Algebroids

There is a canonical sheaf of algebras $\mathfrak{A}\rightarrow \mathfrak{M}$ over $\mathfrak{M}$ . The set of sections of $\mathfrak{A}$ over the $S$ -valued point $x$ of $\mathfrak{M}$ is the $\mathscr{O}(S)$ -algebra $\mathfrak{A}_{x}=\operatorname{End}(x)$ . For $\mathfrak{M}=\mathfrak{Vect}$ , the point $x$ is a vector bundle over $S$ , and $\operatorname{End}(x)$ is the $\mathscr{O}(S)$ -module of endomorphisms of $x$ .

There is also a canonical isomorphism of group sheaves $\mathfrak{A}^{\times }\rightarrow I_{\mathfrak{M}}$ over $\mathfrak{M}$ , where $I_{\mathfrak{M}}$ is the inertia stack of $\mathfrak{M}$ . (Recall that the sections of $I_{\mathfrak{M}}$ over the $S$ -valued point $x$ of $\mathfrak{M}$ are the automorphisms of $x$ , in other words, the units in the algebra of endomorphisms.)

We call a triple $(X,A_{X},\unicode[STIX]{x1D704})$ an algebroid (see Definition 1.38 and Remark 1.44) if $X$ is an algebraic stack, $A_{X}\rightarrow X$ is a representable sheaf of finite $\mathscr{O}$ -algebras over $X$ (or finite type algebras, as we call them; see Definition 1.18), and $\unicode[STIX]{x1D704}:A_{X}^{\times }\rightarrow I_{X}$ is an open immersion of relative group schemes over $X$ , making the canonical diagram

commute.

So, $\mathfrak{M}$ with its canonical sheaf of algebras $\mathfrak{A}$ is an example of an algebroid. In this case, $\unicode[STIX]{x1D704}$ is an isomorphism, yielding what we call a strict algebroid.

Algebroids are generalizations of linear algebraic stacks (they are linear over their coarse moduli spaces, if they are strict). They are slightly more flexible. For example, schemes can be considered as algebroids in a canonical way. If the algebraic stack $X$ is the base of an algebroid, then the connected component $I_{X}^{\circ }$ of its inertia stack $I_{X}$ is the group of units in an algebra. This is the main significance of algebroids for us.

Just like algebraic stacks, algebroids form a 2-category, in which 2-fibred products exist. Whenever $(X,A_{X})$ is an algebroid, and $Y\rightarrow X$ is an inert morphism of algebraic stacks, i.e., $I_{Y}^{\circ }=I_{X}^{\circ }|_{Y}$ (see Definition 1.47), the stack $Y$ is endowed with a natural structure of an algebroid via $A_{X}|_{Y}$ . Examples of inert morphisms include monomorphisms and projections $Z\times X\rightarrow X$ , for schemes $Z$ . A locally closed immersion of algebroids $(Y,A_{Y})\rightarrow (X,A_{X})$ is a morphism where $Y\rightarrow X$ is a locally closed immersion of algebraic stacks, such that $A_{Y}=A_{X}|_{Y}$ . Every scheme $Z$ is an algebroid via the definition $A_{Z}=0_{Z}$ .

A key observation is that if $(X,A)$ is an algebroid, then $(I_{X},I_{A})$ is another algebroid. In fact, $I_{A}$ , the inertia stack of the stack $A$ (the total ‘space’ of the sheaf of algebras $A$ ), is equal to the subalgebra of $A|_{I_{X}}$ fixed under its tautological automorphism. We call $(I_{X},I_{A})$ the inertia algebroid of $(X,A)$ . It comes with a canonical morphism to $(X,A)$ . There is also a semi-simple connected version of the algebroid inertia, denoted by $I^{\circ ,\text{ss}}$ .

The motivic Hall algebra

We define stack functions to be representable morphisms of algebroids $(X,A_{X})\rightarrow (\mathfrak{M},\mathfrak{A})$ , where $X$ is of finite type. The Hall algebra  $K(\mathfrak{M})$ of $\mathfrak{M}$ is the $\mathbb{Q}$ -vector space on the isomorphism classes of stack functions modulo the scissor relations relative  $\mathfrak{M}$ :

$$\begin{eqnarray}[X\rightarrow \mathfrak{M}]=[Z\rightarrow X\rightarrow \mathfrak{M}]+[X\setminus Z\rightarrow X\rightarrow \mathfrak{M}]\end{eqnarray}$$

and the inert bundle relations relative  $\mathfrak{M}$ :

$$\begin{eqnarray}[Y\rightarrow X\rightarrow \mathfrak{M}]=[F\times X\rightarrow X\rightarrow \mathfrak{M}].\end{eqnarray}$$

Here, $[X\rightarrow \mathfrak{M}]$ denotes the Hall algebra element defined by a stack function with base $X$ . Also, $Z\rightarrow X$ is a closed immersion of algebroids, with open complement $X\setminus Z$ , and $Y\rightarrow X$ is an inert fibre bundle (Definition 1.53) with special structure group and fibre $F$ , all endowed with their canonical algebroid structure. (Examples of inert fibre bundles are étale locally trivial ones.)

We have the following structures on the Hall algebra.

1. (i) Module structure. Let $K(\text{Var})$ denote the Grothendieck ring of varieties over $k$ . We denote the motivic weight of the affine line by $q=[\mathbb{A}^{1}]\in K(\text{Var})$ . By $[Z]\cdot [X\rightarrow \mathfrak{M}]=[Z\times X\rightarrow \mathfrak{M}]$ we define a $K(\text{Var})$ -module structure on $K(\mathfrak{M})$ .

2. (ii) Multiplication. By $[X\rightarrow \mathfrak{M}]\cdot [Y\rightarrow \mathfrak{M}]=[X\times Y\rightarrow \mathfrak{M}\times \mathfrak{M}\stackrel{\oplus }{\longrightarrow }\mathfrak{M}]$ we define a commutative multiplication on $K(\mathfrak{M})$ , and $K(\mathfrak{M})$ is a $K(\text{Var})$ -algebra with this multiplication.

3. (iii) Hall product. Using the stack of short exact sequences in $\mathfrak{M}$ , we can define a Hall algebra product $[X\rightarrow \mathfrak{M}]\ast [Y\rightarrow \mathfrak{M}]$ on $K(\mathfrak{M})$ . For details, see § 3. The module $K(\mathfrak{M})$ is a $K(\text{Var})$ -algebra also with respect to the Hall product.

