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I’m working through Vakil’s algebraic geometry text and I’ve been stuck on Exercise 1.6.E (page 52 on http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf.)

Suppose that $F$ is an exact functor. Show that applying $F$ to an exact sequence preserves exactness. For example, if $F$ is covariant and $A’ \to A \to A”$ is exact, then $FA’ \to FA \to FA”$ is exact.

Here’s what I’ve been thinking:

- The smallest subobject $\sum{A_i}$ containing a family of subobjects {$A_i$}
- Equivalent conditions for a preabelian category to be abelian
- Hom is a left-exact functor
- Definition of the image as coker of ker == ker of coker?
- Example of a compact module which is not finitely generated
- The construction of the localization of a category

Let the maps be denoted $f, g$ (so we have $A’ \xrightarrow{f} A \xrightarrow{g} A”$).

We know $F$ is left-exact and right-exact. To use the left-exactness of $F$, we note that $0 \to \ker f \to A \xrightarrow{g} A”$ is exact, so $0 \to F(\ker f) \to FA \xrightarrow{Fg} FA”$ is exact.

However, I’m not quite sure what to do with the $F(\ker f)$ object. (It’d be nice if $F(\ker f) = \ker Ff$ but I don’t see any reason for this to actually be true.)

- What is the definition of a commutative diagram?
- Equivalent characterizations of faithfully exact functors of abelian categories
- What does it mean for pullbacks to preserve monomorphisms?
- The dense topology
- Monoid as a single object category
- Is restriction of scalars a pullback?
- Universal property of tensor products / Categorial expression of bilinearity
- categorical interpretation of quantification
- Co/counter variancy of the Yoneda functor
- Is gcd the right adjoint of something?

We have a diagram

with exact diagonals. Applying $F$ gives a diagram

also with exact diagonals.

Now, note that

\begin{align*}

\DeclareMathOperator{Im}{Im}\Im F(f)

&= \Im\big(F(A^\prime)\to F(\Im f)\to F(A)\big) \\

&\overset{\ast}{=} \Im\big(F(\Im f)\to F(A)\big) \\

&= \DeclareMathOperator{Ker}{Ker}\Ker\big(F(A)\to F(\Im g)\big) \\

&\overset{\circledast}{=} \Ker\big(F(A)\to F(\Im g)\to F(A^{\prime\prime})\big) \\

&= \Ker F(g)

\end{align*}

where $\ast$ holds since $F(A^\prime)\to F(\Im f)$ is epi and $\circledast$ holds since $F(\Im g) \to F(A^{\prime\prime})$ is mono. Hence

$$

F(A^\prime)\to F(A)\to F(A^{\prime\prime})

$$

is exact.

Of course, the above argument extends to prove that an exact functor maps acyclic complexes to acyclic complexes.

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- To Find The Exponential Of a Matrix
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- To find the total no. of six digit numbers that can be formed having property that every succeeding digit is greater than preceding digit.