5.1 Definitions and properties of projective modules

Projective modules are a key concept in homological algebra, allowing us to "lift" along surjective homomorphisms. They're closely related to free modules, which have a basis. Understanding these modules helps us analyze algebraic structures and solve complex problems.

Projective modules have special properties like being direct summands of free modules and splitting exact sequences. These characteristics make them powerful tools for studying module theory and developing advanced algebraic techniques.

Projective and Free Modules

Properties and Definitions

Top images from around the web for Properties and Definitions

• 

  Category:Module theory - Wikimedia Commons View original

  Is this image relevant?

• 

  projective geometry - Aether Force View original

  Is this image relevant?

• 

  Category:Homological algebra - Wikimedia Commons View original

  Is this image relevant?

• 

  Category:Module theory - Wikimedia Commons View original

  Is this image relevant?

• 

  projective geometry - Aether Force View original

  Is this image relevant?

1 of 3

Top images from around the web for Properties and Definitions

• 

  Category:Module theory - Wikimedia Commons View original

  Is this image relevant?

• 

  projective geometry - Aether Force View original

  Is this image relevant?

• 

  Category:Homological algebra - Wikimedia Commons View original

  Is this image relevant?

• 

  Category:Module theory - Wikimedia Commons View original

  Is this image relevant?

• 

  projective geometry - Aether Force View original

  Is this image relevant?

1 of 3

• Projective module is a module $P$ such that for every surjective homomorphism $f: M \to N$ and every homomorphism $g: P \to N$, there exists a homomorphism $h: P \to M$ such that $f \circ h = g$
• Free module is a module that has a basis, which means it is isomorphic to a direct sum of copies of the base ring $R$ (denoted as $R^{(I)}$ for some index set $I$)
  • Examples of free modules include $\mathbb{R}^n$ over $\mathbb{R}$ and $\mathbb{Z}^n$ over $\mathbb{Z}$
• Projectivity is a property of modules that allows them to be "lifted" along surjective homomorphisms
  • Every free module is projective, but not every projective module is free (e.g., the $\mathbb{Z}$-module $\mathbb{Q}$ is projective but not free)

Dual Basis Lemma

• Dual basis lemma states that if $M$ is a free module with basis $\{x_i\}_{i \in I}$, then there exists a unique family of linear maps $\{f_i: M \to R\}_{i \in I}$ such that $f_i(x_j) = \delta_{ij}$ (Kronecker delta) for all $i, j \in I$
  • The family $\{f_i\}_{i \in I}$ is called the dual basis of $\{x_i\}_{i \in I}$
• Dual basis lemma is useful for proving that every free module is projective and for constructing projective resolutions of modules

Direct Summands and Splitting

Direct Summands

• Direct summand is a submodule $N$ of a module $M$ such that there exists another submodule $N'$ of $M$ with $M = N \oplus N'$ (direct sum)
  • Examples of direct summands include $\mathbb{Z}$ as a direct summand of $\mathbb{Z} \oplus \mathbb{Z}$ and $\mathbb{Q}$ as a direct summand of $\mathbb{Q} \oplus \mathbb{Q}$
• A module $M$ is projective if and only if it is a direct summand of a free module

Splitting Exact Sequences

• Splitting exact sequence is a short exact sequence $0 \to N \to M \to P \to 0$ such that the middle term $M$ is isomorphic to the direct sum of $N$ and $P$ (i.e., $M \cong N \oplus P$)
  • The sequence is said to split if there exists a homomorphism $s: P \to M$ such that $f \circ s = id_P$, where $f: M \to P$ is the surjective homomorphism in the sequence
• A module $P$ is projective if and only if every short exact sequence $0 \to N \to M \to P \to 0$ splits

Lifting Property

• Lifting property is a characteristic of projective modules, stating that for every surjective homomorphism $f: M \to N$ and every homomorphism $g: P \to N$, there exists a homomorphism $h: P \to M$ such that $f \circ h = g$
  • This property allows projective modules to be "lifted" along surjective homomorphisms
• The lifting property is equivalent to the splitting of short exact sequences and the existence of a direct summand in a free module

Key Lemmas

Schanuel's Lemma

• Schanuel's lemma states that if $0 \to K \to P \to M \to 0$ and $0 \to K' \to P' \to M \to 0$ are two short exact sequences with $P$ and $P'$ projective modules, then $K \oplus P' \cong K' \oplus P$
  • This lemma is useful for comparing the kernels of two projective resolutions of the same module
• Schanuel's lemma can be used to prove that the projective dimension of a module is well-defined (i.e., independent of the choice of projective resolution)
  • The projective dimension of a module $M$ is the smallest non-negative integer $n$ such that there exists a projective resolution of $M$ of length $n$, or $\infty$ if no such finite resolution exists