Homological Algebra: Foundations¶
Job: abelian categories, resolutions, and derived functors \(\operatorname{Ext}\) / \(\operatorname{Tor}\); double complexes and spectral sequences — the formal engine behind cohomology theories.
Core¶
Abelian categories¶
An abelian category \(\mathcal{A}\) is additive (Hom-sets abelian groups, bilinear \(\circ\), finite biproducts) such that every morphism has kernel and cokernel, every mono is a kernel, every epi is a cokernel, and the canonical \(\operatorname{coim}f\to\operatorname{im}f\) is an iso.
| Example | Notes |
|---|---|
| \(R\text{-Mod}\), \(\mathbf{Ab}\) | prototypes |
| \(\operatorname{QCoh}(X)\), \(\operatorname{Coh}(X)\) | \(X\) scheme (Noetherian for Coh) |
| sheaves of abelian groups on a space | enough injectives |
| finite abelian groups | not enough projectives/injectives for all constructions |
Exact sequence: \(\cdots\to A^{i-1}\xrightarrow{d^{i-1}}A^i\xrightarrow{d^i}A^{i+1}\to\cdots\) with \(\ker d^i=\operatorname{im}d^{i-1}\). Short exact \(0\to A\to B\to C\to 0\): \(A=\ker(B\to C)\), \(C=\operatorname{coker}(A\to B)\).
Freyd–Mitchell. Small abelian \(\mathcal{A}\) embeds exactly into \(R\text{-Mod}\) for some \(R\) — elementwise diagram chases are valid abstractly.
Additive functors and exactness¶
\(F:\mathcal{A}\to\mathcal{B}\) additive: \(F(f+g)=F(f)+F(g)\).
| Exactness | Preserves |
|---|---|
| Left exact | kernels (and finite limits) |
| Right exact | cokernels (and finite colimits) |
| Exact | short exact sequences |
Standard: \(\operatorname{Hom}(A,-)\) left exact; \(-\otimes_R N\) right exact; sheaf global sections left exact; \(f_*\) left exact on sheaves.
Projectives and injectives¶
\(P\) projective: \(\operatorname{Hom}(P,-)\) exact \(\Leftrightarrow\) lifts through epimorphisms. \(I\) injective: \(\operatorname{Hom}(-,I)\) exact \(\Leftrightarrow\) extends along monomorphisms.
| Category | Projectives | Injectives |
|---|---|---|
| \(R\text{-Mod}\) | direct summands of free | enough injectives always |
| \(\mathbf{Ab}\) | free abelian | divisible groups (Baer) |
| sheaves on \(X\) | often few | enough injectives (Godement) |
Enough projectives: every object is quotient of a projective. Enough injectives: every object embeds in an injective.
Resolutions and derived functors¶
Projective resolution of \(M\): \(\cdots\to P_1\to P_0\to M\to 0\) exact, \(P_i\) projective. Injective resolution: \(0\to M\to I^0\to I^1\to\cdots\).
Comparison theorem. Maps of objects lift to maps of resolutions, unique up to homotopy. Hence homology of \(F(P_\bullet)\) is well-defined.
For \(F\) right exact (enough projectives): left derived \(L_iF(M)=H_i\bigl(F(P_\bullet)\bigr)\).
For \(G\) left exact (enough injectives): right derived \(R^iG(M)=H^i\bigl(G(I^\bullet)\bigr)\).
Long exact sequences from short exact sequences of arguments (horseshoe lemma). \(\delta\)-functor universality characterizes \((R^iG)\) among cohomological functors.
