Category: Teaching

Homological Algebra Course (2025)

Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book. Homological algebra was invented by Eilenberg-MacLane. General category theory (i. e. the theory of arrow-theoretic results) is generally known as abstract nonsense (the terminology is due to Steenrod).
——Serge Lang, Algebra, First edition: page 105; Second edition: page 175

This semester, I will present a series of lectures on homological algebra, and the recordings of the accompanying videos will be made available on Bilibili and YouTube. The main reference textbook is Fundamentals of Algebra: Modules, Categories, Homological Algebra, and Sheaves by Zhi-Jie Chen(陈志杰), published by Higher Education Press (The book is in Chinese). This book is organized in a similar manner to Introduction to Categories, Homological Algebra, and Sheaf Cohomology by Jan R. Strooker (published by Cambridge University Press), with the addition of a chapter on the basics of module theory. We will use both of books as our main reference.

The lecture notes and exercise solutions for the course will be written in English. Handwritten lecture notes will be published as the course progresses, and a LaTeX-based PDF version will be provided upon completion of the course. Also check my notion page (link, note that this page is in Chinese) for more details.

Lecture Notes and Videos

We will closely follow the textbook. The links for the video, lecture notes, and exercises will be updated weekly after each section is covered. Below are the details:

◊ Introduction

§ 0.1 Introduction [Video]

◊ Chapter 1: Modules

  • § 1.1 Definition and Basic Properties of Modules: [Video, Notes, Exercises]
    Definition of modules, Some important examples of modules, Submodules, Quotient modules, Irreducible modules, Cyclic modules, Annihilators, and Torsion modules.
  • § 1.2 Module Homomorphisms: [Video, Notes, Exercise]
    Module homomorphism, Monomorphism, Epimorphism, Isomorphism, Kernel and image, Cokernel and coimage, Hom module, Isomorphism theorems of modules, Exact sequence, Short five lemma, Five lemma, Snake lemma.
  • § 1.3 Direct Sum and Direct Product of Modules: [Video, Notes, Exercise]
    Direct sum of modules, Internal direct sum, Splitting lemma, Universal properties of direct sum and direct product.
  • § 1.4 Free Modules: [Video, Notes, Exercise]
    Definition of free module, Criteria for free module, Rank of free module.
  • § 1.5 Hom and Projective Modules: [Video, Notes, Exercise]
    Definition of projective module, Criterion for projective modules, Relationship between free modules and projective modules, Dual module, Application of the splitting lemma.
  • § 1.6 Injective Modules: [Video, Notes, Exercise]
    Duality, Definition of injective module, Criterion of injective module, Baer’s criterion, Divisible module.
  • § 1.7 Tensor Product and Flat Modules: [Video 1, Video 2, Notes, Exercise]
    Definition of tensor product, Universal property of tensor product, Basic properties of tensor product, Tensor product and Hom, Tensor product of free modules, Definition of flat module, Criteria for flat modules, Properties of flat modules, Flat modules over a PID, Summary of relationships between various modules.
  • § 1.8 Tensor Algebra, Symmetric Algebra, and Exterior Algebra: [Video, Notes, Exercise]
    Tensor algebra, Symmetric algebra, Exterior algebra, Properties of exterior algebra, Symmetric and exterior algebras from free modules.

◊ Chapter 2: Categories

  • § 2.1 Definition of Categories [Video, Notes, Exercise]
    Definition of category, Small category, Locally small category, Subcategory, Full subcategory, Examples of category
  • § 2.2 Functors and Natural Transformations [Video, Notes, Exercise]
    Definition of functors, Covariant functor, Contravariant functor, Opposite category (or dual category), Free functor, Forgetful functor, Hom functor, Tensor functor, and Natural transformation.
  • § 2.3 Products, Coproducts, and Universal Constructions [Video, Notes, Exercise]
    Product, Coproduct, Initial object, Terminal object, Zero object, Pullback, Pushout, Fiber product
  • § 2.4 Representable Functors and Adjoint Functors [Video, Notes, Exercise]
    Representable functor, Hom-functor, Yoneda lemma, Adjoint functor
  • § 2.5 Abelian Categories [Video, Notes, Exercise]
    Monomorphism, Epimorphism, Equalizer, Coequalizer, Kernel, Cokernel, Additive Category, Abelian Category, Additive Functor, Mitchell’s Embedding Theorem.
  • § 2.6 Limits and Colimits [Video, Notes, Exercise]
    Directed index set, Direct system, Direct limit, Inverse System, Inverse limit, Colimit, Limit.

◊ Chapter 3: Homological Algebra

  • § 3.1 Complexes and Homology Modules
  • § 3.2 Long Exact Sequences and Homotopy
  • § 3.3 Decomposition of Modules
  • § 3.4 Derived Functors
  • § 3.5 Tor
  • § 3.6 Ext
  • § 3.7 Homological Dimensions
  • § 3.8 Homology and Cohomology of Groups

◊ Chapter 4: Sheaves and Their Cohomology

  • § 4.1 Presheaves and Sheaves
  • § 4.2 Category of Sheaves
  • § 4.3 Base Change
  • § 4.4 Soft, Flasque, and Injective Sheaves
  • § 4.5 Sheaf Cohomology
  • § 4.6 Čech Cohomology
  • § 4.7 Overview of Spectral Sequences

Recommended Reading

The following are some textbooks and lecture notes on homological algebra.

