Editing Module homomorphism (section)

== Exact sequences ==
Consider a sequence of module homomorphisms
:<math>\cdots \overset{f_3}\longrightarrow M_2 \overset{f_2}\longrightarrow M_1 \overset{f_1}\longrightarrow M_0 \overset{f_0}\longrightarrow M_{-1} \overset{f_{-1}}\longrightarrow \cdots.</math>
Such a sequence is called a [[chain complex]] (or often just complex) if each composition is zero; i.e., <math>f_i \circ f_{i+1} = 0</math> or equivalently the image of <math>f_{i+1}</math> is contained in the kernel of <math>f_i</math>. (If the numbers increase instead of decrease, then it is called a cochain complex; e.g., [[de Rham complex]].) A chain complex is called an [[exact sequence]] if <math>\operatorname{im}(f_{i+1}) = \operatorname{ker}(f_i)</math>. A special case of an exact sequence is a short exact sequence:
:<math>0 \to A \overset{f}\to B \overset{g}\to C \to 0</math>
where <math>f</math> is injective, the kernel of <math>g</math> is the image of <math>f</math> and <math>g</math> is surjective.

Any module homomorphism <math>f : M \to N</math> defines an exact sequence
:<math>0 \to K \to M \overset{f}\to N \to C \to 0,</math>
where <math>K</math> is the kernel of <math>f</math>, and <math>C</math> is the [[cokernel]], that is the quotient of <math>N</math> by the image of <math>f</math>.

In the case of modules over a [[commutative ring]], a sequence is exact if and only if it is exact at all the [[maximal ideal]]s; that is all sequences 
:<math>0 \to A_{\mathfrak{m}} \overset{f}\to B_{\mathfrak{m}} \overset{g}\to C_{\mathfrak{m}} \to 0</math>
are exact, where the subscript <math>{\mathfrak{m}}</math> means the [[localization of a module|localization]] at a maximal ideal <math>{\mathfrak{m}}</math>.

If <math>f : M \to B, g: N \to B</math> are module homomorphisms, then they are said to form a '''fiber square''' (or '''[[pullback square]]'''), denoted by ''M'' ×<sub>''B''</sub> ''N'', if it fits into
:<math>0 \to M \times_{B} N \to M \times N \overset{\phi}\to B \to 0</math>
where <math>\phi(x, y) = f(x) - g(x)</math>.

Example: Let <math>B \subset A</math> be commutative rings, and let ''I'' be the [[annihilator (ring theory)|annihilator]] of the quotient ''B''-module ''A''/''B'' (which is an ideal of ''A''). Then canonical maps <math>A \to A/I, B/I \to A/I</math> form a fiber square with <math>B = A \times_{A/I} B/I.</math>