Ext and Tor¶
$$ \operatorname{Tor}_i^R(M,N)=L_i(-\otimes_R N)(M)=L_i(M\otimes_R-)(N), $$ $$ \operatorname{Ext}^i_R(M,N)=R^i\operatorname{Hom}_R(M,-)(N)=R^i\operatorname{Hom}_R(-,N)(M). $$ Balanced: compute via resolution in either slot. \(\operatorname{Tor}_0\cong\otimes\), \(\operatorname{Ext}^0\cong\operatorname{Hom}\).
| Interpretation | Statement |
|---|---|
| Extensions | \(\operatorname{Ext}^1(C,A)\cong\) equiv. classes of \(0\to A\to B\to C\to 0\) |
| Yoneda Ext | \(\operatorname{Ext}^n\) classes of \(n\)-extensions modulo splicing |
| Flat dim | \(\operatorname{fd}(M)=\max\{i:\operatorname{Tor}_i(M,-)\neq 0\}\) |
| Projective dim | \(\operatorname{pd}(M)=\max\{i:\operatorname{Ext}^i(M,-)\neq 0\}\) |
| Global dim | \(\operatorname{gl.dim}R=\sup_M\operatorname{pd}(M)\) |
Examples. \(\operatorname{Tor}_1^{\mathbb{Z}}(\mathbb{Z}/m,\mathbb{Z}/n)\cong\mathbb{Z}/\gcd(m,n)\); \(\operatorname{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/n,B)\cong B/nB\); \(\operatorname{pd}_{k[x_1,\ldots,x_n]}k=n\) (Hilbert syzygy).
Dimension shifting and change of rings¶
From \(0\to K\to P\to M\to 0\) with \(P\) projective: \(\operatorname{Ext}^{i+1}(M,N)\cong\operatorname{Ext}^i(K,N)\) for \(i\ge 1\). Group cohomology: \(H^i(G,M)=\operatorname{Ext}^i_{\mathbb{Z}[G]}(\mathbb{Z},M)\) — link to representation theory.
Double complexes¶
A double complex (I-bicomplex) \(C^{p,q}\) has \(d_h:C^{p,q}\to C^{p+1,q}\), \(d_v:C^{p,q}\to C^{p,q+1}\) with \(d_h^2=d_v^2=0\) and \(d_h d_v+d_v d_h=0\) (anticommuted squares). Total complex $$ (\operatorname{Tot} C)^n=\bigoplus_{p+q=n}C^{p,q},\qquad d=d_h+d_v. $$ Two filtrations (by \(p\) or by \(q\)) yield two spectral sequences abutting to \(H^*(\operatorname{Tot} C)\) (under boundedness / completeness hyp.).
Standard sources of double complexes:
| Source | \(C^{p,q}\) |
|---|---|
| Cartan–Eilenberg resolution | resolve each term of a complex by injectives/projectives |
| \(\operatorname{Hom}(P_\bullet,I^\bullet)\) | computes \(\operatorname{Ext}\) via totalization |
| \(P_\bullet\otimes Q_\bullet\) | computes \(\operatorname{Tor}\) |
| Čech–derived functor | Čech cochains valued in injective resolutions |
Horseshoe lemma is the special case of building a double complex from a short exact sequence of objects.
Spectral sequences¶
A spectral sequence (cohomological indexing) is a sequence of pages \((E_r,d_r)_{r\ge r_0}\) with \(d_r:E_r^{p,q}\to E_r^{p+r,q-r+1}\), \(d_r^2=0\), and \(E_{r+1}\cong H(E_r,d_r)\). Convergence \(E_r\Rightarrow H^n\) means for large \(r\), \(E_r^{p,q}\) stabilize to graded pieces of a filtration on \(H^{p+q}\).
First-quadrant SS: \(E_2^{p,q}=0\) for \(p<0\) or \(q<0\) \(\Rightarrow\) only finitely many differentials hit each spot; useful edge maps \(E_2^{n,0}\to H^n\leftarrow E_2^{0,n}\).
| Spectral sequence | \(E_2\) (schematic) | Abutment |
|---|---|---|
| Grothendieck | \(R^p F(R^q G(A))\) | \(R^{p+q}(FG)(A)\) |
| Lyndon–Hochschild–Serre | \(H^p(G/N;H^q(N;M))\) | \(H^{p+q}(G;M)\) |
| Universal coeff. (as SS) | \(\operatorname{Ext}/\operatorname{Tor}\) pages | \(H^*(-;G)\) |
| Serre (topology) | \(H^p(B;H^q(F))\) | \(H^{p+q}(E)\) |
Hypercohomology: \(R^i\Gamma\) of a complex of sheaves — total derived functor via Cartan–Eilenberg or \(K\)-injective resolutions; SS compares to cohomology of cohomology sheaves.
