Problem 17 - An Introduction to Homological Algebra

Question

(Base-change for Ext) Let \(f: R \rightarrow S\) be a ring map. Show that there is a first quadrant cohomology spectral sequence $$ E_{2}^{p q}=\operatorname{Ext}_{S}^{p}\left(A, \operatorname{Ext}_{R}^{q}(S, B)\right) \Rightarrow \operatorname{Ext}_{R}^{p+q}(A, B) $$ for every \(S\)-module \(A\) and every \(R\)-module \(B\).

Step-by-Step Solution

Answer

Utilize a double complex formed from a projective resolution of B, tensor with S, and apply Hom. This yields a spectral sequence converging to \( \operatorname{Ext}_{R}^{p+q}(A, B) \).

Step 1: Understanding the Setup

First, let's clarify what the problem entails. You have a ring map \( f: R \rightarrow S \) and two modules, \( A \) which is an \( S \)-module, and \( B \) which is an \( R \)-module. We aim to establish a spectral sequence (a tool in homological algebra) which starts with \( E_2^{pq} = \operatorname{Ext}_{S}^{p}(A, \operatorname{Ext}_{R}^{q}(S, B)) \) and converges to \( \operatorname{Ext}_{R}^{p+q}(A, B) \).

Step 2: Consider the Derived Functors

The problem involves the derived functors Ext and hence higher derived functors. \( \operatorname{Ext} \) measures the 'extension classes', or equivalently, how one module can serve as an extension for another. Here, \( \operatorname{Ext}_{R}^{q}(S, B) \) denotes the right-derived functors of the Hom functor, giving us a module by measuring extensions of \( S \) by \( B \).

Step 3: Construct the Double Complex

By constructing a double complex, we will bridge the connection between different modules. Consider a projective resolution \( P_{ullet} \rightarrow B \) of the \( R \)-module \( B \). Tensor this with \( S \) to obtain \( S \otimes_{R} P_{ullet} \), providing a complex of \( S \)-modules.

Step 4: Tensor Product and Hom Functor

Taking Hom over \( S \) with \( A \), consider the double complex \( \operatorname{Hom}_{S}(A, S \otimes_{R} P_{ullet}) \). This renders \( E_0^{p,q} = \operatorname{Hom}_{S}(A, S \otimes_R P_q) \). Each row is exact except at the zeroth position.

Step 5: First Spectral Sequence

Apply the Hom to get the vertical spectral sequence. Resolve \( A \) to acquire \( E_1^{p,q} = \operatorname{Ext}_{S}^{p}(A, \text{cothomology of } q\text{th column}) \). This gives \( E_2^{p,q} = \operatorname{Ext}_{S}^{p}(A, \operatorname{Ext}_{R}^{q}(S, B)) \).

Step 6: Convergence to \( \operatorname{Ext}_{R}^{p+q}(A, B) \)

Tracing through the resolution process as both spectral sequences converge, we see that the double complex does converge to \( \operatorname{Ext}_{R}^{p+q}(A, B) \). This is because the total complex of this double complex effectively computes \( \operatorname{Hom}_{R}(A, P_{\bullet}) \), which resolves \( B \).

Step 7: Conclusion: Linking It All Together

The spectral sequence linking \( \operatorname{Ext}_{S}^{p}(A, \operatorname{Ext}_{R}^{q}(S, B)) \) converges to the expected \( \operatorname{Ext}_{R}^{p+q}(A, B) \) due to the foundational structure of the double complex. This aligns with classical homological frameworks.

Key Concepts

Ring Map

In algebra, a ring map is a function between two rings that respects the structures of addition and multiplication. Suppose you have two rings, \(R\) and \(S\), with a function \(f: R \rightarrow S\). This map should keep both addition and multiplication intact. What does this mean?
Here are the essential properties that \(f\) must satisfy:

• For every element \(a, b\) in \(R\), \(f(a + b) = f(a) + f(b)\).
• It also needs to hold that \(f(a \cdot b) = f(a) \cdot f(b)\).
• The identity element of \(R\) should map to the identity element of \(S\).

When these conditions are met, \(f\) successfully links the algebraic structures of \(R\) and \(S\), making it a powerful bridge between two sets of numbers with their respective operations.These mappings are essential in module theory and allow one to move freely between different modules, leading to a clearer understanding of extension problems between different structures.

Spectral Sequence

A spectral sequence is a mathematical tool used in homological algebra. Think of it like a book containing layers of pages, where each page contains information aggregated from its predecessors. This aggregation process helps solve complex algebraic problems.
Here are some ways to understand spectral sequences:

• They are arranged in stages or pages \(E_r\), starting typically from \(r = 1\).
• At each stage, you perform computations which simplify the problem, leading to the next page \(E_{r+1}\).
• The goal is to reach a particular page where the desired result "converges," giving insight into the underlying structure or solution.

In the problem, the spectral sequence aims to reveal the relationship between \(\operatorname{Ext}_{S}^{p}(A, \operatorname{Ext}_{R}^{q}(S, B))\) and \(\operatorname{Ext}_{R}^{p+q}(A, B)\). Through many pages of calculations and transitions, it unravels step by step, leading to an enlightening conclusion about the module extensions.

Ext Functor

The Ext functor in homological algebra measures how modules extend each other. It's a complex but significant concept that analyzes how modules can fit together or extend, revealing layers of structure behind these mathematical entities.
Here’s how the Ext functor works:

• The notation \(\operatorname{Ext}_{R}^{n}(M, N)\) represents an extension group, which is essentially a way to classify extensions of a module \(N\) by another module \(M\).
• It sits at the heart of derived functors, which expand tools like Hom into the realm of higher-dimensional analysis.
• The essence is understanding different ways one module can be a piece of a larger module structure, contextualized by exact sequences.

In practical terms, if you have two modules \(M\) and \(N\), and you want to understand all the possible ways in which \(N\) can be built into \(M\), Ext gives you a precise mathematical framework to do this.
In the exercise, the focus was on \(\operatorname{Ext}_{R}^{q}(S, B)\), exploring how module \(S\) interacts with \(B\), and tying it back to the finding of extensions between \(A\) and \(B\).

Module Theory

Module theory is a vital area in algebra that generalizes key ideas from vector spaces to more general rings. While vector spaces deal with fields, modules work with rings, providing a broader and more versatile framework. Understanding module theory involves some key concepts:

• Modules vs. Vector Spaces: Unlike vector spaces, modules are defined over rings, which are less restrictive than fields.
• Simple Modules: A module is considered simple if it has no proper submodules other than zero, analogous to prime numbers in integers.
• Projective and Injective Modules: Projective modules have properties similar to free modules and are useful in constructing and deconstructing modules, while injective modules help in extending modules.
• Together with Spectral Sequences: Combine the flexibility of module theory with the computational strength of spectral sequences to solve complex theoretical puzzles.

In the exercise context, modules \(A\) and \(B\) serve as the ground for the Ext operations and spectral sequences. Understanding their structures and how they interact allows the spectral sequence to find meaningful conclusions about \(\operatorname{Ext}_{R}^{p+q}(A, B)\).
This not only unveils relationships between different algebraic objects but also broadens a mathematician's toolkit in solving extension problems.