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If $V$ and $W$ are vector spaces (over the same field $\mathbb{F}$) and if $U \subseteq V$ is a subspace, then the obvious exact sequence $$ 0 \longrightarrow U \longrightarrow V \longrightarrow V/U \longrightarrow 0 $$ turns into the (non-trivial) exact sequence $$ 0 \longrightarrow U \mathbin{\otimes} W \longrightarrow V \mathbin{\otimes} W \longrightarrow (V/U) \mathbin{\otimes} W \longrightarrow 0. \tag*{$(*)$} $$ Even though I rarely need exact sequences in my work, whenever I'm working with tensor products I find that simple manipulations like this tend to make life a lot easier.

_{(Interestingly, I don't know a textbook that mentions $(*)$ explicitly. Most linear algebra textbooks never even mention exact sequences, and most abstract algebra textbooks hide $(*)$ behind a general statement such as "every projective module is flat".)}

An example that might be useful in virtually all branches of mathematics: If $V$ and $W$ are vector spaces (over the same field $\mathbb{F}$) and if $U \subseteq V$ is a subspace, then the obvious exact sequence $$ 0 \longrightarrow U \longrightarrow V \longrightarrow V/U \longrightarrow 0 $$ turns into the non-trivial exact sequence $$ 0 \longrightarrow U \mathbin{\otimes} W \longrightarrow V \mathbin{\otimes} W \longrightarrow (V/U) \mathbin{\otimes} W \longrightarrow 0. \tag*{$(*)$} $$ (I rarely need exact sequences in my work, but simple manipulations like this make quotients and subspaces of tensor products much easier to deal with.)

If $V$ and $W$ are vector spaces (over the same field $\mathbb{F}$) and if $U \subseteq V$ is a subspace, then the obvious exact sequence $$ 0 \longrightarrow U \longrightarrow V \longrightarrow V/U \longrightarrow 0 $$ turns into the (very useful) exact sequence $$ 0 \longrightarrow U \mathbin{\otimes} W \longrightarrow V \mathbin{\otimes} W \longrightarrow (V/U) \mathbin{\otimes} W \longrightarrow 0. \tag*{$(*)$} $$ Even though I rarely need exact sequences in my work, whenever I'm working with tensor products I find that simple manipulations like this tend to make life a lot easier.

_{(Interestingly, I don't know a textbook that mentions $(*)$ explicitly. Most linear algebra textbooks never even mention exact sequences, and most abstract algebra textbooks hide $(*)$ behind a general statement such as "every projective module is flat".)}

If $V$ and $W$ are vector spaces (over the same field $\mathbb{F}$) and if $U \subseteq V$ is a subspace, then the obvious exact sequence $$ 0 \longrightarrow U \longrightarrow V \longrightarrow V/U \longrightarrow 0 $$ turns into the (non-trivial) exact sequence $$ 0 \longrightarrow U \mathbin{\otimes} W \longrightarrow V \mathbin{\otimes} W \longrightarrow (V/U) \mathbin{\otimes} W \longrightarrow 0. \tag*{$(*)$} $$ Even though I rarely need exact sequences in my work, whenever I'm working with tensor products I find that simple manipulations like this tend to make life a lot easier.

_{(Interestingly, I don't know a textbook that mentions $(*)$ explicitly. Most linear algebra textbooks never even mention exact sequences, and most abstract algebra textbooks hide $(*)$ behind a general statement such as "every projective module is flat".)}

If $V$ and $W$ are vector spaces (over the same field $\mathbb{F}$) and if $U \subseteq V$ is a subspace, then the obvious exact sequence $$ 0 \longrightarrow U \longrightarrow V \longrightarrow V/U \longrightarrow 0 $$ turns into the (very useful) exact sequence $$ 0 \longrightarrow U \mathbin{\otimes} W \longrightarrow V \mathbin{\otimes} W \longrightarrow (V/U) \mathbin{\otimes} W \longrightarrow 0. \tag*{$(*)$} $$ Even though I rarely need exact sequences in my work, whenever I'm working with tensor products I find that simple manipulations like this tend to make life a lot easier. So $(*)$ is a short exact sequence every mathematician should know.

_{(Interestingly, I don't know a textbook that mentions $(*)$ explicitly. Most linear algebra textbooks never even mention exact sequences, and most abstract algebra textbooks hide $(*)$ behind a general statement such as "every projective module is flat".)}

If $V$ and $W$ are vector spaces (over the same field $\mathbb{F}$) and if $U \subseteq V$ is a subspace, then the obvious exact sequence $$ 0 \longrightarrow U \longrightarrow V \longrightarrow V/U \longrightarrow 0 $$ turns into the (very useful) exact sequence $$ 0 \longrightarrow U \mathbin{\otimes} W \longrightarrow V \mathbin{\otimes} W \longrightarrow (V/U) \mathbin{\otimes} W \longrightarrow 0. \tag*{$(*)$} $$ Even though I rarely need exact sequences in my work, whenever I'm working with tensor products I find that simple manipulations like this tend to make life a lot easier.

_{(Interestingly, I don't know a textbook that mentions $(*)$ explicitly. Most linear algebra textbooks never even mention exact sequences, and most abstract algebra textbooks hide $(*)$ behind a general statement such as "every projective module is flat".)}