One-liner on reading SS: always track what \(d_r\) can hit; vanishing lines and dimension shifting often force collapse at \(E_2\) or \(E_3\).
Künneth and universal coefficients (quick)¶
For chain complexes of free abelian groups (or flat modules), Künneth relates \(H(C\otimes D)\) to \(H(C)\otimes H(D)\) plus Tor correction. Universal coefficients: \(0\to H_n(X)\otimes G\to H_n(X;G)\to\operatorname{Tor}(H_{n-1}(X),G)\to0\) (splits noncanonically).
See it¶
Hilbert syzygy. \(R=k[x_1,\ldots,x_n]\): every f.g. module has a free resolution of length \(\le n\). Koszul complex on regular sequences / inductive mapping cones. Consequence: \(\operatorname{gl.dim} R=n\); Auslander–Buchsbaum \(\operatorname{pd} M+\operatorname{depth} M=\operatorname{depth} R\).
Sheaf cohomology as derived functor. \(\Gamma(X,-)\) left exact on \(\mathbf{Ab}(X)\); \(H^i(X,\mathcal{F})=R^i\Gamma(X,\mathcal{F})\). Čech agrees under separated + affine covers for quasi-coherent sheaves — AG cohomology.
Computing \(\operatorname{Ext}\) via double complex. Resolve \(M\) projectively and \(N\) injectively; \(H^*(\operatorname{Tot}\operatorname{Hom}(P_\bullet,I^\bullet))\cong\operatorname{Ext}^*(M,N)\) — balance as collapse of either filtration’s SS.
Check¶
Prove that \(M\) is projective iff \(\operatorname{Ext}^1_R(M,N)=0\) for all \(N\).
\(\Rightarrow\): \(\operatorname{Hom}(M,-)\) exact so higher \(R^i=0\). \(\Leftarrow\): for \(0\to A\to B\to C\to 0\), LES gives surjectivity \(\operatorname{Hom}(M,B)\to\operatorname{Hom}(M,C)\); take \(C=M\), \(\mathrm{id}\) lifts \(\Rightarrow\) projective.
Show \(\operatorname{Tor}_1^R(R/I,R/J)\cong (I\cap J)/IJ\) for ideals in commutative \(R\).
Tensor \(0\to I\to R\to R/I\to 0\) with \(R/J\): identify \(\operatorname{Tor}_1(R/I,R/J)\cong\ker(I/IJ\to R/J)\cong(I\cap J)/IJ\).
What does collapse of a first-quadrant SS at \(E_2\) give?
If \(d_r=0\) for all \(r\ge2\), then \(E_2^{p,q}\cong E_\infty^{p,q}\), so the abutment’s associated graded is \(E_2\). Edge maps often become more informative; extensions among graded pieces may still be nontrivial when reconstructing \(H^n\) from \(\bigoplus_{p+q=n}E_\infty^{p,q}\).
Pitfalls¶
- Projective resolution is not unique — only up to homotopy; homology after \(F\) is invariant.
- \(\operatorname{Ext}\) in \(\operatorname{Coh}(X)\) vs \(\operatorname{QCoh}\) vs \(\mathcal{O}_X\text{-Mod}\) can differ.
- Flat \(\neq\) projective (\(\mathbb{Q}\) over \(\mathbb{Z}\)).
- Unbounded resolutions needed for unbounded derived functors (Spaltenstein).
- SS convergence fails without boundedness/completeness; \(E_2\) does not determine abutment if differentials act.
- Sign conventions in total complexes and SS indexing (\(E^r_{p,q}\) vs \(E_r^{p,q}\)) vary by author.
Refs¶
- Weibel, An Introduction to Homological Algebra
- Rotman, An Introduction to Homological Algebra
- Cartan–Eilenberg, Homological Algebra
- McCleary, A User’s Guide to Spectral Sequences
- Stacks Project, Homological Algebra / Derived Categories
Next¶
Derived categories · HA index · Modules · Category theory · AG cohomology · Algebraic topology