A note of the Cartan determinant conjecture

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Introduction

Finite projective resolutions express simple modules as integer combinations of indecomposable projective modules. In 1954, Eilenberg established the resulting unimodularity of the Cartan matrix for finite-dimensional algebras of finite global dimension [3]. The Cartan determinant conjecture asks whether its sign is always positive. Zacharia proved this for Artin algebras of global dimension at most two in 1983 [6].

An Artin algebra is an associative unital algebra that is finite as a module over a commutative Artin ring. Let $A$ be a nonzero Artin algebra. Choose representatives $S_1,\ldots,S_n$ of its simple right modules and their indecomposable projective covers $P_1,\ldots,P_n$. Its integral Cartan matrix is

where $[P_j:S_i]$ is composition multiplicity. Write gl. dim $A$ for global dimension. If it is finite, choose a finite projective resolution of each $S_j$ and write its degree-$q$ term as $\bigoplus_i P_i^{a_{iqj}}$ for integers $q\geq0$, with nonnegative multiplicities $a_{iqj}$ that vanish beyond the resolution. Taking alternating composition multiplicities gives $C_AQ=I$, where $Q_{ij}=\sum_{q\geq0}(-1)^qa_{iqj}$ and $I$ is the $n\times n$ identity matrix. Hence $\det C_A=\pm1$.

Two reduction methods compare Cartan determinants with those of an idempotent corner or an idempotent quotient. Burgess–Fuller develop matrix reduction using projective resolutions [1 Lemma 1.5]. Ingalls–Paquette compare an algebra with a corner under a finite-global-dimension hypothesis on the Yoneda algebra associated with the deleted idempotent [4 Proposition 6.1]. For representation-finite Artin algebras, Chen–Guo prove the conjecture [2 Theorem 1.1].

2020 Mathematics Subject Classification. Primary 16E10; Secondary 16G10, 16G60.

The proof filters the category of finite projective right $R$-modules by the projective dimensions of the evaluated modules $Pe$. Each positive layer, and a modified initial layer, has an ordinary object quotient whose endomorphism algebra has global dimension at most two. Integer Cartan column operations give a factorization of $\det C_R$ into $\det C_A$ and the determinants of these auxiliary algebras. Zacharia’s theorem determines their signs.

A finitely generated right $A$-module is torsionless if it embeds in a finitely generated projective module. The algebra $A$ is torsionless-finite if there are only finitely many isomorphism classes of indecomposable torsionless right modules; see Ringel [5]. The category of torsionless modules is closed under kernels. Its additive generator therefore gives an endomorphism algebra of global dimension at most two, to which Theorem A applies.

Kernels and Cartan factorization

Fix a commutative Artin ring $K$. All algebras are finite $K$-algebras, and modules are finitely generated right modules unless specified otherwise. Write $\operatorname{mod} B$ for their category, $\operatorname{proj} B$ for its full subcategory of projective modules, $\operatorname{pd}_B N$ for projective dimension, and $\operatorname{gl.dim} B$ for global dimension on all right modules. For integers $q \ge0$, $\operatorname{Ext}^{q}_{B}(-,-)$ is the degree-$q$ derived functor of $\operatorname{Hom}_{B}(-,-)$. We put $\operatorname{pd}_B 0 = -\infty$. The symbol $\ell_K V$ denotes the composition length of a finite $K$-module $V$, and $\operatorname{rad} B$ denotes the Jacobson radical of a ring $B$. Endomorphism rings are multiplied by composition, with the rightmost map applied first; $\operatorname{Hom}_{\mathcal{C}}(X,Y)$ denotes the morphisms from $X$ to $Y$ in a category $\mathcal{C}$.