Quantum Discrete Integral Transform

The integral transform is a powerful mathematical tool that maps a function from its original domain into a new function space through an integration operation (or summation in the discrete case). Typical examples includs the Fourier transform, Laplace transform, Mellin transform, and many others, are indispensable across mathematics, physics, engineering, signal processing, quantum mechanics, and beyond. These transformation often reveals properties of the original function—such as frequency content, growth behavior, or analytic structure—that are more easily analyzed or manipulated in the transformed space. In most cases, the original function can be recovered exactly via an inverse transform.

Mathematically, an integral transform \mathcal{T} is an operator which maps a function {f} into another function \mathcal{T}[f], and two functions do not necessarily have the same domain and range. Usually an integral transform {T} can be written via a corresponding integral kernel {K(p,x)} as

\displaystyle \mathcal{T}[f](p)=\int K(p,x)f(x) dx

Among the vast family of integral transforms, the Fourier transform stands out for its elegance and ubiquity. In the natural units where \hbar = 1, physicists typically define the position-space wave function \psi(x) from its momentum-space counterpart \psi(p) by the (inverse) Fourier transform (chosen to avoid an overall minus sign in the exponent, which is a common convention in quantum Fourier transform):

\mathcal{F}[\psi](x)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{ipx} \psi(p) dp

have the integral kernel {e^{2\pi i xp}/\sqrt{2\pi}}. This single transformation lies at the very heart of quantum mechanics. It converts differential equations into algebraic ones, turns convolutions into products, and recasts purely local position-space information as global momentum content. Momentum-space methods often render seemingly intractable operator equations manifestly solvable.

The discussion that follows extends these classical ideas into the realm of quantum theory through the notion of quantum integral transforms. Since we are concerned with a discrete qubit system, our focus will be on discrete integral transforms. Note that a function {f(x)} can be regarded as a sequence of numbers index by {x} in a continuous index set. The integral transform \mathcal{T}[f] thus looks like a matrix transform with continuous indices. In the discrete context, it turns out that an integral transform becomes a matrix transform, which is us discrete integral transform or simply discrete transform.

Definition 1 (Discrete integral transform) Let {K_{ij}=:K(i,j)} be a matrix with indices {i,j=0,\cdots,N-1} and {\vec{x}=(x_0,\cdots,x_{N-1})^T=:(x(0),\cdots,x(N-1))^{T}} be a vector (discrete function). Then the discrete integral transform is defined as {y(i)=(Kx)(i)=\sum_{j}K(i,j)x(j)} which is in fact a matrix transform

\displaystyle \vec{y}=K\vec{x}. \ \ \ \ \ (1)

Note that we always assume vectors are column vectors. And for the convenience of the generalization we will also assume that {K} is unitary, for which case the invertible discrete integral transform kernel is given by the Hermitian conjugate {K^{\dagger}} of {K}.

For instance, the discrete Fourier transform determined by the kernel matrix {K_{jk}=e^{2\pi i jk/N}/\sqrt{N}} is of the form

\displaystyle y_k=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}e^{2\pi i jk/N}x_j. \ \ \ \ \ (2)

Here we want to mention that in the large {N} limit, the discrete Fourier transform kernel will become the usual Fourier integral kernel and sum is replaced with the integral. This is the reason why we choose {K_{jk}=e^{2\pi i jk/N}/\sqrt{N}} as discrete Fourier transform kernel.

The quantum discrete integral transform is the quantum analogue of discrete integral transform, actually they are exactly the same transform as we will see. Suppose that we have a Hilbert space {\mathcal{H}} with basis states {|0\rangle,\cdots,|N-1\rangle}, for a given discrete integral transform kernel {K_{ij}}, the corresponding quantum discrete integral transform on the basis states is defined as

\displaystyle |j\rangle\overset{QDIT}{\rightarrow}U_K|j\rangle=\sum_{k=0}^{N-1}K_{kj}|k\rangle. \ \ \ \ \ (3)

Then for a state {|\psi\rangle=\sum_{j=0}^{N-1}x_j|j\rangle/\|\mathbf{x}\|},

\displaystyle U_K|\psi\rangle=\frac{1}{\|\mathbf{x}\|}\sum_{j=0}x_jU_K|j\rangle=\frac{1}{\|\mathbf{x}\|}\sum_{k=0}^{N-1}(\sum_{j=0}K_{kj}x_j)|k\rangle=\frac{1}{\|\mathbf{y}\|}\sum_{k=0}^{N-1}y_k|k\rangle \ \ \ \ \ (4)

calculate the discrete integral transform {y_k=\sum_{j=0}^{N-1}K_{kj}x_j}. Notice that {U_K} is unitary matrix, thus {\|\mathbf{x}\|=\|\mathbf{y}\|}. In summary, we have

Definition 2 (Quantum discrete integral transform) Quantum discrete integral transform corresponding to a unitary kernel {K_{kj}} is a unitary transformation {U_K=(K_{kj})}

\displaystyle |j\rangle\overset{QDIT}{\rightarrow}U_K|j\rangle=\sum_{k=0}^{N-1}K_{kj}|k\rangle. \ \ \ \ \ (5)

To carry out the discrete integral transform {y_k=\sum_{j=0}^{N-1}K_{jk}x_j}, we first prepare the state {|\psi\rangle=\sum_{j=0}^{N-1}x_j|j\rangle/\|\mathbf{x}\|} and then apply {U_K} to get {U_K|\psi\rangle}, finally we read out the value {y_k} by measuring in the basis state {|k\rangle}.

To implement a quantum discrete integral transform algorithm, one must design a quantum circuit that realizes the unitary operator $U_K$. This step is generally highly nontrivial and requires careful thought and insightful design. Moreover, the structure of the quantum circuit depends sensitively on the choice of kernel K; different kernels typically lead to very different circuit constructions.