Intuitive meaning of Exact Sequence

I’m currently learning about exact sequences in grad sch Algebra I course, but I really can’t get the intuitive picture of the concept and why it is important at all.

Can anyone explain them for me? Thanks in advance.

Answer

In the linear algebra of Euclidean space (i.e. \mathbb R^n), the consideration of subspaces and their orthogonal complements are fundamental: if V is a subspace of \mathbb R^n then we think of it as filling out “some of” the dimensions in \mathbb R^n, and then its orthogonal complement V^{\perp} fills out the other directions. Together they span \mathbb R^n in a minimial way (i.e. with no redundancies, i.e. \mathbb R^n is the direct sum of V and V^{\perp}).

Now in more general settings (say modules over a ring) we don’t have an inner product and so we can’t form orthogonal complements, but we can still talk about submodules and quotients.

So if A is a submodule of B, then A fills up “some of the directions” in B, and the remaining directions are encoded in B/A.

Now by itself this doesn’t seem like anything new, or worth memorializing with new terminology, but often what happens is that one has a submodule A \subset B, and then a surjection B \to C, given without any a priori relation to each other.

However, if A is precisely the kernel of the map B \to C, then we are (somewhat secretly) in the previous situation: A fills out some of the directions in B, and all the complementary directions are encoded in C.

So we introduce the terminology “\, \, 0 \to A \to B \to C \to 0 is a short exact sequence” to capture this situation.

Since long (i.e. not necessarily short) exact sequences can always be broken
up into a bunch of short exact sequences that are glued together,
getting a feeling for short exact sequences is a good first step.

Of course, you should be coupling your study of these homological concepts with
examples, e.g. short exact sequences arising from tangent and normal bundles
to submanifolds of manifolds, all the important long exact sequences in homology theory (from algebraic topology), and so on; without these examples of naturally occuring set-ups of the “A, B, C” form described above, it won’t be so
easy to get a feel for why this concept was isolated as being a fundamental one.

Attribution
Source : Link , Question Author : finnlim , Answer Author : Tyler

Leave a Comment