4. (iv) Unit. The unit with respect to both products is represented by $1=[\operatorname{Spec}k\stackrel{0}{\longrightarrow }\mathfrak{M}]$ .

5. (v) Inertia endomorphism. The algebroid inertia defines an operator $I:K(\mathfrak{M})\rightarrow K(\mathfrak{M})$ via $I[X\rightarrow \mathfrak{M}]=[I_{X}\rightarrow X\rightarrow \mathfrak{M}]$ . This inertia operator is linear over $K(\text{Var})$ and multiplicative, $I(x\cdot y)=I(x)\cdot I(y)$ , with respect to the commutative multiplication. The same facts hold for the connected semi-simple inertia operator $I^{\circ ,\text{ss}}:K(\mathfrak{M})\rightarrow K(\mathfrak{M})$ .

There is also a ‘non-representable’ version of the Hall algebra, where we drop the representability requirement for stack functions and simply define a stack function to be a morphism of algebroids $X\rightarrow \mathfrak{M}$ , with $X$ of finite type. The representable Hall algebra is a subalgebra (with respect to both products) of the non-representable one. Our results on the diagonalizability of the various operators $I$ , $I^{\circ ,\text{ss}}$ , $E_{n}$ hold true also in the non-representable Hall algebra, but the algebraic results on the structure of the Hall algebra need representability. For simplicity, we restrict ourselves therefore to the representable case from the beginning.

Usually, when defining the Hall algebra of $\mathfrak{M}$ , one requires the bundle relations also for non-inert morphisms. The connected inertia operator does not respect such relations, and we therefore do not include them.

Example

A stack function $X\rightarrow \mathfrak{Vect}$ is the same thing as an algebroid $(X,A)$ together with a faithful representation, i.e., a vector bundle $V$ over $X$ together with a monomorphism of algebras $A\rightarrow \operatorname{End}(V)$ , making the canonical diagram

commute.

Examples of stack functions with values in $\mathfrak{Vect}$ include subalgebras $A\subset M_{n\times n}$ . (The induced morphism of algebroids is $(BA^{\times },A^{\times }\backslash A)\rightarrow (B\operatorname{GL}_{n},\operatorname{GL}_{n}\backslash M_{n\times n})$ .) The elements of $K(\mathfrak{Vect})$ defined by $A\subset M_{n\times n}$ and $B\subset M_{n\times n}$ are equal if and only if $A$ and $B$ are conjugate in $M_{n\times n}$ .

The subalgebra $U$ of $K(\mathfrak{Vect})$ with respect to the Hall product, generated by the $[n]=[B\operatorname{GL}_{n}\rightarrow \mathfrak{Vect}]$ , is free on these elements $[n]$ , for $n>0$ , as a unitary $\mathbb{Q}$ -algebra. In the literature, $U$ is known as the Hopf algebra of non-commutative symmetric functions; see [Reference CartierCar07, Example 4.1(F)].

(If we add the (non-inert) vector bundle relations relative to $\mathfrak{Vect}$ , see, e.g., [Reference BridgelandBri12], we get

$$\begin{eqnarray}[\unicode[STIX]{x1D706}_{1}]\ast \ldots \ast [\unicode[STIX]{x1D706}_{r}]=\frac{[\operatorname{GL}_{n}]}{[P(\unicode[STIX]{x1D706})]}[n]=\binom{n}{\unicode[STIX]{x1D706}_{1}\ldots \unicode[STIX]{x1D706}_{n}}_{q}[n].\end{eqnarray}$$

Here $n=\sum \unicode[STIX]{x1D706}_{i}$ , and $\binom{n}{\unicode[STIX]{x1D706}_{1}\ldots \unicode[STIX]{x1D706}_{n}}_{q}$ denotes the $q$ -deformed multinomial coefficient, which gives the motivic weight of the flag variety of type $\unicode[STIX]{x1D706}$ . We have also denoted the parabolic subgroup of $\operatorname{GL}_{n}$ of type $\unicode[STIX]{x1D706}$ by $P(\unicode[STIX]{x1D706})$ . Hence, the $\mathbb{Q}$ -algebra obtained by dividing $U$ by the vector bundle relations is the commutative polynomial algebra over $\mathbb{Q}$ , on the symbols $[1],[2],[3],\ldots$  . This is the Hopf algebra of symmetric functions.)

The spectrum of semi-simple inertia

The main point of this work is to study the spectral theory of the semi-simple inertia operator $I^{\circ ,\text{ss}}$ on $K(\mathfrak{M})$ .

Before announcing our results, let us do a few sample calculations. They contain some of the central ideas of this paper. Only strict algebroids will occur, so we write $I^{\text{ss}}$ instead of $I^{\circ ,\text{ss}}$ .

We consider $\mathfrak{M}=\mathfrak{Vect}$ . The linear stack of line bundles defines the stack function $[B\operatorname{GL}_{1}\rightarrow \mathfrak{Vect}]\in K(\mathfrak{Vect})$ . We have

$$\begin{eqnarray}\displaystyle I^{\text{ss}}[B\operatorname{GL}_{1}\rightarrow \mathfrak{Vect}] & = & \displaystyle [\operatorname{GL}_{1}^{\text{ss}}\times \,B\operatorname{GL}_{1}\rightarrow \mathfrak{Vect}]\nonumber\\ \displaystyle & = & \displaystyle [\operatorname{GL}_{1}\times \,B\operatorname{GL}_{1}\rightarrow \mathfrak{Vect}]\nonumber\\ \displaystyle & = & \displaystyle (q-1)[B\operatorname{GL}_{1}\rightarrow \mathfrak{Vect}].\nonumber\end{eqnarray}$$

This proves that $[B\operatorname{GL}_{1}\rightarrow \mathfrak{Vect}]$ is an eigenvector of $I^{\text{ss}}$ , with corresponding eigenvalue $(q-1)\in K(\text{Var})$ .

Because $I^{\text{ss}}$ is an algebra morphism with respect to the commutative product, it immediately follows that every $(q-1)^{r}$ for $r\geqslant 0$ is an eigenvalue of $I^{\text{ss}}$ , with corresponding eigenvector $[B\operatorname{GL}_{1}^{r}\rightarrow B\operatorname{GL}_{n}\rightarrow \mathfrak{Vect}]\in K(\mathfrak{Vect})$ .

