The Preposition

Above:

a. Has often the same meaning as over and can be contrasted with below:

In Mr. Rami's room there is a clock above (over) the fireplace.

b. Mentioned earlier:

c. Most important of:

Think about what I have told you, but above all, don't breathe a word of it to Henry.

Across: "Across" means "From one side to other".

The child ran across the road.

Run across can also mean "meet unexpectedly":

I ran across our friend Smith yesterday.

After: "After" is generally used to denote time or order.

I'll see you after dinner.

To look after: to care for.

If my wife goes away for a week, who’s going to look after me and the children?

After all: In spite of what you thought:

You can see I was right after all.

Against: "Against" expresses the idea of:

a. Opposition:

He who is not for us in against us.

b. Support:

He rested his bicycle against the wall.

c. To avoid the danger from:

My house is insured against fire.