Is it standard for a math research paper to include less detail in proofs than a typical textbook would? I am a fourth-year undergraduate student in Mathematics. Recently, I read a mathematics paper. It seems that the proof in the paper is not as convincing as what one might find in a textbook. For instance, the authors skip some calculations and arguments in the proof. I feel that the detail in the textbook is better. In addition, the instructor in my undergraduate year always checked the completeness and the detail of the proof in exam and homework.

Is this a standard practice in writing mathematics papers? What is the advantage (if any) of skipping some calculation and argument?

I plan to apply to a graduate school in the future. If “the skill of minimizing the proof in math papers” is important, then how do I learn this habit and unlearn the old habits of my undergraduate years?
Yes, it's normal. Homeworks and exams are written to prove that the writer has certain skills; papers are written to prove something new. The reader's skills are not under question, so a different style of writing is appropriate. Also, journals used to have stricter page limits than they do now, so there was quite some pressure to be terse.  Conversely, somebody who has a hundred exam scripts to mark needs all the details to be spelled out because they don't have more than a few minutes to give to each script.

It is conventional to omit "routine" calculations that the reader should be able to do themself. For example, one might just assert that a certain function reaches its maximum at x=2p/(1-pq) and assume that the reader is capable of setting the derivative to zero and solving. The reader will typically trust the writer (and the peer-reviewers!) to have done the calculation correctly.

In my view, some authors take this too far and omit calculations which can take hours or days to reconstruct, which is a royal pain when trying to adapt or extend the result. Over time, as you read more research papers, you'll learn what is an appropriate level of detail: the big hints come when you start to co-author papers with your advisor.