# Directed algebras and quadratic presentations

*Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).*

A finitely presented algebra \(R\) is usually infinite-dimensional, while the counterexample needs a finite-dimensional algebra. This lesson encodes a presentation of \(R\) in a finite-dimensional algebra \(B\) with three vertices: an \(R\)-module \(Y\) becomes the \(B\)-module \(M(Y)\) with \(Y\) at each vertex, and the arrows record the action of the generators (Section 3). The encoding is the homogenization principle that Neeman, Ranicki and Schofield use for universal localization [NRS, Section 1]; we only need its elementary part. The algebra \(B\) has global dimension at most two (Proposition 2.2), which later splits a complex with cohomology in two degrees three apart. The construction follows [OpenAI-FD, Section 3].

Throughout, \(k\) is a field and \(R\) a finitely presented \(k\)-algebra.

## 1. Quadratic presentations

**Lemma 1.1.** \(R\) has a presentation \(R=k\langle x_1,\dots,x_d\rangle/(p_1,\dots,p_t)\) in which every \(p_j\) is a polynomial of degree at most two in the noncommuting variables \(x_h\), possibly with constant and linear terms.

**Proof.** Start from any finite presentation. Each relation is a finite linear combination of monomials. For every monomial \(y_1y_2\cdots y_l\) of degree \(l\ge3\) occurring in a relation, add new generators \(w_2,\dots,w_{l-1}\) with the quadratic relations \(w_2=y_1y_2\) and \(w_{q}=w_{q-1}y_q\) for \(3\le q\le l-1\), and replace the monomial by \(w_{l-1}y_l\). All relations have degree at most two. The new generators are determined by the old ones, so eliminating them recovers the original presentation: the inclusion of the old generators induces an isomorphism of the two presented algebras. \(\square\)

Fix such a presentation. Products in \(R\) and products of arrows below are written in the order of composition, from right to left.

## 2. The algebra B

Consider the quiver with vertices \(0,1,2\), arrows \(s,a_1,\dots,a_d\) from \(0\) to \(1\), and arrows \(s',a'_1,\dots,a'_d\) from \(1\) to \(2\). Let \(\mathcal A_{01}\) and \(\mathcal A_{12}\) be these two sets of arrows. A path of length two is a product \(ba\) with \(a\in\mathcal A_{01}\) and \(b\in\mathcal A_{12}\). The *homogenization* of a polynomial of degree at most two is the linear combination of paths of length two obtained by the substitutions
\[
x_hx_l\mapsto a'_ha_l,\quad x_h\mapsto s'a_h,\quad1\mapsto s's .
\]
Let \(B\) be the path algebra of the quiver modulo the ideal generated by the elements (2.1),
\[
\begin{gathered}
a'_hs-s'a_h\quad(1\le h\le d),\\
\widetilde p_j\quad(1\le j\le t),
\end{gathered}
\]
where \(\widetilde p_j\) is the homogenization of \(p_j\). Let \(\mathbf e_0,\mathbf e_1,\mathbf e_2\) be the trivial paths. An arrow \(a\colon i\to j\) lies in \(\mathbf e_jB\mathbf e_i\).

There are no paths of length three, so the path algebra is finite-dimensional, with basis the three trivial paths, the \(2(d+1)\) arrows and the \((d+1)^2\) paths of length two. All relations (2.1) are combinations of paths of length two from \(0\) to \(2\). Fix a basis \(\mathcal R\) of the space they span; each \(\rho\in\mathcal R\) is a combination \(\rho=\sum_{a,b}\rho_{ba}\,ba\) with scalars \(\rho_{ba}\). Since no nonzero path can be multiplied onto a path of length two, the ideal generated by (2.1) is exactly the span of \(\mathcal R\), and \(B\) has basis: the trivial paths, the arrows, and the classes of a complement of \(\operatorname{span}\mathcal R\) among the paths of length two.

**Definition 2.1.** A finite-dimensional algebra \(\Lambda\) is *directed* with respect to complete orthogonal idempotents \(\varepsilon_0,\dots,\varepsilon_{n-1}\) (so \(1=\sum_i\varepsilon_i\)) if \(\varepsilon_i\Lambda\varepsilon_i=k\varepsilon_i\) for every \(i\) and \(\varepsilon_j\Lambda\varepsilon_i=0\) for \(j<i\).

**Proposition 2.2.** A directed algebra with \(n\) idempotents has global dimension at most \(n-1\), for left modules and for right modules. In particular \(B\) has global dimension at most two on both sides.