These are not the only eigenvalues of $I^{\text{ss}}$ . In fact, let us consider the stack function of all rank 2 vector bundles $[B\operatorname{GL}_{2}\rightarrow \mathfrak{Vect}]$ . Recall that the inertia stack of $B\operatorname{GL}_{2}$ is the quotient stack $\operatorname{GL}_{2}/_{\text{ad}}\operatorname{GL}_{2}$ , where $\operatorname{GL}_{2}$ acts on itself by the adjoint action. The semi-simple part of $\operatorname{GL}_{2}$ decomposes as $\operatorname{GL}_{2}^{\text{eq}}\sqcup \operatorname{GL}_{2}^{\text{neq}}$ , according to whether the two eigenvalues of an element of $\operatorname{GL}_{2}$ are equal or not equal. By the scissor relations, we have

$$\begin{eqnarray}\displaystyle I^{\text{ss}}[B\operatorname{GL}_{2}] & = & \displaystyle [\operatorname{GL}_{2}^{\text{eq}}/_{\text{ad}}\operatorname{GL}_{2}]+[\operatorname{GL}_{2}^{\text{neq}}/_{\text{ad}}\operatorname{GL}_{2}]\nonumber\\ \displaystyle & = & \displaystyle [\unicode[STIX]{x1D6E5}\times B\operatorname{GL}_{2}]+[T^{\ast }/_{\text{ad}}N]\nonumber\\ \displaystyle & = & \displaystyle (q-1)[B\operatorname{GL}_{2}]+x.\nonumber\end{eqnarray}$$

Here, $\unicode[STIX]{x1D6E5}$ is the one-parameter subgroup of scalar matrices, and $T$ is the maximal torus of diagonal matrices in $\operatorname{GL}_{2}$ . Further notation: $T^{\ast }=T\setminus \unicode[STIX]{x1D6E5}$ , $N$ is the normalizer of $T$ in $\operatorname{GL}_{2}$ , and $x=[T^{\ast }/_{\text{ad}}N]$ .

Next, we calculate $I^{\text{ss}}x$ . In fact, we have $I_{T^{\ast }/N}^{\text{ss}}=I_{T^{\ast }/N}=(T^{\ast }\times T)/N$ , by the ‘stabilizer formula’ for the inertia stack of a quotient stack

$$\begin{eqnarray}I_{Y/G}=\{(y,g)\in Y\times G\mid yg=y\}/G.\end{eqnarray}$$

We note that $N=T\rtimes \mathbb{Z}_{2}$ acts on $T^{\ast }\times T$ diagonally, via its quotient $\mathbb{Z}_{2}$ , by swapping the entries of $T$ . We embed $T$ into $\mathbb{A}^{2}$ equivariantly with respect to $\mathbb{Z}_{2}$ and then decompose $\mathbb{A}^{2}$ as $T\sqcup (\operatorname{GL}_{1}\times 0)\sqcup (0\times \operatorname{GL}_{1})\sqcup (0,0)$ . This gives

(1) $$\begin{eqnarray}[(T^{\ast }\times \mathbb{A}^{2})/N]=[(T^{\ast }\times T)/N]+[T^{\ast }\times (\operatorname{GL}_{1}\times \,0\sqcup 0\times \operatorname{GL}_{1})/N]+[T^{\ast }\times (0,0)/N].\end{eqnarray}$$

We have a pullback diagram of algebroids.

It shows that the vector bundle $(T^{\ast }\times \mathbb{A}^{2})/N\rightarrow T^{\ast }/N$ is a pullback of the vector bundle $(T^{\ast }\times \mathbb{A}^{2})/\mathbb{Z}_{2}\rightarrow T^{\ast }/\mathbb{Z}_{2}$ . The latter is a vector bundle over a scheme, and is therefore Zariski-locally trivial by Hilbert’s Theorem 90. The same is then true for any pullback bundle. Hence, we conclude that

$$\begin{eqnarray}[(T^{\ast }\times \mathbb{A}^{2})/N]=q^{2}[T^{\ast }/N],\end{eqnarray}$$

using the scissor relations, or the inert bundle relations. So, from (1), we conclude that

$$\begin{eqnarray}q^{2}x=I^{\text{ss}}x+(q-1)[T^{\ast }/T]+x,\end{eqnarray}$$

which we solve for $I^{\text{ss}}x$ to get

$$\begin{eqnarray}I^{\text{ss}}x=(q^{2}-1)x-(q-1)^{2}(q-2)[BT].\end{eqnarray}$$

We already know that $I^{\text{ss}}[BT]=(q-1)^{2}[BT]$ , and so we conclude that $[B\operatorname{GL}_{2}]$ , $x$ , and $[BT]$ generate an $I^{\text{ss}}$ -invariant subspace of $K(\mathfrak{Vect})$ , and the matrix of $I^{\text{ss}}$ on this subspace is

$$\begin{eqnarray}\left(\begin{array}{@{}ccc@{}}q-1 & 0 & 0\\ 1 & q^{2}-1 & 0\\ 0 & -(q-1)^{2}(q-2) & (q-1)^{2}\end{array}\right).\end{eqnarray}$$

This matrix is lower triangular, with distinct scalars on the diagonal, and is therefore diagonalizable over the field $\mathbb{Q}(q)$ . So, on this subspace, $I^{\text{ss}}$ is diagonalizable, with eigenvalues $(q-1)$ , $(q^{2}-1)$ , and $(q-1)^{2}$ . If we decompose $[B\operatorname{GL}_{2}]$ as a sum of eigenvectors, we get the eigenvectors

$$\begin{eqnarray}\displaystyle \begin{array}{@{}lcl@{}}[B\operatorname{GL}_{2}]-{\displaystyle \frac{1}{q(q-1)}}x-{\displaystyle \frac{1}{q}}[BT]\quad & \text{with eigenvalue}\ & (q-1),\\ {\displaystyle \frac{1}{q(q-1)}}x-{\displaystyle \frac{q-2}{2q}}[BT]\quad & \text{with eigenvalue}\ & (q^{2}-1),\\ {\textstyle \frac{1}{2}}[BT]\quad & \text{with eigenvalue}\ & (q-1)^{2}.\end{array} & & \displaystyle \nonumber\end{eqnarray}$$

A very important observation is that when we add together the eigencomponents whose eigenvalues have the same order of vanishing at $q=1$ , we get coefficients in $\mathbb{Q}$ , instead of $\mathbb{Q}(q)$ . In the above example, we add together the components of $[B\operatorname{GL}_{2}]$ with eigenvalues $(q-1)$ and $(q^{2}-1)$ to obtain $[B\operatorname{GL}_{2}]-\frac{1}{2}[BT]$ .

