Perhaps after the contravariant relationship between commutative algebra and algebraic geometry
This is a major edit of varieties, people starting equating algebra with geometrythe original post after receiving helpful comments.But I notice that
It is often the case when one algebraizes a certain geometry, one adds an additional structure to make a problem more tractable. When one attempts to forget this added structure, this leads to symmetries. One then needs to look at the original solution of the tractable case and modulo cases congruent under symmetry.
The simplest example is that linear algebra is the geometry of when we study free module with an added basismodules. In fact for much of We add a basis to make the interest application of problem tractable, reducing to linear algebra. However, one has when attempting to forget this added structure make basis-free statements of linear spaces and only look for maps, one then must talk about matrices up to similarity, and this is the congruence relation. The change of base are symmetries.
Another example might be in trying to get that algebraically closed field containing is the reals, one must make a choice use of which is $i$ and which is $-i$. Effectively one must choose between spectral sequences. The original grading of the almost complex structures $J$ and $-J$. In graded ring may not be amenable to computation. So we add the endstructure of a filtration to introduce another grading. Again, one needds to forget this results in structure if one wants the conjugation involution on original grades of the complex field and this symmetry is ultilized by pde people when attacking problems on complex manifoldsring.
