# Algebra: Best mental images

I’m curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on itself. I know that a categorical approach is becoming more mainstream. For me, lattice theory is my fallback.

Lattice theory is useful to remember the diamond morphism theorem and lattice morphism theorem. Whenever I need to remember if a group can be expressed as a semidirect product I look for two subgroups where their meet is the bottom of the subgroup lattice, the join is the top of this lattice and one of those subgroups is contained in the normal sublattice. I find this easier than to remember the formal definition since I’ve translated it to relations that are spacial in the lattice. Now I’m studying ideal theory and commutative algebra.

I think of the zero set $\mathbb(V)$ as an up-set of the smallest prime ideal containing the element. I’m curious if this is a general way others have gone about thinking “algebraically”.

Examples! All abstract theories were motiviated historically by specific examples. They are still motivating today, offer nice mental images and tell us what the study of abstract objects is all about.

As for groups, I think of automorphism groups of geometric objects, as well as of fundamental groups of nice spaces, or more simply of the group corresponding to a puzzle such as Rubik’s cube. The conjugation $$a−1baa^{-1} b a$$ means that I first make a setup move $$aa$$, then make my actual move $$bb$$, and then I have to reset with $$a−1a^{-1}$$. This intuition is also useful for general and more abstract groups. The product is just some kind of concatenation of doing something, and the inverse just reverses the action. The commutator $$aba−1b−1a b a^{-1} b^{-1}$$ measures the overlap of the moves $$aa$$ and $$bb$$.

A groupoid is just a bunch of groups which communicate with each other. There are many puzzles which correspond to groupoids, for example the Square One.

For monoids, I think of endomorphism monoids of geometric objects. The product may be imagined as above. We only cannot go backwards. Every choice is ultimate.

As for commutative rings, I think of the ring of nice functions on a nice space. In algebraic geometry one learns that in fact every commutative ring is the ring of regular functions on its spectrum. Thus one may always imagine elements of a commutative ring as functions. For functions we have a lot of experience and intuition since the school days.

Modules over a commutative ring $$RR$$ can be added ($$⊕\oplus$$) and multiplied ($$⊗\otimes$$) much as if they would constitute a ring. We have the distributivity law $$M⊗⊕iNi=⊕i(M⊗Ni)M \otimes \oplus_i N_i = \oplus_i (M \otimes N_i)$$. In fact they constitute a $$22$$-ring (see my paper with A. Chirvasitu), or more simply (decategorified) the isomorphism classes of finitely generated modules constitute a semiring (actually this idea leads to K-theory $$K0(R)K_0(R)$$). In some sense we can calculate with modules as if they were numbers. The natural number $$nn$$ corresponds to the free module $$RnR^n$$. Many isomorphisms between modules decategorify to combinatorial proofs, for example $$Λn(X⊕Y)=⨁p+q=nΛp(X)⊗Λq(Y)\Lambda^n(X \oplus Y) = \bigoplus_{p+q=n} \Lambda^p(X) \otimes \Lambda^q(Y)$$ decategorifies to the Vandermonde identity $$\binom{x+y}{n} = \sum_{p+q=n} \binom{x}{p} \cdot \binom{y}{q}\binom{x+y}{n} = \sum_{p+q=n} \binom{x}{p} \cdot \binom{y}{q}$$. Besides, in commutative algebra it becomes important to realize that $$RR$$-modules are in fact quasi-coherent modules on $$\mathrm{Spec}(R)\mathrm{Spec}(R)$$, and therefore may be treated as bundles. In particular the fiber $$M \otimes_R \mathrm{Quot}(R/\mathfrak{p})M \otimes_R \mathrm{Quot}(R/\mathfrak{p})$$ serves as a first approximation for $$MM$$ at some prime ideal $$\mathfrak{p}\mathfrak{p}$$. After that one might continue with the thickenings $$M \otimes_R R/\mathfrak{p}^nM \otimes_R R/\mathfrak{p}^n$$, with the formal fiber $$M \otimes_R \hat{R_{\mathfrak{p}}}M \otimes_R \hat{R_{\mathfrak{p}}}$$, with the stalk $$M \otimes_R R_{\mathfrak{p}}M \otimes_R R_{\mathfrak{p}}$$ and finally with the localizations $$M \otimes_R R_fM \otimes_R R_f$$ for $$f \notin \mathfrak{p}f \notin \mathfrak{p}$$, which exactly captures the local behaviour of $$MM$$ around $$\mathfrak{p}\mathfrak{p}$$.

As for graded modules in contrast to modules, we just add another dimension which is given by the grading. For many basic considerations, a graded module may be visualized as a long line of dots, representing the homogeneous components. A homomorphism between graded modules of degree $$00$$ is a long ladder. For degree $$nn$$ you have to bend the rungs.

As for sheaves, I often imagine them as two-dimensional objects where one dimension is given by the open subsets and the other one is given by the sections on these open subsets. This also helps to understand and remember some basic notion of sheaf theory, such as surjective homomorphisms and flabby sheaves. Another nice visualization for sheaves is given by their equivalent definition as étale spaces. In fact, from this point of view the notions of sections, germs and stalks make the agricultural analogy just perfect.

In general, one of the main ideas of algebraic geometry is to study geometric objects by algebra, but it is also vice versa: Algebraic objects can be understood by means of geometry. Of course there are lots of other areas of mathematics following these lines, for example noncommutative geometry and geometric group theory.

I could also add mental images for all these categorical notions (colimits as generalized suprema, natural transformations as homotopies, adjoint functors as homotopy equivalences, monoidal categories as categorified rings, etc.), but this answer is long enough.