## Abstract

We investigate the variety of commuting matrices.
We classify its components for any number of matrices of size at most 7.
We prove that starting from quadruples of *r*, *n*.

**Funding statement: **Joachim Jelisiejew was supported by Polish National Science Center, project 2017/26/D/ST1/00755 and by the START fellowship of the Foundation for Polish Science. Klemen Šivic is partially supported by Slovenian Research Agency (ARRS), grant numbers N1-0103 and P1-0222.

## A Functorial approach to comparison between C n ( 𝕄 d ) and Quot r d

The bijection on points obtained in
Lemma 3.4 is not enough to compare
singularities of

For a *A*, an *A*-point of a scheme *X* is just a
morphism *X* is reasonable (for example,
locally of finite type), the
*X* and closed points of *X* agree.
The set of *A*-points is denoted by

The idea of the functor of points is that for every *A* the set of
*X*, so it is easier
to deal with sets of *A*-points for every *A* than with
*X* viewed as a locally ringed topological space.

## Example A.1.

What is an *A*-point of *A*-point of *A*. Conversely, having
such a matrix *A*-point of *A*. Similarly, since *A*-point of *A* and with invertible determinant which is
exactly an element of

## Example A.2.

The scheme

given by the
quadratic equations as explained in Section 3.1. An *A*-point
of

This gives an *n*-tuple
*A*-point of *n*-tuple of *A*.

For simplicity of notation, let *defined* as a functor of
points.

## Example A.3 ([12, Chapter 5]).

We define the scheme *A* the set of *A*-points of

For example, *K* is a kernel of a surjection of locally
free *A*-modules, so it is a locally free *A*-module as well.
Warning: the

A technically less demanding way of proving that

## Example A.4.

Fix *d* elements of the module *F* that are “monomials”,
i.e., that have the form *A*, we define
*A*-linear base of

Having discussed the existence of

## Lemma A.5.

*Let A be a *

*This bijection gives rise to an isomorphism of functors.*

## Proof.

The proof works exactly as in the case

Arguing as in Lemma 3.6 but for *A*-points, we get a
map of functors *G*, a *(Zariski local) principal G-bundle*

*G*-action fiberwise that locally trivializes: for every point

*U*of

*t*and a

*G*-equivariant isomorphism

*U*.

## Corollary A.6.

*There is a morphism of schemes *

*This map makes *

## Proof.

By Lemma A.5, the map above
is a map of functors, so by Yoneda’s
Lemma [11, Lemma VI.1] it gives a morphism of
schemes *p* is a principal *B*-module of rank *d*. Shrink *Z* so that
*B*-module and choose an isomorphism
*B*-modules.
The preimage *A*-point of this preimage is
a morphism

and an *A*-point of
*A*-point
gives an *A*-modules
*K* and

Summing up, an *A*-point of *A*-modules

Let *A*-module *A*-point of *A*-point gives an isomorphism of

We can repeat the argument of Lemma A.5 for

## Lemma A.7.

*Let A be a *

There is no scheme *X* whose *A*-points correspond to locally free *A*-modules. However, there is such an algebraic stack (see [46] for introduction to stacks) and it is called

## Corollary A.8.

*The variety *

## Corollary A.9.

*The variety *

*M*has obstruction group

## Acknowledgements

We very much thank Nathan Ilten for coding and sharing an experimental
version of his VersalDeformations package for *Macaulay2*
which allows one to compute deformations of modules. We thank Joseph
Landsberg, Maciej Gałązka, Hang Huang, and the referee for suggesting several
improvements to the text.

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**Received:**2021-07-28

**Revised:**2022-03-01

**Published Online:**2022-05-25

**Published in Print:**2022-07-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston