Thanks for the question!
The relationship between the duchess and the jeweler in Virginia Woolf's short story is complex. Before we place this relationship under the microscope, it might be helpful to briefly examine each character individually.
The Jeweler is a wealthy London merchant named Oliver Bacon who lives surrounded by servants in a garish mansion. Even though Mr. Bacon is "the richest jeweler in England," he comes from humble roots. At one point, the jeweler muses that he "began life in a filthy little alley." The jeweler's rags to riches backstory is important as we explore his relationship with the Duchess.
The Duchess of Lambourne is described as "past her prime" and "very fat." However, despite the Duchess's puffed, peacock-like appearance, she is an influential and important person in the English aristocracy. The Duchess boasts a large network of aristocratic and powerful friends and wields her prestige as bait in her interactions with the Jeweler, who lacks the same social pedigree despite his vast wealth.
The relationship between Mr. Bacon and the Duchess is described below in the following excerpt from the story:
They were friends, yet enemies; he was master, she was mistress; each cheated the other, each needed the other, each feared the other, each felt this and knew this every time they touched hands thus in the little back room . . . .
The Duchess is a regular customer who pawns her valuables to Mr. Bacon. Because of this, the jeweler is privy to potentially damaging information about the Duchess, including clear evidence that she has a gambling problem. Mr. Bacon is also in love with the Duchess's daughter, Diana. This proves to be a conundrum for the jeweler, as he mistrusts the Duchess. When she shows him the pearls, he instantly suspects trickery:
But real was it, or false? Was she lying again? Did she dare?
This shows that the Duchess has a history of deception. Nevertheless, due to her social connections, Mr. Bacon writes an astoundingly large check for pearls he believes (and are soon confirmed) to be fake. Despite his expansive wealth, Oliver Bacon's money can't buy him the social prestige he seeks.
I hope this helps!