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Math Notes

University Mathematics

ConceptComplete

Exact Sequences and the Hom/Tensor Functors

Exact sequences capture the algebraic structure of module maps, and the Hom and tensor functors are the fundamental tools for constructing new modules from old.

Exact Sequences

Definition9.4Exact sequence

A sequence of RR-module homomorphisms β‹―β†’Miβˆ’1β†’fiβˆ’1Miβ†’fiMi+1β†’β‹―\cdots \to M_{i-1} \xrightarrow{f_{i-1}} M_i \xrightarrow{f_i} M_{i+1} \to \cdots is exact at MiM_i if im(fiβˆ’1)=ker⁑(fi)\mathrm{im}(f_{i-1}) = \ker(f_i). A short exact sequence is 0β†’Aβ†’fBβ†’gCβ†’00 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 where ff is injective, gg is surjective, and im(f)=ker⁑(g)\mathrm{im}(f) = \ker(g).

ExampleShort exact sequences
  1. 0→Z→×2Z→Z/2Z→00 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0.
  2. 0β†’Iβ†ͺRβ†’R/Iβ†’00 \to I \hookrightarrow R \to R/I \to 0 for any ideal II.
  3. 0β†’ker⁑fβ†ͺMβ†’fNβ†’coker(f)β†’00 \to \ker f \hookrightarrow M \xrightarrow{f} N \to \mathrm{coker}(f) \to 0.

The Hom Functor

Definition9.5Hom module

For RR-modules M,NM, N: HomR(M,N)={f:Mβ†’N:fΒ isΒ R-linear}\mathrm{Hom}_R(M, N) = \{f: M \to N : f \text{ is } R\text{-linear}\} is an abelian group (and an RR-module when RR is commutative). The functor HomR(M,βˆ’)\mathrm{Hom}_R(M, -) is left exact: it sends 0β†’Aβ†’Bβ†’Cβ†’00 \to A \to B \to C \to 0 to 0β†’Hom(M,A)β†’Hom(M,B)β†’Hom(M,C)0 \to \mathrm{Hom}(M,A) \to \mathrm{Hom}(M,B) \to \mathrm{Hom}(M,C) (exact, but the last map need not be surjective).

The Tensor Product

Definition9.6Tensor product

The tensor product MβŠ—RNM \otimes_R N of a right RR-module MM and a left RR-module NN is the abelian group generated by symbols mβŠ—nm \otimes n subject to:

  • (m1+m2)βŠ—n=m1βŠ—n+m2βŠ—n(m_1 + m_2) \otimes n = m_1 \otimes n + m_2 \otimes n,
  • mβŠ—(n1+n2)=mβŠ—n1+mβŠ—n2m \otimes (n_1 + n_2) = m \otimes n_1 + m \otimes n_2,
  • mrβŠ—n=mβŠ—rnmr \otimes n = m \otimes rn for r∈Rr \in R.

The functor MβŠ—Rβˆ’M \otimes_R - is right exact: it sends Aβ†’Bβ†’Cβ†’0A \to B \to C \to 0 to MβŠ—Aβ†’MβŠ—Bβ†’MβŠ—Cβ†’0M \otimes A \to M \otimes B \to M \otimes C \to 0.

ExampleTensor product computations
  1. Z/mZβŠ—ZZ/nZβ‰…Z/gcd⁑(m,n)Z\mathbb{Z}/m\mathbb{Z} \otimes_\mathbb{Z} \mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/\gcd(m,n)\mathbb{Z}.
  2. RβŠ—RMβ‰…MR \otimes_R M \cong M for any RR-module MM.
  3. QβŠ—ZZ/nZ=0\mathbb{Q} \otimes_\mathbb{Z} \mathbb{Z}/n\mathbb{Z} = 0 (tensoring torsion with a divisible module kills it).
  4. k[x]βŠ—kk[y]β‰…k[x,y]k[x] \otimes_k k[y] \cong k[x,y].
RemarkDerived functors

The failure of left/right exactness leads to derived functors: ExtRn(M,N)\mathrm{Ext}^n_R(M,N) measures the failure of Hom\mathrm{Hom} to be exact, and TornR(M,N)\mathrm{Tor}^R_n(M,N) measures the failure of βŠ—\otimes to be exact. These are central objects in homological algebra and algebraic topology.