Homological Commutative Algebra¶

Job: projective and global dimension over commutative rings; Auslander–Buchsbaum; Koszul homology; pointer to local duality — where Ext/Tor meet Spec.

Core¶

Assume $R$ commutative Noetherian unless noted; $M$ finitely generated. Background: homological algebra, foundations.

Projective and global dimension¶

Projective dimension $$ \operatorname{pd}_R M=\min\bigl{n:\exists\text{ projective resolution of length }n\bigr}\in\mathbb{N}\cup{\infty}. $$ Equivalently $\operatorname{pd} M\le n$ iff $\operatorname{Ext}^{n+1}_R(M,-)=0$ iff $\operatorname{Ext}^{n+1}_R(M,N)=0$ for all f.g. $N$ (Noetherian).

Injective / flat dimensions $\operatorname{id}$, $\operatorname{fd}$ analogously via Ext in the other slot / Tor.

Global dimension $$ \operatorname{gl.dim} R=\sup_M\operatorname{pd}_R M=\sup_M\operatorname{id}_R M $$ (sup over all modules, or f.g. modules when $R$ Noetherian — equal in that case). Weak global dimension uses flat dim.

Ring	$\operatorname{gl.dim}$
field	$0$
PID not a field	$1$
$k[x_1,\ldots,x_n]$	$n$ (Hilbert syzygy)
regular local, $\dim n$	$n$ (Serre)
$\mathbb{Z}/4\mathbb{Z}$	$\infty$

Serre’s theorem. Noetherian local $R$ is regular $\Leftrightarrow\operatorname{gl.dim} R<\infty$ $\Leftrightarrow\operatorname{pd}_R k<\infty$ ($k=R/\mathfrak{m}$) $\Leftrightarrow\operatorname{pd}_R k=\dim R$.

Auslander–Buchsbaum formula¶

Theorem (Auslander–Buchsbaum). $(R,\mathfrak{m})$ Noetherian local, $M$ f.g. nonzero with $\operatorname{pd}_R M<\infty$: $$ \operatorname{pd}_R M+\operatorname{depth} M=\operatorname{depth} R. $$ In particular if $R$ is regular (or CM with finite pd), depth and pd trade off exactly.

Corollary. Over regular local $R$, $M$ is free iff $\operatorname{Ext}^i_R(M,R)=0$ for all $i>0$ (depth $M=\dim R$), more precisely: finite pd + maximal depth $\Rightarrow$ free when $R$ regular.

Koszul complex¶

For $x=(x_1,\ldots,x_r)\in R^r$ and $M$ an $R$-module, the Koszul complex $K_\bullet(x;M)$ is $$ K_\bullet(x)={\bigwedge}^\bullet R^r\quad\text{with }d(e_{i_1}\wedge\cdots\wedge e_{i_p})=\sum_j(-1)^{j+1}x_{i_j}\,e_{i_1}\wedge\cdots\hat\cdots\wedge e_{i_p}, $$ tensored with $M$. Homology $H_i(x;M)=H_i(K_\bullet(x;M))$.

Fact	Statement
$H_0(x;M)$	$M/(x)M$
Regular sequence	$x$ $M$-regular $\Rightarrow H_i(x;M)=0$ for $i>0$ (and conversely under mild hyp., e.g. $M$ f.g., $R$ local, $x\subset\mathfrak{m}$)
Depth	$\operatorname{depth}_{(x)}M=\min\{i:H^{i}(x;M)\neq 0\}$ via cohomological Koszul / Ext
Exterior	$K_\bullet(x)$ is a free resolution of $R/(x)$ iff $x$ is regular

Koszul is the standard resolution of the residue field over a regular local ring for a regular system of parameters: $\operatorname{pd}_R k=\dim R$.

Change of rings and spectral sequences¶

If $R\to S$, spectral sequences relate $\operatorname{Tor}^R$, $\operatorname{Tor}^S$ and $\operatorname{Ext}$. Standard: for $x$ regular on $R$, relating pd over $R$ and $R/(x)$. Hilbert syzygy follows by iterating Koszul / mapping cones on $k[x_1,\ldots,x_n]$.

