Exact sequences
Exact sequences (Tag D-BJYH3)
Let \(R\) be a ring. A sequence of \(R\)-modules and \(R\)-linear maps \[ M_0 \mapsvia{f_1} M_1 \mapsvia{f_2} \cdots \mapsvia{f_n} M_n \] is exact at \(M_i\), for \(0 < i < n\), if \(\im f_i = \ker f_{i+1}\); it is exact if it is exact at \(M_i\) for every \(0 < i < n\).
A sequence of \(R\)-module maps \[ 0 \mapsvia{d_1} A \mapsvia{d_2} B \mapsvia{d_3} C \mapsvia{d_4} 0 \] is exact if and only if \(\im d_i = \ker d_{i+1}\) for \(i = 1, 2, 3\), that is, if and only if \(d_2\) is injective, \(\im d_2 = \ker d_3\), and \(d_3\) is surjective.
Left and right splittings of a short exact sequence (Tag D-3JJJN)
Let \(R\) be a ring, and let \[ \xi\colon 0 \to A \mapsvia{d_1} B \mapsvia{d_2} C \to 0 \] be a short exact sequence of \(R\)-modules. A right splitting of \(\xi\) is an \(R\)-linear map \(s\colon C\to B\) with \(d_2 \circ s = \id_{C}\). A left splitting of \(\xi\) is an \(R\)-linear map \(t\colon B\to A\) with \(t \circ d_1 = \id_A\).
Splitting lemma for short exact sequences (Tag PR-OODAV)
Let \(R\) be a ring and let \(\xi\colon 0 \to A \mapsvia{d_1} B \mapsvia{d_2} C \to 0\) be a short exact sequence of \(R\)-modules. The following are equivalent:
\(\xi\) admits a right-splitting: an \(R\)-linear \(s\colon C\to B\) with \(d_2\circ s=\id_C\).
\(\xi\) admits a left-splitting: an \(R\)-linear \(t\colon B\to A\) with \(t\circ d_1=\id_A\).
There is an isomorphism \(\varphi\colon B\to A\oplus C\) with \(\varphi\circ d_1=\iota_A\) and \(\pi_C\circ\varphi=d_2\), where \(\iota_A\colon A\to A\oplus C\) is the inclusion and \(\pi_C\colon A\oplus C\to C\) is the projection.
A projective or injective module characterizes splitting for all sequences at once: \(C\) is projective if and only if every short exact sequence of \(R\)-modules \(0\to A'\to B'\to C\to 0\) splits, and \(A\) is injective if and only if every short exact sequence \(0\to A\to B'\to C'\to 0\) splits.
A single short exact sequence can split with \(C\) not projective. Over \(\ZZ\), the sequence \(0\to \ZZ/2\ZZ \to \ZZ/2\ZZ \oplus \ZZ/2\ZZ \to \ZZ/2\ZZ\to 0\) with the inclusion of the first summand and the projection onto the second splits, but \(\ZZ/2\ZZ\) is not a projective \(\ZZ\)-module, since the surjection \(\ZZ\to\ZZ/2\ZZ\) has no section.
(Sufficient conditions for splitting)
A short exact sequence \(0\to A\xrightarrow{f} B\xrightarrow{g} C\to 0\) of \(R\)-modules splits, and then \(B \cong A\oplus C\), if \(C\) is projective or if \(A\) is injective. Every free module is projective, so the sequence splits when \(C\) is free. Over a PID, applied to \(0\to M_t\to M\to M/M_t\to 0\) with \(M/M_t\) free, this splits off the free part of a finitely generated module \(M\).
Tensor products
For positive integers \(m\) and \(n\), \(\ZZ/m \tensor_\ZZ \ZZ/n \cong \ZZ/\gcd(m,n)\). In particular \(\ZZ/2 \tensor_\ZZ \ZZ/3 = 0\).
Tensoring the exact sequence \(\ZZ \xrightarrow{n} \ZZ \to \ZZ/n \to 0\) with \(\ZZ/m\) gives, by right exactness, the exact sequence \(\ZZ/m \xrightarrow{n} \ZZ/m \to \ZZ/n\tensor_\ZZ \ZZ/m \to 0\). Hence \(\ZZ/n\tensor_\ZZ\ZZ/m \cong (\ZZ/m)/n(\ZZ/m) \cong \ZZ/\gcd(m,n)\).
(Right exactness)
For every \(R\)-module \(M\), the functor \(-\tensor_R M\) is right exact, and it is exact if and only if \(M\) is flat. Multiplication by \(2\) on \(\ZZ\) is injective, and tensored with \(\ZZ/2\) it is the zero map on \(\ZZ/2\), which is not injective; the kernel is \(\Tor_1^\ZZ(\ZZ/2,\ZZ/2)\cong\ZZ/2\).