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Math Notes

University Mathematics

TheoremComplete

Ext and Tor - Applications

Ext and Tor have profound applications in commutative algebra, algebraic geometry, and topology.

Theorem5.18Depth and Ext

For a Noetherian local ring (R,m,k)(R, \mathfrak{m}, k) and finitely generated RR-module MM: depthR(M)=inf⁡{n:ExtRn(k,M)≠0}\text{depth}_R(M) = \inf\{n : \text{Ext}^n_R(k, M) \neq 0\}

This characterization of depth via Ext is fundamental in local algebra.

ExampleCohen-Macaulay Property

A Noetherian local ring RR of dimension dd is Cohen-Macaulay if and only if: depthR(R)=d\text{depth}_R(R) = d

Equivalently, ExtRi(k,R)=0\text{Ext}^i_R(k, R) = 0 for i<di < d.

Theorem5.19Serre's Intersection Formula

For finitely generated modules M,NM, N over a regular local ring RR with M⊗RNM \otimes_R N having finite length: ∑i(−1)ilength(ToriR(M,N))=e(M)⋅e(N)\sum_i (-1)^i \text{length}(\text{Tor}_i^R(M, N)) = e(M) \cdot e(N)

where e(−)e(-) denotes the Hilbert-Samuel multiplicity.

Remark

Serre's formula connects homological algebra (via Tor) to intersection theory in algebraic geometry, showing that Tor groups measure intersection multiplicity.

Theorem5.20Bass Numbers and Injective Dimension

For a Noetherian local ring (R,m,k)(R, \mathfrak{m}, k) and module MM, the Bass numbers are: μi(M)=dim⁡kExtRi(k,M)\mu^i(M) = \dim_k \text{Ext}^i_R(k, M)

These measure the structure of minimal injective resolutions.

ExampleTor and Flat Base Change

For a ring homomorphism R→SR \to S and RR-modules M,NM, N: TornS(M⊗RS,N⊗RS)≅TornR(M,N)⊗RS\text{Tor}_n^S(M \otimes_R S, N \otimes_R S) \cong \text{Tor}_n^R(M, N) \otimes_R S

when SS is flat over RR. This is crucial for base change in algebraic geometry.

Theorem5.21Auslander-Buchsbaum-Serre Theorem

A Noetherian local ring RR is regular if and only if every finitely generated RR-module has finite projective dimension.

Remark

This theorem connects the geometric notion of regularity (non-singularity) to the homological property of finite global dimension, one of the deepest results in commutative algebra.

ExampleExt in Algebraic Geometry

For coherent sheaves F,G\mathcal{F}, \mathcal{G} on a scheme XX: Exti(F,G)\mathcal{E}xt^i(\mathcal{F}, \mathcal{G})

are sheaves whose stalks compute local Ext groups, providing a sheafification of the Ext functor essential for studying coherent sheaf cohomology.