Consider the variety with coordinate ring $k[x,y]/(xy-1)$ where $k=\mathbb{C}$ for instance. If a point $(a,b)$ lives on the variety, then scheme-theoretically, this means that the ideal $(x-a,y-b)$ contains the ideal $(xy-1)$.
To directly see this containment, note that if $ab = 1$, then $$(x-a)(y-b) + b(x-a) + a(y-b) = xy-1$$
In other words, letting $A = x-a$ and $B = y-b$, we got the polynomial expression $$AB + bA + aB$$ I want to know whether there is some way to understand this expression in $A,B$ (and I guess $a,b$ as well) better, and whether this idea leads anywhere in terms of general theory, i.e. does this polynomial fall out of some algebro-geometric construction? Also, is there a general way to determine this expression for, say, a variety defined by a single polynomial (reducible or not) in $n$ variables? Any thoughts would be appreciated.
The pair of polynomials $A$ and $B$ form a Groebner basis of the ideal $(A,B)$.
The relation $A(B+b)+aB-(xy-1)=0$ is called a syzygy between the generators $A,B,xy-1$ of the ideal $(A,B,xy-1)$.
Another way to look at it is that $xy-1=A(B+b)+aB + 0$ is the division of $xy-1$ by the Groebner basis $A,B$, which leaves remainder $0$.