For an object $W$ in an additive category, $\operatorname{add} W$ is the full subcategory of direct summands of finite direct sums of $W$. For a collection of objects, $\operatorname{add}$ has the same closure meaning. All additive subcategories below are closed under isomorphisms and direct summands. We use $K$-linear, idempotent-complete Krull–Schmidt categories: every idempotent splits, and every object is a finite direct sum of indecomposable objects with local endomorphism rings, uniquely up to isomorphism and permutation. A local ring has a unique maximal right ideal, its Jacobson radical. The Hom spaces have finite $K$-length, and there are finitely many indecomposable isomorphism classes. A basic additive generator is the direct sum of one representative of each such class. The zero category is allowed as an auxiliary category; its Cartan determinant is the empty determinant $1$.

If $\mathcal{C} = \operatorname{add} W$, the functor $\operatorname{Hom}_{\mathcal{C}}(W,-)$ sends $Y$ to $\operatorname{Hom}_{\mathcal{C}}(W,Y)$ and a map $f:Y\to Y'$ to $t\mapsto ft$. Its values are right $\operatorname{End}_{\mathcal{C}}(W)$-modules by $t\cdot b = tb$. It identifies $\mathcal{C}$ with $\operatorname{proj}\operatorname{End}_{\mathcal{C}}(W)$. Indeed, it is fully faithful on finite sums of $W$, hence on their direct summands, and every finite projective module is such a summand of a finite free module. A categorical kernel of $f:X\to Y$ is a morphism $u:T\to X$ with $fu=0$ such that every $v:V\to X$ with $fv=0$ factors uniquely through $u$.

Proof. Put $B=\operatorname{End}_{\mathcal{C}}(W)$. A finite right $B$-module $N$ has a projective presentation induced by a morphism $f:X_1\to X_0$ in $\mathcal{C}$. The kernel property gives an exact sequence

Its first three modules are projective, so $\operatorname{pd}_B N\le2$. In particular, every simple right $B$-module has projective dimension at most two. This also bounds arbitrary modules: the radical of $B$ is nilpotent, every module has a finite filtration by its radical powers with semisimple factors, and direct sums of projective resolutions of simple modules resolve those factors in length at most two. Successive extensions preserve this bound. ∎

Cartan entries can be computed using lengths over $K$. Let $X_1,\ldots,X_n$ represent the indecomposable isomorphism classes of $\mathcal{C}$, take $W=\bigoplus_i X_i$, and let $L_i$ be the simple top of $\operatorname{Hom}(W,X_i)$ over $B=\operatorname{End}(W)$. Define the residual division ring

Let $\varepsilon_i\in B$ be the projection of $W$ onto $X_i$ followed by inclusion. The right projective $\operatorname{Hom}(W,X_j)$ is $\varepsilon_jB$; its simple top is $\varepsilon_jB/\varepsilon_j\operatorname{rad}B$. The corner $\varepsilon_iB\varepsilon_i$ is $\operatorname{End}(X_i)$, and the quotient $B/\operatorname{rad}B$ is the product of the division rings $\Delta_i$. Thus $N\mapsto N\varepsilon_i$ is an exact component functor, with value $\Delta_i$ on $L_i$ and zero on the other simples. Applying it to a composition series, and using $\operatorname{Hom}(W,X_j)\varepsilon_i\cong\operatorname{Hom}(X_i,X_j)$, gives

Thus these ratios, with sources indexing rows and targets indexing columns, give $C_B$. The same formula on a subset of the objects gives the Cartan matrix of their endomorphism algebra. Replacing an additive generator by its basic generator is a Morita equivalence and preserves the Cartan matrix up to simultaneous relabelling.

Let $\mathcal{L}\subseteq\mathcal{C}$ be an additive subcategory. A right $\mathcal{L}$-approximation of $Y$ is a morphism $p_Y:U_Y\to Y$, with $U_Y\in\mathcal{L}$, through which every morphism from an object of $\mathcal{L}$ to $Y$ factors. Such approximations exist here: take generators over $K$ of the finitely many spaces $\operatorname{Hom}(V,Y)$, where $V$ runs over the indecomposable objects of $\mathcal{L}$, and combine their maps into a morphism from a finite direct sum of these objects.

