Problem 10 - An Introduction to Homological Algebra

Question

(Eilenberg-Moore) Let \(f: B \rightarrow C\) be a map of filtered chain complexes. For each \(r \geq 0\), define a filtration on the mapping cone cone \((f)\) 1.5.1 by $$ F_{p} \text { cone }(f)_{n}=F_{p-r} B_{n-1} \oplus F_{p} C_{n} . $$ Show that \(E_{p}^{r}(\) cone \(f)\) is the mapping cone of \(f^{r}: E_{p}^{r}(B) \rightarrow E_{p}^{r}(C) .\) By \(1.5 .2\) this gives a long exact sequence $$ \cdots E_{p+r}^{r}(\text { cone } f) \rightarrow E_{p}^{r}(B) \rightarrow E_{p}^{r}(C) \rightarrow E_{p}^{r}(\text { cone } f) \cdots $$

Step-by-Step Solution

Answer

The spectral sequence \(E_p^r(\text{cone}(f))\) is the mapping cone of \(f^r: E_p^r(B) \rightarrow E_p^r(C)\).

Step 1: Understand the Mapping Cone Construction

To comprehend the solution, we first need to examine what the mapping cone of a chain map \(f: B \rightarrow C\) is. The mapping cone \(\text{cone}(f)\) is a chain complex constructed from the complexes \(B\) and \(C\) by setting \(\text{cone}(f)_n = B_{n-1} \oplus C_n\). Its differential is given by \(d(b, c) = (-db, f(b) + dc)\).

Step 2: Define the Filtration on the Mapping Cone

We are given a filtration \(F_p \text{ cone }(f)_n = F_{p-r} B_{n-1} \oplus F_p C_n\). This preserves the structure of the mapping cone filtered by the differences in the filtration levels of \(B\) and \(C\).

Step 3: Use Spectral Sequence on Filtered Complexes

For a filtered complex, the associated spectral sequence \(E_p^r\) approximates the homology of the total complex. Specifically, the \(r ext{-th}\) page of a spectral sequence is derived by related exact sequences and filtration steps on the differentials.

Step 4: Mapping Cone Induces a Long Exact Sequence

The construction of a mapping cone leads to a long exact sequence in homology related to \(B\) and \(C\). Using spectral sequence theory, particularly by constructions like \(1.5.2\), we attain a long exact sequence by the mapping cone property on spectral sequences of complexes.

Step 5: Relate to Spectral Sequences of B and C

\(E_p^r(\text{cone}(f))\) corresponds to the cone of the map \(f^r: E_p^r(B) \rightarrow E_p^r(C)\) because part of the filtration setup serves to approximate the successive approximations of homologies of complexes involved. These adjustments make the homology analogs consistent upon filtration comparison.

Key Concepts

Filtered Chain Complexes

Filtered chain complexes are a powerful tool in homological algebra. Essentially, a chain complex is a sequence of abelian groups or modules connected by differential maps. The term 'filtered' refers to an additional structure where each component has a filtration, which is a nested sequence of subgroups. So in simple terms, within each chain complex, you have layers, much like an onion. Each layer fits into the next one in a systematic nested way.
By giving this extra layering, we can break down complex calculations into simpler steps. Filtered chain complexes provide a systematic framework to 'peel back' layers of information. This makes them particularly useful when dealing with complex mappings like those found in mapping cones or spectral sequences.

Mapping Cone

The mapping cone, commonly represented as \( \text{cone}(f) \), is an important construction in homological algebra. It is a way to capture the difference between two chain complexes connected by a map \( f: B \to C \). To visualize, imagine you have two sequences of numbers (or spaces) and a function connecting them. The mapping cone then becomes a new complex that reflects the effects of this connection.
The mapping cone \( \text{cone}(f)_n = B_{n-1} \oplus C_n \) consists of pairs from these sequences. The differential operation \( d \) acts on these pairs, taking into account both the differential on \( B \) and the map \( f \). The formula \( d(b, c) = (-db, f(b) + dc) \) reflects this operation. This construction not only captures the interaction of \( B \) and \( C \), but also prepares the stage for related analytical tools like spectral sequences.

Spectral Sequence

Spectral sequences are a way to solve complex problems step-by-step, akin to unwrapping a multi-layered present. In the context of filtered chain complexes, a spectral sequence provides a method for approximating the homology of a complex. By breaking down the chain complex into 'pages', we can understand its structure more profoundly with each successive page offering a clearer view.
The idea is that instead of attempting to solve the whole problem at once, you deal with parts of it in stages called pages, denoted by \( E_p^r \). These pages evolve through an iterative process resembling a passage from raw outline to refined solution. By doing this, the spectral sequence converts a potentially daunting task into manageable parts, simplifying the analysis of complicated homological structures.

Long Exact Sequence

A long exact sequence is a sequence of algebraic objects, such as groups or modules, where each map's kernel equals the previous map's image. When dealing with mapping cones, they naturally lead to long exact sequences. This is important because these sequences reveal vital information about the homological properties of the involved complexes.
The long exact sequence allows us to track changes within structures post-mapping. In the exercise, creating a mapping cone gives rise to such a sequence, functioning almost like a map for navigation. It outlines the route from one part of the complex to another, showing how components behave and interact. Ultimately, a long exact sequence provides an overarching look at the homology and builds pathways through the intricate landscape of algebraic topology.