Local duality (pointer)¶

For complete local Cohen–Macaulay $(R,\mathfrak{m})$ of dim $n$ with dualizing module $\omega_R$ (or $R$ Gorenstein: $\omega_R\cong R$): $$ \operatorname{Ext}^i_R(M,\omega_R)^\vee \cong H^{n-i}_\mathfrak{m}(M), $$ Matlis dual $(-)^\vee=\operatorname{Hom}_R(-,E(k))$, and $H^\bullet_\mathfrak{m}$ local cohomology. For regular $R=k[[x_1,\ldots,x_n]]$, $\omega_R\cong R$, recovering classical duality between Ext and local cohomology of $M$.

Full dualizing complexes and Grothendieck duality: AG cohomology, derived categories.

Depth via Ext¶

$$ \operatorname{depth} M=\min{i:\operatorname{Ext}^i_R(k,M)\neq 0} $$ for $M$ f.g. over Noetherian local $(R,\mathfrak{m})$, $k=R/\mathfrak{m}$. Links regular sequences to the vanishing of low Ext against the residue field.

See it¶

Hilbert syzygy as gl.dim. $R=k[x_1,\ldots,x_n]$: Koszul complex on $(x_1,\ldots,x_n)$ resolves $k$, length $n$. For arbitrary f.g. $M$, take a free presentation and use the Hilbert syzygy theorem (or gr and regular sequence arguments) to get a free resolution of length $\le n$. Hence $\operatorname{gl.dim} R=n$. Over $R_\mathfrak{m}$ regular local of dim $n$, same bound; Auslander–Buchsbaum then reads $\operatorname{pd} M=n-\operatorname{depth} M$.

Complete intersection. If $R=S/(f_1,\ldots,f_c)$ with $S$ regular local and $f$ a regular sequence, then $\operatorname{pd}_S R=c$ and Tate / Gulliksen resolutions control $\operatorname{Ext}_R(k,k)$ — infinite pd over $R$ is still “generated by” $c$ relations (CI rings).

Check¶

Compute $\operatorname{pd}_{k[x,y]} k$ and verify Auslander–Buchsbaum after localizing at $(x,y)$.

Koszul on $x,y$ resolves $k$: $\operatorname{pd}=2$. Localize at $\mathfrak{m}=(x,y)$: $R=k[x,y]_\mathfrak{m}$ regular, $\operatorname{depth} R=2$, $\operatorname{depth} k=0$, so $2+0=2$.

Show that if $\operatorname{pd}_R M=0$ then $M$ is projective; give a f.g. example over $R=\mathbb{Z}/6\mathbb{Z}$ that is projective but not free.

$\operatorname{pd}=0$ means $M$ is projective by definition of resolution length. Over $\mathbb{Z}/6\cong\mathbb{Z}/2\times\mathbb{Z}/3$, the factor $\mathbb{Z}/2$ is projective as a module (idempotent splitting) but not free of constant rank in the naive $R^n$ sense as an $R$-module with a single generator annihilated by $3$ — free modules are $(\mathbb{Z}/6)^n$; $\mathbb{Z}/2\cong R/(3)$ is projective via the Chinese remainder decomposition $R\cong R e_1\oplus R e_2$.

Pitfalls¶

Finite global dimension $\neq$ hereditary except in low dim: hereditary means $\operatorname{gl.dim}\le 1$.
Auslander–Buchsbaum requires finite pd — without it the formula is meaningless (both sides may fail to match naive expectations).
Koszul homology can be nonzero even if each $x_i$ is a nonzerodivisor separately — order and joint regularity matter; regular sequence is ordered (though for ideals in local CM rings permutations often work).
Local duality needs dualizing modules / completeness for the Matlis dual form; non-CM rings need dualizing complexes.

Refs¶

Bruns–Herzog, Cohen–Macaulay Rings
Matsumura, Commutative Ring Theory (homological chapters)
Weibel, An Introduction to Homological Algebra
Eisenbud, Commutative Algebra (Ch. on free resolutions)
Stacks Project, Dualizing Complexes

Next¶

Foundations · CA index · Homological algebra · Derived categories · AG cohomology · Modules

Ring	\(\operatorname{gl.dim}\)
field	\(0\)
PID not a field	\(1\)
\(k[x_1,\ldots,x_n]\)	\(n\) (Hilbert syzygy)
regular local, \(\dim n\)	\(n\) (Serre)
\(\mathbb{Z}/4\mathbb{Z}\)	\(\infty\)