The object quotient $\mathcal{Q}=\mathcal{C}/[\mathcal{L}]$ has the same objects as $\mathcal{C}$, with

where $[\mathcal{L}](X,Y)$ consists of maps factoring through objects of $\mathcal{L}$. Composition is induced from $\mathcal{C}$. The class of a map $f$ will be denoted by $\overline{f}$. For $f:X\to Y$ and $p:U\to Y$, the map $(f,p):X\oplus U\to Y$ has restrictions $f$ and $p$ on the two summands.

Proof. Fix $f:X\to Y$, and write $T=\ker(f,p_Y)$. Let $h:T\to X$ and $v:T\to U_Y$ be the components of its kernel morphism. Then $fh=-p_Yv$, so $\overline{f}\overline{h}=0$.

For a map $g:V\to X$ with $\overline{f}\overline{g}=0$, write $fg=ab$ with $b:V\to V'$ and $a:V'\to Y$, where $V'\in\mathcal{L}$. By approximation, $a=p_Ys$ for some $s:V'\to U_Y$. Thus $(g,-sb):V\to X\oplus U_Y$ factors through $T$, giving a factorization of $\overline{g}$ through $\overline{h}$.

For uniqueness, suppose $k:V\to T$ and $hk=cb$ factor through $V'\in\mathcal{L}$, with $c:V'\to X$. Lift $fc$ through $p_Y$, writing $fc=p_Yt$. The morphism $(c,-t)$ induces $j:V'\to T$, and $h(k-jb)=0$. Let $\kappa:\ker p_Y\to U_Y$ be the kernel map. The pair $(0,\kappa)$ induces $i:\ker p_Y\to T$. If $w:V\to T$ satisfies $hw=0$, then $p_Yvw=-fhw=0$. There is a unique $s:V\to\ker p_Y$ with $\kappa s=vw$. The pairs $(h,v)w$ and $(h,v)js$ are equal; hence the kernel morphism $(h,v)$ is monic, so $w=0$. Thus $i$ is the kernel of $h$. In particular, $k-jb$ factors through $\ker p_Y\in\mathcal{L}$. Both $jb$ and $k-jb$ vanish in $\mathcal{Q}$, so $\overline{k}=0$. Hence $\overline{h}$ is a categorical kernel of $\overline{f}$.

Decompose each object using its indecomposable summands in $\mathcal{C}$. The summands in $\mathcal{L}$ become zero in $\mathcal{Q}$. For a summand $X$ outside $\mathcal{L}$, the ideal $[\mathcal{L}](X,X)$ contains no invertible map, since an invertible factorization would make $X$ a summand of an object of $\mathcal{L}$. It is therefore contained in the radical of the local ring $\operatorname{End}_{\mathcal{C}}(X)$. The quotient endomorphism ring remains local, with the same residual division ring. Distinct surviving summands remain nonisomorphic: mutually inverse quotient maps lift to maps whose composites differ from identities by radical elements, and an invertible composite makes one indecomposable object a summand of the other.

Let $V$ be the sum of the surviving representatives and set $D=\operatorname{End}_{\mathcal{Q}}(V)$. The projections onto these summands give a complete set of pairwise nonisomorphic primitive idempotents in the Artin algebra $D$. Thus all finite projective $D$-modules are represented by direct sums of those objects. Full faithfulness of $\operatorname{Hom}_{\mathcal{Q}}(V,-)$ on these sums gives $\mathcal{Q}\simeq\operatorname{proj}D$, including idempotent completeness. The kernel assertion and Lemma 2.1 imply $\operatorname{gl.dim}D\leq2$.