Another important observation is that diagonalizing $I^{\text{ss}}$ does not, despite appearances, require us to invert $(q-1)$ . In fact, the algebroid $x$ appearing in the above argument is divisible by $(q-1)$ , although the quotient is not a strict algebroid any longer. To see this, consider the action of $\mathbb{G}_{m}$ on $T^{\ast }$ by left multiplication. It commutes with the adjoint action by $N$ . Let us denote the quotient $\mathbb{G}_{m}\backslash T^{\ast }$ by $\widetilde{T}^{\ast }$ . We can endow the stack $\widetilde{T}^{\ast }/N$ with the structure of an algebroid, making the quotient map $T^{\ast }/N\rightarrow \widetilde{T}^{\ast }/N$ into an inert morphism of algebroids with fibre $\mathbb{G}_{m}$ . (For the details, see Proposition 1.61.) The inert bundle relations then imply that $x=[T^{\ast }/N]$ is indeed divisible by $(q-1)$ .

The matrices in $\widetilde{T}^{\ast }$ of trace zero have the full group $N$ as stabilizer. Decomposing $\widetilde{T}^{\ast }$ into matrices of trace zero, and matrices of non-zero trace, we see that

$$\begin{eqnarray}\widetilde{T}^{\ast }/N=BN\sqcup (\mathbb{G}_{m}\setminus \{1\})\times BT.\end{eqnarray}$$

Hence, $\widetilde{T}^{\ast }/N$ is not a strict algebroid, as $N$ is not connected, and is not the group of units in any algebra. This is, in fact, the reason for considering non-strict algebroids at all. (For more on this, see Example 2.18.)

(In the above calculations, we have suppressed the algebra part $A$ of the various algebroids $(X,A)$ . We leave it to the reader to supply the natural algebra for each algebroid mentioned.)

Results

We now summarize the main results of this paper.

Theorem 1 (Diagonalizability of $I^{\circ ,\text{ss}}$ ).

The operator $I^{\circ ,\text{ss}}$ on

$$\begin{eqnarray}K(\mathfrak{M})(q)=K(\mathfrak{M})\otimes _{\mathbb{Q}[q]}\mathbb{Q}(q)\end{eqnarray}$$

is diagonalizable, the eigenvalues are indexed by partitions $\unicode[STIX]{x1D706}$ , and the eigenvalue corresponding to the partition $\unicode[STIX]{x1D706}$ is the cyclotomic polynomial

$$\begin{eqnarray}\mathscr{Q}(\unicode[STIX]{x1D706})=\prod (q^{\unicode[STIX]{x1D706}_{i}}-1).\end{eqnarray}$$

In other words, we have a direct sum decomposition

$$\begin{eqnarray}K(\mathfrak{M})(q)=\bigoplus _{\unicode[STIX]{x1D706}}K^{\unicode[STIX]{x1D706}}(\mathfrak{M})\end{eqnarray}$$

into subspaces invariant under $I^{\circ ,\text{ss}}$ , and $I^{\circ ,\text{ss}}|_{K^{\unicode[STIX]{x1D706}}(\mathfrak{M})}$ is multiplication by $\mathscr{Q}(\unicode[STIX]{x1D706})$ .

The same theorem holds for the operator $I^{\text{ss}}$ in the context of strict algebroids. We also prove a stronger version avoiding denominators divisible by $(q-1)$ , but this version only works for algebroids.

The proof of this theorem is a generalization of the above sample calculation for the stack of rank 2 vector bundles. One goal of § 1 is to set up the necessary notation.

Theorem 2 (Graded structure of $K(\mathfrak{M})$ ).

There is a direct sum decomposition

(2) $$\begin{eqnarray}K(\mathfrak{M})=\bigoplus _{r\geqslant 0}K^{r}(\mathfrak{M})\end{eqnarray}$$

such that

$$\begin{eqnarray}K^{r}(\mathfrak{M})(q)=\bigoplus _{\operatorname{ ord}_{q=1}\mathscr{Q}(\unicode[STIX]{x1D706})=r}K^{\unicode[STIX]{x1D706}}(\mathfrak{M}).\end{eqnarray}$$

Moreover, the commutative product is graded with respect to (2).

Again, the same theorem holds in the context of strict algebroids.

The fact about the gradedness of the commutative product is expected from the fact that the semi-simple inertia respects the commutative product (it follows from this fact over $\mathbb{Q}(q)$ , but is true over $\mathbb{Q}$ ).

Geometrically, the descending filtration $K^{{\geqslant}r}(\mathfrak{M})$ induced by the grading (2) can be described as follows: $K^{{\geqslant}r}(\mathfrak{M})$ is the $\mathbb{Q}$ -span of all stack functions $[X\rightarrow \mathfrak{M}]$ , for which the algebra of global sections $\unicode[STIX]{x1D6E4}(X,A_{X})$ admits at least $r$ orthogonal non-zero central idempotents, where $A_{X}$ is the algebra of the algebroid $(X,A_{X})$ .

The direct summands $K^{r}(\mathfrak{M})$ are the common eigenspaces of the family of commuting operators $(E_{n})_{n\geqslant 0}$ , where $E_{n}(X)$ is the stack of decompositions of $1\in A_{X}$ into a sum of $n$ orthogonal labelled idempotents. The eigenvalues of the operators $E_{n}$ are integers, and the whole family of operators $(E_{n})$ is diagonalizable over $\mathbb{Q}$ . The proof of this fact proceeds by proving that the $(E_{n})$ preserve the descending filtration described geometrically above and have distinct integer diagonal entries.