**Proof.** Call a left module \(M\) *supported in degrees at least \(p\)* if \(\varepsilon_jM=0\) for \(j<p\). For such an \(M\), consider
\[
\bigoplus_{i\ge p}\Lambda\varepsilon_i\otimes_k\varepsilon_iM\longrightarrow M,\quad c\otimes v\mapsto cv .
\]
Its source is projective, being a direct sum of copies of the projective modules \(\Lambda\varepsilon_i\), whether or not \(M\) is finitely generated. It is surjective, since every \(v\in M\) equals \(\sum_{i\ge p}\varepsilon_iv\). Apply \(\varepsilon_j\) for \(j\le p\): the summand \(\Lambda\varepsilon_i\otimes\varepsilon_iM\) contributes \(\varepsilon_j\Lambda\varepsilon_i\otimes\varepsilon_iM\), which is \(0\) for \(i>j\); so for \(j<p\) nothing remains, and for \(j=p\) only the summand \(i=p\) remains, and it maps \(k\varepsilon_p\otimes\varepsilon_pM\) isomorphically onto \(\varepsilon_pM\). Hence the kernel \(K\) satisfies \(\varepsilon_jK=0\) for \(j\le p\): it is supported in degrees at least \(p+1\).

Starting with an arbitrary module (\(p=0\)) and repeating this \(n-1\) times gives an exact sequence
\[
\begin{gathered}
0\to K_{n-1}\to Q_{n-2}\to\cdots\\
\cdots\to Q_0\to M\to0
\end{gathered}
\]
with projective \(Q_i\) and \(K_{n-1}\) supported in degree \(n-1\) alone. Such a module equals \(\varepsilon_{n-1}K_{n-1}\), and \(\Lambda\varepsilon_{n-1}=\varepsilon_{n-1}\Lambda\varepsilon_{n-1}=k\varepsilon_{n-1}\), because \(\varepsilon_j\Lambda\varepsilon_{n-1}=0\) for \(j<n-1\). So the map \(\Lambda\varepsilon_{n-1}\otimes_k\varepsilon_{n-1}K_{n-1}\to K_{n-1}\) is an isomorphism, and \(K_{n-1}\) is projective. Thus every left module has projective dimension at most \(n-1\) (for \(n=1\), every module is projective). For right modules, apply the argument to the opposite algebra, which is directed with respect to the idempotents in reverse order. The algebra \(B\) is directed with respect to \(\mathbf e_0,\mathbf e_1,\mathbf e_2\): the only paths from a vertex to itself are trivial, and no path goes from \(i\) to \(j<i\). \(\square\)

So \(\operatorname{Ext}^q_B(M,M')=0\) for all \(B\)-modules \(M,M'\) and all \(q\ge3\) (core course [Category Theory and Homological Methods](https://kokunoyumeto.github.io/program-matematika-indonesia/en/#course-D80); [Li, *Methods of Algebra*, Volume 2, Proposition 4.9.6 and Definition–Proposition 4.9.7](https://github.com/KokunoYumeto/methods-of-algebra-volume-2-en)).

## 3. Modules from R-modules

A left \(B\)-module \(M\) is the same as three vector spaces \(\mathbf e_0M,\mathbf e_1M,\mathbf e_2M\) with linear maps for the arrows that satisfy the relations (2.1); the action of a path is the composite of its arrows.

**Definition 3.1.** For a left \(R\)-module \(Y\), let \(M(Y)\) be the \(B\)-module with \(\mathbf e_iM(Y)=Y\) for \(i=0,1,2\), with \(s\) and \(s'\) acting as the identity, and with \(a_h\) and \(a'_h\) acting as \(x_h\).

The relations hold: \(a'_hs\) and \(s'a_h\) both act as \(x_h\), and \(\widetilde p_j\) acts as \(p_j(x_1,\dots,x_d)\), which is \(0\) on \(Y\), because the homogenization replaces \(x_hx_l\), \(x_h\) and \(1\) by paths acting as \(x_hx_l\), \(x_h\) and the identity. A map of \(R\)-modules gives the same map at the three vertices, a \(B\)-module map; so \(M\) is an additive functor, faithful and exact, with \(\dim M(Y)=3\dim Y\). Conversely, a \(B\)-module \(N\) in which \(s\) and \(s'\) act invertibly carries an \(R\)-action on \(\mathbf e_0N\): \(x_h\) acts as \(s^{-1}a_h\), and the relations of \(R\) hold because
\[
\begin{gathered}
(s's)^{-1}a'_ha_l=s^{-1}a_hs^{-1}a_l,\\
(s's)^{-1}s'a_h=s^{-1}a_h,
\end{gathered}
\]
using \(a'_h=s'a_hs^{-1}\). The same computation in a category of complexes is the starting point of the next lesson.