For the determinant, let $W$ be the basic generator of $\mathcal{C}$ and put $P_Y=\operatorname{Hom}_{\mathcal{C}}(W,Y)$. Let $I_Y=[\mathcal{L}](W,Y)$, the image of $\operatorname{Hom}(W,p_Y)$. The actual kernel gives

In the group of finite $\Gamma$-modules modulo short exact sequences, write $[N]$ for the class of a module $N$. The sequence yields

Order the indecomposable objects with those of $\mathcal{L}$ first. Keep the corresponding Cartan columns and replace each remaining column by the displayed combination. Only columns indexed by $\mathcal{L}$ are added to it, so the transformation has determinant one. The new high columns vanish on the low rows, since every map from a low object factors through $\mathcal{L}$. The low diagonal block is $C_{\Gamma_{\mathcal{L}}}$ by (1). The high diagonal block is $C_D$: the surviving residual division rings are unchanged, and its component spaces are exactly $\mathcal{Q}(X_i,X_j)$. The transformed matrix is therefore

which proves (2).

The filtration of a corner

Proof of Theorem A. Put $A=eRe$ and $a=\operatorname{gl.dim}A<\infty$. For $\mathcal{E}=\operatorname{proj}R$, evaluation at $e$ is the exact functor

It preserves kernels by restriction to the $e$-components. If a map is surjective and $y=ye$ is in its target, choose a preimage $x$; then $xe$ maps to $y$. Hence $F$ also preserves surjections and is exact.

The category $\mathcal{E}$ is closed under module kernels. Indeed, for a map $g:P_1\longrightarrow P_0$ between projectives, put $T=\ker g$. For any right $R$-module $V$, the short exact sequences with middle terms $P_1$ and $P_0$ give

Thus $T$ is projective; it is finitely generated because $R$ is Artinian.

For integers $0\leq m\leq a$, define

These are finite additive subcategories, $\mathcal{P}\subseteq\mathcal{E}_0$, and $\mathcal{E}_a=\mathcal{E}$. Let $\Gamma_m$ be the endomorphism algebra of a basic generator of $\mathcal{E}_m$.

For $Q\in\operatorname{proj}A$, the induced module $Q\otimes_A eR$ belongs to $\mathcal{P}$: a splitting of $Q$ from a finite free $A$-module tensors to a splitting from a finite sum of $eR$. For $Y\in\operatorname{proj}R$ we have

The first bijection sends $u$ to $q \mapsto u(q \otimes e)$; its inverse sends $v$ to $q \otimes er \mapsto v(q)r$. In the second isomorphism, $q \otimes ere$ maps to $qere$. Thus $F$ restricts to an equivalence $\mathcal{P} \simeq\operatorname{proj} A$.

Positive layers. Fix $1 \le m \le a$ and $Y \in\mathcal{E}_m$. Choose a right $\mathcal{E}_{m-1}$-approximation $p_Y : U_Y \to Y$. Since $\mathcal{P} \subseteq\mathcal{E}_{m-1}$, (3) lifts a projective cover of $F Y$ to a map from an object of $\mathcal{P}$, which then factors through $p_Y$. Hence $F p_Y$ is surjective.

The module kernel $K_Y = \ker_R p_Y$ belongs to $\operatorname{proj} R$, and exactness of $F$ gives

The projective-dimension bound for this sequence is

For every right $A$-module $V$ and integer $q \ge1$, the long exact Ext sequence has a segment

The outer terms vanish for $q > m - 1$, so $K_Y \in\mathcal{E}_{m-1}$.

For $f : X \to Y$ in $\mathcal{E}_m$, the kernel $T = \ker_R(f,p_Y)$ is again projective over $R$. Its evaluation fits into

The last map is surjective, and the same bound gives $\operatorname{pd}_A F T \le m$. Thus $T \in\mathcal{E}_m$. Lemma 2.2 now applies to $\mathcal{E}_{m-1} \subseteq\mathcal{E}_m$. Write $D_m$ for the endomorphism algebra of the surviving basic generator of $\mathcal{E}_m/\mathcal{E}_{m-1}$. We obtain

The initial layer. Let

The kernel of any map between its objects is projective over $R$ and evaluates to zero. Lemma 2.1 implies that the endomorphism algebra $\mathcal{Z}$ of a basic generator of $\mathcal{Z}$ has global dimension at most two.