It turns out that the ascending filtration $K^{{\leqslant}n}(\mathfrak{M})$ associated with the grading in the above theorem can be described as

$$\begin{eqnarray}K^{{\leqslant}n}(\mathfrak{M})=\ker E_{n+1}.\end{eqnarray}$$

Let us also point out that

$$\begin{eqnarray}K^{0}(\mathfrak{M})=K(\text{DM}),\end{eqnarray}$$

where $K(\text{DM})$ is the Grothendieck $\mathbb{Q}$ -algebra of Deligne–Mumford stacks (see § 2). For this reason, $K(\text{DM})$ is a more natural ring of scalars than $K(\text{Var})$ , and we will use $K(\text{DM})$ throughout. (On the other hand, in the context of strict algebroids, we have $K^{0}(\mathfrak{M})=K(\text{Var})$ .)

If we denote by $\unicode[STIX]{x1D70B}_{r}:K(\mathfrak{M})\rightarrow K(\mathfrak{M})$ the projection operator onto the summand $K^{r}(\mathfrak{M})$ and form the generating series $\unicode[STIX]{x1D70B}_{t}=\sum _{r\geqslant 0}\unicode[STIX]{x1D70B}_{r}t^{r}$ , then we have

(3) $$\begin{eqnarray}\unicode[STIX]{x1D70B}_{t}=\mathop{\sum }_{n\geqslant 0}\binom{t}{n}E_{n}.\end{eqnarray}$$

All the above results could be proved for pairs $(X,A)$ of algebraic stacks $X$ endowed with finite type algebras $A$ , instead of algebroids or strict algebroids. One simply replaces $I^{\circ ,\text{ss}}$ by $A^{\times ,\text{ss}}$ .

Theorem 3 (Filtered nature of the Hall algebra).

The Hall product is filtered with respect to the filtration $K^{{\leqslant}r}(\mathfrak{M})$ induced by the grading (2). Moreover, for the associated graded algebra, we have

$$\begin{eqnarray}\text{gr}(K(\mathfrak{M}),\ast )=(K(\mathfrak{M}),\cdot ).\end{eqnarray}$$

In other words, if $x\in K^{{\leqslant}r}(\mathfrak{M})$ and $y\in K^{{\leqslant}s}(\mathfrak{M})$ , then $x\ast y\in K^{{\leqslant}r+s}(\mathfrak{M})$ , and

$$\begin{eqnarray}x\ast y\equiv x\cdot y\quad \text{mod}~K^{{\leqslant}r+s-1}(\mathfrak{M}).\end{eqnarray}$$

The proof of this theorem uses not much more than some simple combinatorics involving relabelling of direct sum decompositions, and compatibilities between direct sum decompositions of short exact sequences and splittings of short exact sequences.

The theorem implies that the one parameter family of algebras $(\mathscr{K}(\mathfrak{M}),\ast )$ given by the Rees construction

$$\begin{eqnarray}\mathscr{K}(\mathfrak{M})=\bigoplus _{n\geqslant 0}t^{n}K^{{\leqslant}n}(\mathfrak{M})\end{eqnarray}$$

is a deformation quantization of (i.e., a one-parameter flat family of algebras with special fibre) the commutative algebra $(K(\mathfrak{M}),\cdot )$ . Hence, the graded algebra $(K(\mathfrak{M}),\cdot )$ inherits a Poisson bracket $\{,\}$ of degree $-1$ . In particular, $K^{1}(\mathfrak{M})$ is a Lie algebra, and it turns out that the Lie bracket on $K^{1}(\mathfrak{M})$ is equal to the commutator bracket associated to $\ast$ .

The filtered algebra $(K(\mathfrak{M}),\ast )$ is a filtered quantization of the Poisson algebra $(K(\mathfrak{M}),\cdot ,\{,\})$ , as defined in [Reference SchedlerSch16]. Of course, we have more structure here, as our filtration comes from a natural grading, but it is not clear to us what this additional structure on our filtered quantization is useful for.

Following Joyce [Reference JoyceJoy07a], we call $K^{1}(\mathfrak{M})$ the Lie algebra of virtually indecomposable elements of $K(\mathfrak{M})$ , with the notation $K^{\text{vir}}(\mathfrak{M})=K^{1}(\mathfrak{M})$ .

We denote the projection onto $K^{\text{vir}}(\mathfrak{M})$ by $\unicode[STIX]{x1D70B}^{\text{vir}}$ . With this notation, we have, as a special case of (3),

$$\begin{eqnarray}\unicode[STIX]{x1D70B}^{\text{vir}}=\mathop{\sum }_{n>0}\frac{(-1)^{n+1}}{n}E_{n}.\end{eqnarray}$$

In terms of eigenspaces of semi-simple inertia, we have

$$\begin{eqnarray}K^{\text{vir}}(\mathfrak{M})(q)=K^{(q-1)}(\mathfrak{M})\oplus K^{(q^{2}-1)}(\mathfrak{M})\oplus K^{(q^{3}-1)}(\mathfrak{M})\oplus \ldots .\end{eqnarray}$$

Theorem 4 (Hall algebra logarithms).

Let $\mathfrak{N}\subset \mathfrak{M}$ be a ‘small enough’ substack, closed under extensions and direct summands, and not intersecting $\operatorname{Spec}k\stackrel{0}{\longrightarrow }\mathfrak{M}$ . Then

$$\begin{eqnarray}\unicode[STIX]{x1D700}_{t}[\mathfrak{N}]=\mathop{\sum }_{n\geqslant 0}\binom{t}{n}[\mathfrak{N}]^{\ast n}\in \hat{\mathscr{K}}(\mathfrak{M})_{+}.\end{eqnarray}$$

In particular, the $\ast$ -logarithm

$$\begin{eqnarray}\unicode[STIX]{x1D700}[\mathfrak{N}]=\mathop{\sum }_{n\geqslant 1}\frac{(-1)^{n+1}}{n}[\mathfrak{N}]^{\ast n}\in \hat{K}^{\text{vir}}(\mathfrak{M})_{+}\end{eqnarray}$$

is virtually indecomposable.

For the precise definition of ‘small enough’, see § 3.3. For example, if $\mathfrak{M}=\mathfrak{Coh}_{Y}$ for a curve $Y$ , we could take $\mathfrak{N}$ to consist of all non-zero semi-stable vector bundles of a fixed slope. Since $\mathfrak{N}$ is typically not of finite type, to make sense of $[\mathfrak{N}]$ we have to pass to a certain completion $\hat{K}(\mathfrak{M})_{+}$ of $K(\mathfrak{M})$ . See § 3.3 for details.

Theorem 5 (No poles).

