# Contents

## Idea

Since generally an equation is the statement of equality $\varphi \left(x\right)=\psi \left(y\right)$ of two functions $\varphi$ and $\psi$ of variables $x$ and $y$, so a linear equation is an equation between linear functions.

### In 1-category theory

Typically here a linear function is taken to mean an $R$-linear map over some given ring $R$, hence a homomorphism $\varphi :X\to Z$ or $\psi :Y\to Z$ of $R$-modules $X,Y,Z\in R$Mod. If here $Z$ is an $R$-module of rank greater than 1, one also speaks of a system of linear equations.

Specifically if $R=k$ is a field then these are linear maps of $k$-vector spaces and hence in this case a linear equation is a statement of equality of two vectors $\varphi \left(x\right)=\psi \left(y\right)$ in some vector space $Z$ that depend linearly on vectors $x$ in a vector spaces $X$ and $y\in Y$.

Frequently this is considered specifically for the case that $g$ is a constant function, hence just a fixed vector. In this case the linear equation becomes $\varphi \left(x\right)=g$ for $x\in X$. If moreover $\varphi$ here is represented or representable by a matrix this is typically written as

$\varphi \cdot \stackrel{⇀}{x}=\stackrel{⇀}{g}\phantom{\rule{thinmathspace}{0ex}},$\phi \cdot\vec x = \vec g \,,

which is the form that one finds in standard textbooks on linear algebra. If $\stackrel{⇀}{g}=0$ here this is called a homogeneous linear equation.

But linear equations make sense and are important in the greater generality where $R$ is not necessarily a field, and in fact in contexts more general than that of $R$-modules even. For instance natural isomorphisms between linear functors are a kind of categorification of linear equations.

### In $\left(\infty ,1\right)$-category theory

Indeed, as discussed at equation, in the formal logic of type theory a an equation as above is a judgement of the form

$x:X,y:Y⊢\left(\varphi \left(x\right)=\psi \left(y\right)\right):\mathrm{Type}$x : X , y : Y \vdash (\phi(x) = \psi(y)) : Type

whose solution space is the dependent sum

$\mathrm{Sol}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}≔⊢\sum _{\genfrac{}{}{0}{}{x:X}{y:Y}}\left(\varphi \left(x\right)=\psi \left(y\right)\right):\mathrm{Type}$Sol \;\; \coloneqq \vdash \sum_{{x : X} \atop {y : Y}} (\phi(x) = \psi(y)) : Type

and reading this in fact in homotopy type theory says that $X,Y,\mathrm{Sol}$ are homotopy types.

The generalization of a ring $R$ to a homotopy type is an E-∞-ring and that of an $R$-module $X,Y$ is a module spectrum.

Accordingly a linear equation in homotopy(type) homotopy theory is a statement of equivalence between elements of an $R$-module spectrum that depend $R$-linearly on other $R$-module spectra. More precisely, as discussed at equation, the solution space to such an ”$\infty$-linear equation” is the homotopy pullback

$\begin{array}{ccc}X{×}_{Z}^{\infty }Y& \to & Y\\ ↓& ⇙& {↓}^{\psi }\\ X& \underset{\varphi }{\to }& Z\end{array}$\array{ X \times_Z^\infty Y &\to& Y \\ \downarrow &\swArrow& \downarrow^{\mathrlap{\psi}} \\ X &\underset{\phi}{\to}& Z }

in an (∞,1)-category of $R$-∞-modules.

## Properties

### Solution spaces of homogeneous $R$-linear equations

We discuss solution space of homogeneous linear equations in the general context of linear algebra over a ring $R$ (not necessarily a field).

So let $R$ be a ring and let $N\in R$Mod be an $R$-module.

Let ${n}_{0},{n}_{1}\in ℕ$ and let $K=\left({K}_{ij}\right)$ be an an ${n}_{0}×{n}_{1}$ matrix with entries in $R$. By matrix multiplication this defines a linear function

$K\cdot \left(-\right):{N}^{{n}_{0}}\to {N}^{{n}_{1}}\phantom{\rule{thinmathspace}{0ex}}.$K \cdot (-) : N^{n_0} \to N^{n_1} \,.

which takes the element $\stackrel{⇀}{u}=\left({u}_{1},\cdots ,{u}_{{n}_{0}}\right)\in {N}^{{n}_{0}}$ to the element $K\cdot \stackrel{⇀}{u}$ with

$\left(K\cdot \stackrel{⇀}{u}{\right)}_{i}=\sum _{j=1}^{{n}_{0}}{K}_{ij}\cdot {u}_{j}\phantom{\rule{thinmathspace}{0ex}}.$(K \cdot \vec u)_i = \sum_{j = 1}^{n_0} K_{i j}\cdot u_j \,.

Consider dually the linear map

$\left(-\right)\cdot {K}^{T}:{R}^{{n}_{1}}\to {R}^{{n}_{0}}$(-) \cdot K^T : R^{n_1} \to R^{n_0}

on the free modules over $R$.

Consider the quotient module of ${R}^{{n}_{1}}$ by the image of this map

${R}^{{n}_{1}}/\left({R}^{{n}_{0}}\cdot {K}^{T}\right)\phantom{\rule{thinmathspace}{0ex}},$R^{n_1}/ (R^{n_0} \cdot K^T) \,,

hence the cokernel of the map, fitting in the exact sequence

${R}^{{n}_{1}}\stackrel{\left(-\right)\cdot K}{\to }{R}^{{n}_{0}}\to {R}^{{n}_{1}}/\left({R}^{{n}_{1}}\cdot {K}^{T}\right)\to 0$R^{n_1} \stackrel{(-)\cdot K}{\to} R^{n_0} \to R^{n_1}/(R^{n_1}\cdot K^T) \to 0

Here the morphism on the left is also called the inclusion of the syzygies of the module on the right.

Applying the left exact hom functor ${\mathrm{Hom}}_{R\mathrm{Mod}}\left(-,N\right)$ to this yields exact sequence

$0\to {\mathrm{Hom}}_{R\mathrm{Mod}}\left({R}^{{n}_{1}}/\left({R}^{{n}_{0}}\cdot {K}^{T}\right),N\right)\to {N}^{{n}_{0}}\stackrel{K\cdot \left(-\right)}{\to }{N}^{{n}_{1}}\phantom{\rule{thinmathspace}{0ex}}.$0 \to Hom_{R Mod}(R^{n_1}/(R^{n_0}\cdot K^T), N) \to N^{n_0} \stackrel{K \cdot(-)}{\to} N^{n_1} \,.

This identifies ${\mathrm{Hom}}_{R\mathrm{Mod}}\left({R}^{{n}_{1}}/\left({R}^{{n}_{0}}\cdot K\right),N\right)$ as the space of solutions of the homogeneous linear equation $K\cdot \stackrel{⇀}{u}=0$.

(…)

### Relation to syzygies and projective resolutions of modules

For $R$ a ring, there is close relation between

1. $R$-linear equations in finitely many variables;

2. finitely generated $R$-modules;

3. syzygies in these $R$-modules

4. and projective resolutions of these $R$-modules.

These relations we discuss in the following.

(…)

## References

Discussion in the context of syzygies and projective resolutions of modules is for instance in section 4.5 of

Revised on September 24, 2012 23:07:11 by Tim Porter (95.147.237.35)