I assume that you are referring to the decadence of the South physically, economically and socially, as opposed to moral decadence—for that takes place without concern for a specific location in the country or the world.
In William Faulkner's "A Rose for Emily," there are several instances that show the decadence in the South. "Decadence" is defined as:
...the act or process of falling into an inferior condition or state; deterioration; decay
First, we can look to the decline of the town in which Miss Emily Grierson and her family have lived for many years. When Miss Emily dies, people go to her home—a place in which few have set foot since her father died many years before. The narrator infers that the home had once been beautiful, "set on what had once been our most select street..."
...But garages and cotton gins had encroached and obliterated even the august names of that neighborhood; only Miss Emily's house was left, lifting its stubborn and coquettish decay above the cotton wagons and the gasoline pumps—an eyesore among eyesores.
This passage infers to a second form of decadence from the standpoint of southern society. The grand old ways have disappeared: cotton gins have replaced slaves in the fields. (Slaves made it possible for the elite of the South to live lives of leisure and opulence.) Gasoline-powered conveyances (cars, etc.) have replaced horses. While horses were used for transportation and farming or plantation work, owning expensive horses and going for a casual ride not only spoke of affluence, but also gave the wealthy the opportunity to demonstrate their social and financial success, being seen out in a carriage:
...they passed on Sunday afternoon in the glittering buggy, Miss Emily with her head high and Homer Barron with his hat cocked and a cigar between his teeth, reins and whip in a yellow glove.
We may also find decadence, specifically economic and/or social "deterioration," when Miss Emily offers "china-painting" lessons. She might well make money for providing this service, but she also would have attempted to keep at bay any changes leading away from the graceful, "civilized" living of the old South, to more modern pursuits. For while Miss Emily was something of a rebellious woman herself (with regard to her "unseemly" relationship with Homer Barron), she would also have been slow to move away from time-honored traditions of southern society. In her home:
She fitted up a studio in one of the downstairs rooms, where the daughters and granddaughters of Colonel Sartoris' contemporaries were sent to her with the same regularity and in the same spirit that they were sent to church on Sunday...
The deterioration the reader notes appears not only in the once-grand homes that now turn toward decay, but also in the way the young move away from the graceful traditions of southern living: the honored word of an agreement (Colonel Sartoris' promise to remit Miss Emily's taxes), and a loss of respect for women—the forthright way one of the Board of Alderman—a "member of the rising generation—wants to confront Miss Emily about the spell emanating from her house. He is reminded by one of the older generation (a "graybeard"):
"Dammit, sir," Judge Stevens said, "will you accuse a lady to her face of smelling bad?"