Let $K(\text{St})$ be the Grothendieck $\mathbb{Q}$ -algebra of algebraic stacks, modulo all bundle relations with special structure group (inert or not). Consider the map

$$\begin{eqnarray}\displaystyle \int :K(\mathfrak{M}) & \longrightarrow & \displaystyle K(\text{St})\nonumber\\ \displaystyle \hspace{0.0pt}[(X,A)\rightarrow (\mathfrak{M},\mathfrak{A})] & \longmapsto & \displaystyle [X],\nonumber\end{eqnarray}$$

which forgets the structure map to $(\mathfrak{M},\mathfrak{A})$ , and the algebroid structure over the stack $X$ . If $x\in K^{{\leqslant}r}(\mathfrak{M})$ , then $(q-1)^{r}\int x$ is a regular element of $K(\text{St})$ ; i.e., under the identification

$$\begin{eqnarray}K(\text{St})=K(\text{Var})\biggl[\frac{1}{q},\frac{1}{q-1},\frac{1}{q^{2}-1},\ldots \biggr],\end{eqnarray}$$

it can be written with a denominator that does not vanish at $q=1$ .

Moreover, suppose we have a grading monoid $\unicode[STIX]{x1D6E4}$ for $\mathfrak{M}$ :

$$\begin{eqnarray}\mathfrak{M}=\coprod _{\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}}\mathfrak{M}_{\unicode[STIX]{x1D6FE}}.\end{eqnarray}$$

We say that $x$ has ‘degree’ $\unicode[STIX]{x1D6FE}$ if $x\in \mathfrak{M}_{\unicode[STIX]{x1D6FE}}$ . We need $\unicode[STIX]{x1D6E4}$ to be endowed with a $\mathbb{Z}$ -valued bilinear form $\unicode[STIX]{x1D712}$ such that for every object $x$ in $\mathfrak{M}_{\unicode[STIX]{x1D6FE}}$ and $y$ in $\mathfrak{M}_{\unicode[STIX]{x1D6FD}}$ :

1. (i) every extension of $y$ by $x$ is in $\mathfrak{M}_{\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D6FE}}$ ;

2. (ii) the stack of extensions of $y$ by $x$ is the quotient of a vector space $E_{1}$ by another vector space $E_{0}$ , acting trivially, such that $\dim E_{0}-\dim E_{1}=\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FE})$ .

For the precise assumptions, see § 4.2. They are satisfied if $\mathfrak{M}=\mathfrak{Rep}_{Q}$ , or if $\mathfrak{M}=\mathfrak{Coh}_{Y}$ and $Y$ is a smooth curve, or, more generally, if $\mathfrak{M}$ is hereditary.

Then we have a commutative diagram as follows.

Here we use the $\unicode[STIX]{x1D6E4}$ -graded integral, which is essentially a generating function, indexed by $\unicode[STIX]{x1D6E4}$ , of the integrals of the components of degree $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}$ of a given stack function.

The upper horizontal arrow in this diagram is the specialization map, which exists because of Theorem 3. The left vertical arrow exists by the ‘no poles’ theorem (Theorem 5). It is a standard fact that this map is an algebra morphism, i.e., it respects the $\ast$ -product, if the target $K(\text{St})_{\text{reg}}[\unicode[STIX]{x1D6E4}]$ is endowed with its $q$ -deformed product twisted by $\unicode[STIX]{x1D712}$ . It is a formal consequence of the commutativity of this diagram that the right vertical map is a morphism of Poisson algebras if the target $K(\text{St})_{\text{reg}}/(q-1)[\unicode[STIX]{x1D6E4}]$ is endowed with its bracket induced by $\unicode[STIX]{x1D712}$ . We call the right vertical map the semi-classical motivic integral.

In particular, we deduce that

(4) $$\begin{eqnarray}\displaystyle \int _{q=1}:K^{\text{vir}}(\mathfrak{M}) & \longrightarrow & \displaystyle K(\text{St})_{\text{reg}}/(q-1)[\unicode[STIX]{x1D6E4}]\nonumber\\ \displaystyle x_{\unicode[STIX]{x1D6FE}} & \longmapsto & \displaystyle \biggl((q-1)\int x_{\unicode[STIX]{x1D6FE}}\biggr)\biggr|_{q=1}u^{\unicode[STIX]{x1D6FE}}=\operatorname{res}_{q=1}\biggl(\int x_{\unicode[STIX]{x1D6FE}}\biggr)u^{\unicode[STIX]{x1D6FE}}\end{eqnarray}$$

is a morphism of Lie algebras. The target $K(\text{St})_{\text{reg}}/(q-1)$ of the semi-classical motivic integral is equal to $K(\text{Var})/((q-1)+\operatorname{Ann}(q-1))$ .

The proof of the no poles theorem combines the above results about the diagonalizability of $I^{\circ ,\text{ss}}$ , especially in its form avoiding denominators divisible by $(q-1)$ , with the result that for an algebroid $(X,A)$ , the stack $I_{X}^{\circ ,\text{ss}}$ has regular motivic weight, i.e.,

$$\begin{eqnarray}[I_{X}^{\circ ,\text{ss}}]\in K(\text{St})_{\text{reg}}.\end{eqnarray}$$

We think of this as a motivic version of Burnside’s lemma. The more natural-looking conjecture that for an algebraic stack $X$ , the motivic weight of $I_{X}$ is contained in $K(\text{Var})\subset K(\text{St})$ is most likely false.

Discussion

To produce counting invariants for $\mathfrak{M}$ , we need to look for subcategories $\mathfrak{N}\subset \mathfrak{M}$ , to which we can apply Theorem 4, giving us virtually indecomposable elements $\unicode[STIX]{x1D700}[\mathfrak{N}]$ , to which we can apply the integral (4), yielding generating functions with coefficients in $K(\text{St})_{\text{reg}}/(q-1)$ . (See Remark 4.12 for details.) We can apply the Euler characteristic to these elements of $K(\text{St})_{\text{reg}}/(q-1)$ to obtain rational numbers. In the hereditary case, the fact that (4) is a morphism of Lie algebras gives the relations among generating functions one is interested in. This leads to wall-crossing formulas, and other results. For details we refer to the works of Joyce, Joyce–Song, and many others.

To deal with the Calabi–Yau 3 case, one needs to insert the correct motivic vanishing cycle weights, to define the integral. This is done by Joyce and Song in [Reference Joyce and SongJS12]. We leave it to future work to supply the details in our context.