## 4. Exercises

**Exercise 4.1** (easy). Take \(R=k[x]/(x^2-1)\), presented by one generator and the relation \(x^2-1\). Write down the quiver and the relations of \(B\), and compute \(\dim B\).

**Exercise 4.2** (easy). Show that the path algebra of the quiver \(0\to1\to2\) (one arrow each) modulo the path of length two has global dimension exactly two, so the bound of Proposition 2.2 can be attained.

**Exercise 4.3** (medium). Show that the functor \(M\) of Definition 3.1 is full: every \(B\)-module map \(M(Y)\to M(Y')\) has the form \(M(f)\) for an \(R\)-module map \(f\).

**Exercise 4.4** (medium). Show that an algebra \(k\langle x_1,\dots,x_d\rangle/I\), with \(I\) generated by homogeneous polynomials of degree two, has no idempotents other than \(0\) and \(1\). Conclude that \(k\times k\) has no such presentation, which is why Lemma 1.1 keeps constant and linear terms.

## 5. Solutions

**4.1.** The quiver has arrows \(s,a_1\colon0\to1\) and \(s',a'_1\colon1\to2\). The relations are \(a'_1s-s'a_1\) and the homogenization \(a'_1a_1-s's\) of \(x^2-1\). The path algebra has dimension \(3+4+4=11\), and the two relations are independent combinations of the four paths of length two, so \(\dim B=9\).

**4.2.** Let \(\Lambda\) be this algebra, with arrows \(c_1\colon0\to1\) and \(c_2\colon1\to2\). The left projective \(\Lambda\mathbf e_0\) has basis \(\mathbf e_0\) and \(c_1\), since the path \(c_2c_1\) is zero. The simple module \(T_0\) at vertex \(0\) has the minimal resolution \(0\to\Lambda\mathbf e_2\to\Lambda\mathbf e_1\to\Lambda\mathbf e_0\to T_0\to0\): the kernel of \(\Lambda\mathbf e_0\to T_0\) is spanned by \(c_1\) and isomorphic to the simple module at vertex \(1\), whose projective cover \(\Lambda\mathbf e_1\) (basis \(\mathbf e_1,c_2\)) has kernel spanned by \(c_2\), isomorphic to \(\Lambda\mathbf e_2=k\mathbf e_2\). The resolution is minimal with three nonzero terms, so \(\operatorname{pd}T_0=2\) by Theorem 4.3 of [Finitistic dimensions and minimal resolutions](finitistic-dimensions-and-minimal-resolutions.md). With Proposition 2.2, the global dimension is exactly two.

**4.3.** A \(B\)-module map \(g\colon M(Y)\to M(Y')\) consists of linear maps \(g_0,g_1,g_2\colon Y\to Y'\) commuting with the arrows. Commuting with \(s\) and \(s'\), which act as identities, gives \(g_0=g_1=g_2=:f\). Commuting with \(a_h\) gives \(fx_h=x_hf\), so \(f\) is \(R\)-linear, and \(g=M(f)\).

**4.4.** The ideal \(I\) is spanned by homogeneous elements, so \(\Lambda=k\langle x_1,\dots,x_d\rangle/I\) is graded by degree, with \(\Lambda_0=k\). Write an idempotent as \(e=e_0+e_+\) with \(e_0\in k\) and \(e_+\) of positive degree. Comparing degree-zero parts of \(e^2=e\) gives \(e_0\in\{0,1\}\). If \(e_0=0\), then \(e=e^n\) has no components of degree less than \(n\), for every \(n\); since \(e\) has finitely many homogeneous components, \(e=0\). If \(e_0=1\), apply this to the idempotent \(1-e\). The algebra \(k\times k\) has the idempotent \((1,0)\), so it has no such presentation; it is \(k[x]/(x^2-1)\) with \(x=(1,-1)\), a presentation with a constant term, which the homogenization records with the arrows \(s\) and \(s'\).

## References

- [OpenAI-FD] OpenAI, *An algebra of infinite little finitistic dimension*, OpenAI Math Release preprint, 23 September 2026. https://github.com/openai/math/tree/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026
- [NRS] A. Neeman, A. Ranicki and A. Schofield, *Representations of algebras as universal localizations*, Mathematical Proceedings of the Cambridge Philosophical Society 136 (2004); arXiv:math/0205034. https://arxiv.org/abs/math/0205034
- [Li] W.-W. Li, *Methods of Algebra*, Volume 2, English edition. https://github.com/KokunoYumeto/methods-of-algebra-volume-2-en