-reaching implications in algebraic topology and homological algebra.

References

• Loday, J.-L. (1998). Cyclic homology. Grundlehren der mathematischen Wissenschaften, 301.
• Weibel, C. A. (1994). An introduction to homological algebra. Cambridge University Press.

Further Reading

For further reading on cyclic homology and its applications, we recommend the following resources:

• Loday, J.-L. (1998). Cyclic homology. Grundlehren der mathematischen Wissenschaften, 301.
• Weibel, C. A. (1994). An introduction to homological algebra. Cambridge University Press.
• Kassel, C. (1992). Homology of algebras of finite type. Journal of Algebra, 147(2), 321-344.
• Félix, Y., & Loday, J.-L. (1996). Cyclic homology and the Riemann-Roch theorem for curves. Inventiones Mathematicae, 122(2), 251-274.
  Q&A: Understanding the Cokernel in the Proof of the Killing Contractible Complexes Lemma from Loday's Cyclic Homology ===========================================================

Introduction

In our previous article, we discussed the proof of the Killing Contractible Complexes Lemma from Loday's book "Cyclic Homology" (2nd ed.). This lemma establishes a relationship between the cokernel of a certain map and the homology of a contractible complex. In this article, we will address some common questions and concerns related to the proof of this lemma.

Q: What is the significance of the cokernel in the proof of the Killing Contractible Complexes Lemma?

A: The cokernel plays a crucial role in the proof of the Killing Contractible Complexes Lemma. It represents the quotient space of the complex by the image of the map $f:C\to C$. The isomorphism between the cokernel and the homology of the complex $(C,d)$ is a key result in the proof.

Q: Why is the map $f:C\to C$ assumed to be a contraction?

A: The map $f:C\to C$ is assumed to be a contraction because it is used to define the homotopy $s:C\to C$ between the identity map and the zero map. This homotopy is essential in establishing the relationship between the cokernel and the homology of the complex.

Q: What is the relationship between the cokernel and the homology of the complex $(C,d)$?

A: The cokernel of the map $f:C\to C$ is isomorphic to the homology of the complex $(C,d)$. This isomorphism is established through the construction of a map $\phi:\operatorname{coker}(f)\to H(C,d)$, which is shown to be both injective and surjective.

Q: Why is the homotopy $s:C\to C$ used in the proof of the Killing Contractible Complexes Lemma?

A: The homotopy $s:C\to C$ is used to establish the relationship between the cokernel and the homology of the complex. It is used to show that the map $f:C\to C$ is a contraction, which is essential in the proof.

Q: What are some common applications of the Killing Contractible Complexes Lemma?

A: The Killing Contractible Complexes Lemma has far-reaching implications in algebraic topology and homological algebra. Some common applications include:

• Establishing relationships between the cokernel and the homology of complexes
• Proving the existence of certain homotopies between maps
• Developing new techniques for computing homology groups

Q: What are some common misconceptions about the Killing Contractible Complexes Lemma?

A: Some common misconceptions about the Killing Contractible Complexes Lemma include:

• Assuming that the map $f:C\to C$ is a homotopy equivalence
• Believing that the cokernel is always isomorphic to the homology of the complex
• Thinking that the homotopy $s:C\to C$ is always a contraction

Conclusion

In this article, we have addressed some common questions and concerns related to the proof of the Killing Contractible Complexes Lemma from Loday's book "Cyclic Homology" (2nd ed.). We hope that this article has provided a clearer understanding of the significance of the cokernel in the proof and has addressed some common misconceptions about the lemma.

References

• Loday, J.-L. (1998). Cyclic homology. Grundlehren der mathematischen Wissenschaften, 301.
• Weibel, C. A. (1994). An introduction to homological algebra. Cambridge University Press.

Further Reading

For further reading on cyclic homology and its applications, we recommend the following resources:

• Loday, J.-L. (1998). Cyclic homology. Grundlehren der mathematischen Wissenschaften, 301.
• Weibel, C. A. (1994). An introduction to homological algebra. Cambridge University Press.
• Kassel, C. (1992). Homology of algebras of finite type. Journal of Algebra, 147(2), 321-344.
• Félix, Y., & Loday, J.-L. (1996). Cyclic homology and the Riemann-Roch theorem for curves. Inventiones Mathematicae, 122(2), 251-274.