The work of Joyce on configurations in abelian categories contains results which correspond to ours, but his definitions are more ad hoc. In fact, one reason for writing the present article was to give a more conceptual treatment of Joyce’s results. We do not prove that our notion of ‘virtual indecomposable object’ coincides with Joyce’s (except for in the case of $\mathfrak{M}=\mathfrak{Vect}$ , see the appendix), but instead prove that our notion has the same properties as Joyce’s and is just as useful. (Of course, the counting invariants we obtain are the same as the ones obtained by Joyce: the formula $\int _{q=1}\log _{\ast }(1+[\mathfrak{N}])$ , see (39), does not involve explicitly the definition of virtually indecomposable object.)

We think of the Lie algebra $K^{\text{vir}}(\mathfrak{M})$ as an analogue of the Lie algebra of primitive elements in a cocommutative Hopf algebra. In fact, one may ask whether $(K(\mathfrak{M}),\ast )$ is equal to the universal enveloping algebra of the Lie algebra $K^{\text{vir}}(\mathfrak{M})$ . To deduce such a statement from structure theorems for Hopf algebras, one would need to enhance $K(\mathfrak{M})$ to a cocommutative Hopf algebra. We have not been able to construct the necessary coproduct. We view the family of operators $(E_{n})$ as something of a replacement. It lets us prove at least some of the results expected of a cocommutative Hopf algebra, in particular Theorems 3 and 4.

1.1 Algebraic stacks

Let us briefly summarize our conventions about algebraic stacks.

We choose a noetherian base ring $R$ (commutative and with unit), and we fix our base category $\mathscr{S}$ to be the category of $R$ -schemes, endowed with the étale topology. Over $\mathscr{S}$ we have a canonical sheaf of $R$ -algebras $\mathscr{O}_{\mathscr{S}}$ ; it is represented by $\mathbb{A}^{1}=\mathbb{A}_{\operatorname{Spec}R}^{1}$ and called the structure sheaf.

We will assume our algebraic stacks to be locally of finite type. Thus, an algebraic stack is a stack over the site $\mathscr{S}$ that admits a presentation by a groupoid $X_{1}\rightrightarrows X_{0}$ , where $X_{0}$ and $X_{1}$ are algebraic spaces, locally of finite type over $R$ , the source and target morphisms $s,t:X_{1}\rightarrow X_{0}$ are smooth, and the diagonal $X_{1}\rightarrow X_{0}\times X_{0}$ is of finite type. In fact, all algebraic stacks we encounter will have affine diagonal.

By a stratification of an algebraic stack $X$ , we mean a morphism of algebraic stacks $X^{\prime }\rightarrow X$ that is a surjective monomorphism and that admits a finite decomposition $X^{\prime }=\coprod _{i}X_{i}$ such that every $X_{i}\rightarrow X$ is a locally closed embedding of algebraic stacks.

If $G$ is an algebraic group acting on the algebraic space $X$ , we will denote the quotient stack by $X/G$ , because we fear the more common notation $[X/G]$ would lead to confusion with the notation for elements of various $K$ -groups of schemes and stacks.

Suppose $G\rightarrow X$ is a relative group scheme over the stack $X$ . The connected component of $G$ , denoted by $G^{\circ }$ , is the subsheaf of $G$ defined by requiring a section $g\in G(S)$ to factor through $G^{\circ }(S)$ if and only if for all points (equivalently geometric points) $s$ of $S$ , we have $g(s)\in G_{s}^{\circ }$ . If $G\rightarrow X$ is smooth, the connected component $G^{\circ }\subset G$ is represented by an open substack of $G$ , which is a smooth group scheme with geometrically connected fibres over $X$ . (See [Reference Artin, Bertin, Demazure, Grothendieck, Gabriel, Raynaud and SerreSGA3, Exposé VI $_{\text{B}}$ , Théoreme 3.10].)

If the inertia stack $I_{X}$ of an algebraic stack $X$ is smooth over $X$ , the connected component $I_{X}^{\circ }$ is an algebraic stack. We can apply the rigidification construction (see, for example, [Reference Abramovich, Olsson and VistoliAOV08, Appendix]) to $I_{X}^{\circ }\subset I_{X}$ and obtain a (uniquely determined) Deligne–Mumford stack $\overline{X}$ , together with a morphism $X\rightarrow \overline{X}$ , making $X$ a connected gerbe over $\overline{X}$ (which means that the relative inertia of $X$ over $\overline{X}$ has connected fibres). The structure morphism $X\rightarrow \overline{X}$ is smooth.

A gerbe $X\rightarrow \overline{X}$ is an isotrivial gerbe if it admits a section over a finite étale $\overline{X}$ -stack. If $X\rightarrow \overline{X}$ is a smooth gerbe over a Deligne–Mumford stack, there exists a stratification $\overline{X}^{\prime }\rightarrow \overline{X}$ such that the restriction of the gerbe $X$ to $\overline{X}^{\prime }$ is isotrivial. (This follows from the fact that a quasi-finite morphism of Deligne–Mumford stacks is generically finite. This, in turn, follows from Zariski’s main theorem [Reference Laumon and Moret-BaillyLMB00, §16].)

Let us also remark that every Deligne–Mumford stack admits a stratification by integral normal Deligne–Mumford stacks, although we do not use this fact.

Coherent sheaves

A sheaf $\mathscr{F}$ of $\mathscr{O}_{X}$ -modules is locally coherent if for every $x/U$ , the sheaf $\mathscr{F}_{U}$ is a coherent sheaf of $\mathscr{O}_{U_{\acute{\text{e}}\text{t}}}$ -modules. (This terminology is inspired by [Sta15, Tag 06WJ].) It is cartesian if all compatibility morphisms $\unicode[STIX]{x1D6FC}^{\ast }:f^{\ast }\mathscr{F}_{U}\rightarrow \mathscr{F}_{V}$ are isomorphisms of sheaves of $\mathscr{O}_{V_{\acute{\text{e}}\text{t}}}$ -modules. A sheaf that is both locally coherent and cartesian is coherent. The coherent sheaf $\mathscr{F}$ is said to be locally free coherent if every $\mathscr{F}_{U}$ is locally free. (This is equivalent to $\mathscr{F}$ being locally free.)

