Problem 9 - An Introduction to Homological Algebra

Question

(Shifting or Décalage) Given a filtration \(F\) on a chain complex \(C\), define two new filtrations \(F\) and \(\operatorname{Dec} F\) on \(C\) by \(\tilde{F}_{p} C_{n}=F_{p-n} C_{n}\) and \((\operatorname{Dec} F)_{p} C_{n}=\left\\{x \in F_{p+n} C_{n}: d x \in F_{p+n-1} C_{n-1}\right\\}\). Show that the spectral sequences for these three filtrations are isomorphic after reindexing: \(E_{p q}^{r}(F) \cong\) \(E_{p+n, q-n}^{r+1}(\tilde{F})\) for \(r \geq 0\), and \(E_{p q}^{r}(F) \cong E_{p-n, q+n}^{r-1}(\operatorname{Dec} F)\) for \(r \geq 2\).

Step-by-Step Solution

Answer

Spectral sequences with these filtrations are isomorphic when indices are appropriately reindexed: \(E_{p q}^{r}(F) \cong E_{p+n, q-n}^{r+1}(\tilde{F})\) and \(E_{p q}^{r}(F) \cong E_{p-n, q+n}^{r-1}(\operatorname{Dec} F)\).

Step 1: Understanding Given Filtration Definitions

Given a filtration \(F\) on a chain complex \(C\), two new filtrations are defined. \(\tilde{F}_p C_n = F_{p-n} C_n\) shifts the original filtration indices. \((\operatorname{Dec} F)_p C_n = \{x \in F_{p+n} C_n : dx \in F_{p+n-1} C_{n-1}\}\) is the truncated filtration considering the differential condition.

Step 2: Establish Index Shift for \(\tilde{F}\)

For \(\tilde{F}\), observe the relations between differentials and filtrations before reindexing spectral sequences. With filter \(\tilde{F}_p C_n = F_{p-n} C_n\), elements at \(r\)-th filtration stage become elements in the higher index \((r+1)\)-th stage with indices shifted by \(n\).

Step 3: Reindex Spectral Sequence for \(\tilde{F}\)

The indexed spectral sequence \(E_{p q}^{r}(F)\) correlates with \(E_{p+n, q-n}^{r+1}(\tilde{F})\) by adjusting the index shifts. This reindexing involves changing both filtration and corresponding complex indices by \(n\) to see the correspondence, enhancing the series by the required indices.

Step 4: Align \(\operatorname{Dec} F\) using Differential Condition

For \(\operatorname{Dec} F\), the condition \(dx \in F_{p+n-1} C_{n-1}\) ensures truncation after shifting and confirms adherence to indices aligning between \(F\) and reindexed \(\operatorname{Dec} F\) with respect to the reduced space impacted by the differential operator applied.

Step 5: Reindex Spectral Sequence for \(\operatorname{Dec} F\)

The relationship between filtered terms in the sequence \(E_{p q}^{r}(F)\) and \(E_{p-n, q+n}^{r-1}(\operatorname{Dec} F)\) is established through index realignment, accounting for differential application and truncation effect balances within \(r \geq 2\), depicting concordance in the sequence indices.

Key Concepts

Chain Complex Filtration

To get started with understanding spectral sequences and their isomorphisms, it's crucial to first examine the role of filtration in a chain complex. A **chain complex** is essentially a sequence of abelian groups or modules connected by differential maps. These complexes help in studying algebraic topology and homological algebra. A **filtration** in this context is a way to decompose a chain complex into a nested sequence of subcomplexes indexed by integers. Think of it as a way of categorizing or organizing the components of a chain complex.

A filtration is usually denoted by \( F_p C_n \), where \( C_n \) represents the \( n \)-th component of the chain complex, and \( p \) indexes the level of the filtration. When you apply a filtration, you're essentially dividing the chain complex into parts, each part sitting above or below a certain level \( p \). This is like having a hierarchy, where each level builds upon the previous. Such organization allows us to systematically explore and deduce properties of the chain complex, eventually affecting how the spectral sequence is derived from the complex itself. Understanding this foundation will make the later topics like index shifting much clearer.

Index Shifting in Algebra

In the realm of algebra and chain complexes, **index shifting** is an important concept, particularly when working with spectral sequences. Index shifting involves adjusting the indices,

for example from \( p \) to \( p-n \) or \( p+n \), to form new insights or achieve isomorphisms.

• For \( \tilde{F}_p C_n = F_{p-n} C_n \), we shift the filtration backwards by \( n \). This shifting repositions elements effectively into a different 'layer' of the complex without fundamentally changing their structure. This process ensures that the relationships in a spectral sequence remain consistent.
• Conversely, the \( (\operatorname{Dec} F)_p C_n = \{x \in F_{p+n} C_n : dx \in F_{p+n-1} C_{n-1}\} \) approach shifts forward by \( n \), implementing more constraints by considering differentials. The differential constraint demands that elements must satisfy a specific condition with respect to the differential \( d \).

The power of index shifting lies in its ability to translate complex relationships into manageable analysis tasks. By recalibrating indices, one simplifies comparisons and aligns structures that may initially appear disparate, rendering complex mathematical constructs more navigable.

Differential Filtration

In various filtering techniques on a chain complex, **differential filtration** considers the role of differentials. Differentials, typically represented as \( d \), are operations mapping elements in \( C_n \) to \( C_{n-1} \). This map respects the chain's algebraic structure and satisfies \( d^2 = 0 \), which essentially means applying \( d \) twice results in zero.

When constructing a new filtration like \( (\operatorname{Dec} F) \), the differential plays a pivotal part in the filtration's definition. Here, not only is an element considered an element of a certain part of the filtration \( F_{p+n} C_n \), but it must also adhere to the stipulation that its image under the differential \( d \) remains in another defined filtration \( F_{p+n-1} C_{n-1} \). This condition keeps the structure and constraints of the chain complex intact while shifting indices.

The differential condition can be thought of as a form of 'filtering on steroids' because it incorporates additional structural checks into the filtration process, ensuring that only elements conforming to a specific behavior under differential transformation are considered. This methodology deepens our understanding of the interplay between elements within the chain complex. Besides providing constraints, it helps maintain coherence within spectral sequences across shifts. This intricate balancing act is crucial for maintaining the integrity of complex mathematical structures when applying index shifts or other filtration alterations.