Commutative Algebra
Subhasis,Joydeep,Subhankar
March 10, 2025
Subhasis,Joydeep,Subhankar Commutative Algebra March 10, 2025 1/9
Introduction
An exact sequence is a sequence of module homomorphisms that describes how modules
relate to each other.
In module theory, exact sequences and tensor products play a vital role in understanding
module structures.
Our project explores the theory, properties, and examples of exact sequences of modules
and tensor product of modules.
Subhasis,Joydeep,Subhankar Commutative Algebra March 10, 2025 2/9
Exact sequence
Definition (exact sequence)
Let R be a commutative ring . Then a sequence of R − modules and R − homomorphisms
i f fi−1
of the type ......... → Mi −
→ Mi−1 −−→ Mi−2 → .........is called exact sequence if
Im fi = ker fi−1 ∀ i .
Definition (short exact sequence)
Let M1 , M2 , M3 be three R − modules. Then an exact sequence of the type
f g
0 → M1 − → M2 − → M3 → 0 is called a short exact sequence.
Subhasis,Joydeep,Subhankar Commutative Algebra March 10, 2025 3/9
Example of exact sequence
Remark
Let M and N are R − modules.Then
f
1 0→M−
→ N is exact iff f is injective.
f
2 M−
→ N → 0 is exact iff f is surjective.
Examples
i p
1 The sequence 0 → 2Z → − Z−→ Z2 → 0 is an exact sequence, where i is the inclusion map
and p is the projection map Z onto Z2 = Z/2Z.
i p
2 For any two R − modules M and N, the sequence 0 → M → − M ⊕N − → N → 0,
is exact, where i is the inclusion map given by i(x) = (x, 0) and p is the projection
given by p(x, y ) = y , x ∈ M, y ∈ N.
Subhasis,Joydeep,Subhankar Commutative Algebra March 10, 2025 4/9
Splitting sequence
Definition (splitting sequence)
f g
An exact sequence 0 → M1 −
→ M2 −
→ M3 → 0 of R − modules splits, if there exists a
R − homomorphism h : M3 → M2 such that g ◦ h = IdM3 , the identity map on M3 .
Example
i p
For any two R − modules M and N, the sequence 0 → M → − M ⊕N − → N → 0,
is a split exact sequence with the splitting map h : N → M ⊕ N given by
h(y ) = (0, y ).
Subhasis,Joydeep,Subhankar Commutative Algebra March 10, 2025 5/9
Splitting sequence continued
Proposition
f g
If for any three R − modules M , P and N, the sequence 0→M−
→P−
→ N → 0 is a
split exact sequence, then P ∼
= M ⊕ N.
Corollary
f g
If for any three R − modules M , P and N, the sequence 0 → M − →P− → N → 0 is a
split exact sequence , then there exists an R −homomorphism s : P → M such that s◦f = IdM .
Subhasis,Joydeep,Subhankar Commutative Algebra March 10, 2025 6/9
Exactness on HomR (M, N)
Let M, N1 , N2 , N3 are R − modules. Then we know that HomR (M, Ni ) are R − modules
(i = 1, 2, 3).
f g
Let N1 −
→ N2 −
→ N3
and also M be fixed R − module. Any homomorphism f : N1 → N2
of R − modules, induces a homomorphism f∗ : HomR (M, N1 ) → HomR (M, N2 )
given by f∗ (α) = f ◦ α, α ∈ HomR (M, N1 ).
We consider g∗ : HomR (M, N2 ) → HomR (M, N3 ).Then (gf )∗ = g∗ f∗ , g ∈ HomR (N2 , N3 ).
f∗ g∗
HomR (M, N1 ) −
→ HomR (M, N2 ) −→ HomR (M, N3 ).
Subhasis,Joydeep,Subhankar Commutative Algebra March 10, 2025 7/9
Exactness on HomR (M, N) continued
Proposition
f g
For any given R − module M and an exact sequence 0 → N1 − → N2 −
→ N3 → 0
f∗ g∗
the induced sequence 0 → HomR (M, N1 ) −→ HomR (M, N2 ) −→ HomR (M, N3 )
is exact.
Remark : HomR (M, N) is not right exact.
Example
×2 π
The sequence 0 → Z −−→ Z −
→ Z/2Z → 0 is exact, but
∗ f g∗
0 → Hom(Z/2Z, Z) −
→ Hom(Z/2Z, Z) −→ Hom(Z/2Z, Z/2Z) → 0 is not exact.
Because Hom(Z/2Z, Z) ∼
= 0.
Subhasis,Joydeep,Subhankar Commutative Algebra March 10, 2025 8/9
Projective modules
Certain types of modules P have the property that for any surjective homomorphism
g : M → N, the induced homomorphism g∗ : HomR (P, M) → HomR (P, N) is surjective.
These are the projective modules defined as follows :
Definition (Projective module)
An R − module P is said to be projective if for any surjective map of R − modules
f : M → N, the induced map f∗ : HomR (P, M) → HomR (P, N) is surjective.
Proposition
f g
An R − module P is projective iff every exact sequence 0 → M −
→N−
→ P → 0 splits,
where M and N are R − modules.
Subhasis,Joydeep,Subhankar Commutative Algebra March 10, 2025 9/9