Set $\mathcal{L}_0 = \operatorname{add}(\mathcal{P} \cup\mathcal{Z})$ inside $\mathcal{E}_0$. The two subcategories have no common nonzero indecomposable object, since $F$ is an equivalence on $\mathcal{P}$ and vanishes on $\mathcal{Z}$. Moreover,

Therefore the Cartan matrix of the endomorphism algebra of a basic generator of $\mathcal{L}_0$ is block triangular, with diagonal blocks $C_A$ and $C_{\mathcal{Z}}$. Here the first block uses $\operatorname{End}_R(eR) \cong eRe$, where $x \in eRe$ acts by left multiplication, and Morita invariance removes repeated summands of $eR$. Its determinant is $\det C_A \det C_{\mathcal{Z}}$.

For $Y \in\mathcal{E}_0$, the module $F Y$ is projective over $A$. Define

Every map $Q \otimes_A eR \to Y$ corresponds under (3) to $Q \to F Y$ and therefore factors through $p_Y$ by tensoring that map with $eR$. Also $F p_Y$ is an isomorphism. Choose a right $\mathcal{Z}$-approximation $z_Y : Z_Y \to Y$ and form

This is a right $\mathcal{L}_0$-approximation. Its kernel is projective over $R$ and evaluates to zero because $F q_Y$ is an isomorphism. Hence $\ker q_Y \in\mathcal{Z} \subseteq\mathcal{L}_0$.

For $f : X \to Y$ in $\mathcal{E}_0$, the kernel of $(f,q_Y)$ is projective over $R$. After evaluation, projection onto the first component identifies the kernel of $(Ff,Fq_Y)$ with $FX$; its inverse is $x \mapsto(x,-(Fq_Y)^{-1}Ff(x))$. It is therefore projective over $A$, so $\ker(f,q_Y) \in\mathcal{E}_0$. Applying Lemma 2.2 to $\mathcal{L}_0 \subseteq\mathcal{E}_0$ gives an Artin algebra $D_0$ with

Since $\mathcal{E}_a=\operatorname{proj} R$, its basic generator has endomorphism algebra Morita equivalent to $R$. Multiplying (5) and (6) yields

Every nonzero algebra among $R,Z,D_0,\ldots,D_a$ has global dimension at most two. Zacharia’s theorem [6] gives determinant one for each of them. Zero categories contribute the empty determinant one. Thus (7) gives $\det C_A=1$. □

Finite submodule categories

For a finite right module $M$ over an Artin algebra $B$, let $\operatorname{Sub}(M)$ be the full subcategory of finite modules isomorphic to submodules of finite direct sums of $M$. It is closed under direct sums, direct summands, and module kernels: a kernel is a submodule of its source, which already embeds in a finite sum of $M$. We call $\operatorname{Sub}(M)$ finite if it has finitely many indecomposable isomorphism classes.

Proof. Choose an additive generator $W$ of $\operatorname{Sub}(M)$ containing $M$ as a direct summand; adjoining $M$ to a basic generator achieves this. The category $\operatorname{add} W=\operatorname{Sub}(M)$ has its actual module kernels, so Lemma 2.1 gives $\operatorname{gl.dim}\operatorname{End}_B(W)\le2$. Let $e$ be the projection of $W$ onto $M$ followed by inclusion. Extension by zero on the complementary summand identifies

Apply Theorem A. □

Proof of Theorem B. The torsionless right $A$-modules are precisely $\operatorname{Sub}(A_A)$: every finite projective module is a summand of a finite sum of copies of $A$, and every such sum is projective. Torsionless-finiteness makes this category finite. Corollary 4.1, applied to $M=A_A$, gives the result because left multiplication identifies $\operatorname{End}_A(A_A)$ with $A$. □

AI contributions

The mathematical arguments were developed with AI assistance in ChatGPT. Codex assisted in checking the arguments and cited inputs and in preparing and revising this manuscript.

Zhili College, Tsinghua University, Beijing 100084, China

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