Inspector Slack is the classic disdainful detective that you see over and over again in cozy mysteries. Christie herself didn't use him that much, nor did she develop him beyond a one-note caricature. He was ambitious, prejudiced and only moderately competent. (The one thing he wasn't was a slacker.) He didn't have a real relationship with Miss Marple. He didn't want one. He didn't even provide much opposition to her. He just ignored her, along with ignoring the truth, and thereby provided the need for her to intervene.
When the BBC made the wonderful adaptations starring Joan Hickson, they cast David Horovitch, and they beefed up the role to be more of a foil for Miss Marple. It was necessary because drama is about external conflict - a conflict among characters and their motivations. They didn't really change Slack so much as just let out the emotions that you know had to be there.
The Slack as written is a boring character. As presented by Horovitch, he is a little more interesting. He's not merely annoyed at this woman he'd rather not even acknowledge, he's got a full range of peevish resentments. Slack is a man who lives in a predictable pecking order, and he believes in it. He has position and authority, and he should be able to send little old ladies on their way with a pat on the head. But he's also an underling, and when his superiors force him to work with that silly old puss, he does it. He does it with resentment. He may even feel a little threatened by her success, but he does as he's told.
Because in the end, he and Miss Marple are on the same side. And when she uncovers the crook, he's the first to leap forward and stop the guy from getting away. And in the end, Slack does have to give her a little grudging respect.
I think, when it comes to writing a character like Slack - one who is really there to just acknowledge that the police should be doing this job - you have two choices. You can write like Christie did and just get him off stage as quickly and neatly as possible, and keep him out of the way. There is a certain risk to this, however. While Christie never bothered me much, I can say that I have been annoyed at many other writers who treated their police characters too dismissively.
I think a good example is in Phillip Pullman's Ruby In The Smoke. I really liked that book, and it's been a while since I read it, but I was really annoyed when the heroine dismissed the detective as quickly and thoroughly as he dismissed her. My immediate thought was that he had perfectly good reasons to dismiss her. Maybe he shouldn't have, but she was a child, and a civilian. She has something to prove to him. He, on the other hand, deserved a little respect for the fact that that was his job. I don't remember the scene exactly, but I do remember suddenly losing all respect for the heroine. He may have been a jerk, but she came across as an ignorant little brat.
A little more characterization can fix a situation like that. The more the character is an individual, the more leeway you have to judge them poorly. Otherwise you might just want to leave the judgment out, just don't characterize at all.
ADDENDUM: when I first wrote this post, it had been a very long time since I read Murder At The Vicarage, and I had not remembered that she had given him quite a bit of attention in that book. I recently reread Vicarage, and was very pleased to see that she gave Slack his due. Although he was arrogant and dismissive, he was also competent, and often a step ahead of the Vicar who narrates the book.
Next time I think I'll talk about the policeman as nemesis.
(The other entries in the series are Columbo Ex-Machina, Policeman As Nemesis, and Policeman as Community Member.)