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I recently learned about Cohomology and was wondering about the relation between Homology and Cohomology for a given complex, $M^*$. Aluffi mentions that one can switch from one to the other by changing the order of the indices ($M^i \rightarrow M^{-i}$). I do not understand how switching the order of the indices guarantees us morphisms between each object of the complex if the complex consists of objects of an Abelian Category (I understand it for R-mod because abelian groups have group homomorphisms always).

My background is strictly algebraic, I do not know any Algebraic Topology or Manifolds.

Yunus Syed
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1 Answers1

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the relation between Homology and Cohomology for a given complex

There's no such thing as homology and cohomology of a given complex: your complex is either a chain complex, and then it has has homology, or a cochain complex, and then it has cohomology. This is just terminology (coming from geometry, where some things naturally form chain complexes and some things form cochain complexes).

So a complex is a collection of objects numbered by $i\in \mathbb{Z}$ and morphisms between them (also known as differentials) such that $d\circ d = 0$. $$\cdots \to \bullet \to \bullet \to \bullet \to \bullet \to \bullet \to \bullet \to \bullet \to \cdots$$ There are two possibilities:

  • Differentials decrease the degree: $d_i\colon M_i \to M_{i-1}$. Then $(M_\bullet,d_\bullet)$ is called a chain complex, and we associate to it the corresponding homology $H_i (M_\bullet, d_\bullet)$.

  • Differentials increase the degree: $d^i\colon M^i \to M^{i+1}$. Then $(M^\bullet, d^\bullet)$ is called a cochain complex, and we associate to it cohomology $H^i (M^\bullet, d^\bullet)$.

Now if we have, say, a chain complex $(M_i, d_i)_{i\in \mathbb{Z}}$, we can define a cochain complex $(M^i, d^i)_{i\in \mathbb{Z}}$ by setting $M^i = M_{-i}$ and $$d^i = d_{-i}\colon \underbrace{M_{-i}}_{= M^i} \to \underbrace{M_{-i-1}}_{= M^{i+1}}.$$ This is just a formal renumbering of objects and differentials, and it has nothing to do with the nature of objects. After this renumbering, $$H^i (M^\bullet, d^\bullet) = \frac{\ker (M^i \xrightarrow{d^i} M^{i+1})}{\operatorname{im} (M^{i-1} \xrightarrow{d^{i-1}} M^i)} = \frac{\ker (M_{-i} \xrightarrow{d_{-i}} M_{-i-1})}{\operatorname{im} (M_{-i+1} \xrightarrow{d_{-i+1}} M_{-i})} = H_{-i} (M_\bullet, d_\bullet).$$

(Be careful: when one talks about homology vs. cohomology in geometry, it's not about this renumbering of complexes.)