For example, a groupoid presentation $X_{1}\rightrightarrows X_{0}$ of $X$ and a coherent sheaf $\mathscr{F}_{0}$ on $X_{0}$ , together with an isomorphism $s^{\ast }\mathscr{F}_{0}\rightarrow t^{\ast }\mathscr{F}_{0}$ , satisfying the usual cocycle condition on $X_{2}=X_{1}\times _{X_{0}}X_{1}$ , give rise to a coherent sheaf $\mathscr{F}$ on $X$ . If $\mathscr{F}_{0}$ is locally free, then $\mathscr{F}$ is locally free.

A vector bundle over $X$ is a representable morphism $E\rightarrow X$ endowed with an addition operation and a compatible $\mathbb{G}_{m}$ -action such that the pullback of $E\rightarrow X$ to any scheme $U\rightarrow X$ is a vector bundle on $U$ .

The sheaf of sections of a vector bundle over $X$ is a coherent sheaf. In fact, the notions of vector bundle and locally free coherent sheaf are equivalent, and we will use them interchangeably. (The inverse functor is given by $\mathscr{E}\mapsto \operatorname{Spec}_{X}\operatorname{Sym}_{\mathscr{O}_{X}}\mathscr{E}^{\vee }$ .) The cokernel of a homomorphism of vector bundles is coherent. In fact, every cokernel of a homomorphism of coherent sheaves is coherent.

If the cokernel of a homomorphism of vector bundles is locally free, we call the homomorphism a strict homomorphism of vector bundles. For a strict homomorphism of vector bundles, the image and the kernel, as well as the cokernel, are locally free.

A strict monomorphism of vector bundles is a strict homomorphism whose kernel is zero. A homomorphism of vector bundles is a strict monomorphism/an epimorphism if and only if over every geometric point of $X$ , the induced linear map is injective/surjective. A homomorphism of vector bundles, which is a monomorphism of sheaves, is a strict monomorphism of bundles.

Let $\unicode[STIX]{x1D711}:E\rightarrow F$ be a homomorphism of vector bundles over the algebraic stack $X$ . The flattening stratification $X^{\prime }\rightarrow X$ of $\operatorname{cok}\unicode[STIX]{x1D711}$ serves also as a strictening stratification for $\unicode[STIX]{x1D711}$ . This means that an object of $X(S)$ lifts to $X^{\prime }(S)$ if and only if $\unicode[STIX]{x1D711}_{S}$ is strict.

In general, the kernel (in the category of big sheaves of $\mathscr{O}_{X}$ -modules) of a homomorphism of vector bundles is locally coherent, but not coherent.

By [Sta15, Tag 06WK], a sheaf of $\mathscr{O}_{X}$ -modules $\mathscr{F}$ is coherent if and only if there exists a smooth covering $X_{i}\rightarrow X$ of $X$ by finite type affine schemes $X_{i}$ such that for every $i$ , the restriction $\mathscr{F}_{i}$ of $\mathscr{F}$ to the big étale site of $X_{i}$ is isomorphic to the cokernel of a homomorphism of vector bundles.

Proposition 1.1. Suppose that $\mathscr{F}$ is a coherent sheaf on the algebraic stack $X$ . Then:

1. (i) for every $x/U$ , we have $(\mathscr{F}^{\vee })_{U}=(\mathscr{F}_{U})^{\vee }$ ;

2. (ii) $\mathscr{F}^{\vee }$ is locally coherent;

3. (iii) $\mathscr{F}^{\vee }$ is represented by a an algebraic stack, which is of finite type and affine over $X$ , namely, $\operatorname{Spec}_{X}\operatorname{Sym}_{\mathscr{O}_{X}}\mathscr{F}$ ;

4. (iv) the canonical homomorphism $\mathscr{F}\rightarrow \mathscr{F}^{\vee \vee }$ is an isomorphism of sheaves of $\mathscr{O}_{X}$ -modules;

5. (v) if $\mathscr{F}$ is locally free coherent, then $\mathscr{F}^{\vee }$ is a vector bundle.

Moreover, the functor $\mathscr{F}\mapsto \mathscr{F}^{\vee }$ is a fully faithful functor from the category of coherent sheaves to the category of locally coherent sheaves of $\mathscr{O}_{X}$ -modules. It maps right exact sequences of coherent sheaves to left exact sequences of locally coherent sheaves.

Proof. The canonical homomorphism $\mathscr{H}om(\mathscr{F},\mathscr{G})_{U}\rightarrow \mathscr{H}om(\mathscr{F}_{U},\mathscr{G}_{U})$ is an isomorphism for all $U$ in $\mathscr{S}$ , if $\mathscr{F}$ is cartesian. This implies the first claim.

The second claim follows from the first (see also [Sta15, Tag 06WM]).

For the third claim, see [Reference Laumon and Moret-BaillyLMB00, (14.2.6)].

For the fourth claim, notice that both $\mathscr{F}^{\vee }$ and $\mathscr{O}_{X}$ are representable and affine over $X$ , namely, $\mathscr{F}^{\vee }=\operatorname{Spec}_{X}\operatorname{Sym}_{\mathscr{O}_{X}}\mathscr{F}$ and $\mathscr{O}_{X}=\mathbb{A}_{X}^{1}$ . They are also both endowed with $\mathbb{G}_{m}$ -actions, via scalar multiplication. We consider the big sheaf of $\mathbb{G}_{m}$ -equivariant $X$ -morphisms from $\operatorname{Spec}_{X}\operatorname{Sym}_{\mathscr{O}_{X}}\mathscr{F}$ to $\mathbb{A}_{X}^{1}$ , denoted by $\operatorname{Hom̲}_{\mathbb{G}_{m}}(\mathscr{F}^{\vee },\mathbb{A}^{1})$ . The sections of $\operatorname{Hom̲}_{\mathbb{G}_{m}}(\mathscr{F}^{\vee },\mathbb{A}^{1})$ over $S\rightarrow X$ are the $\mathbb{G}_{m}$ -equivariant $S$ -morphisms $\operatorname{Spec}_{S}\operatorname{Sym}_{\mathscr{O}_{S}}\mathscr{F}_{S}\rightarrow \mathbb{A}_{S}^{1}$ or, equivalently, the morphisms of sheaves of $\mathscr{O}_{S}$ -algebras $\mathscr{O}_{S}[t]\rightarrow \operatorname{Sym}_{\mathscr{O}_{S}}\mathscr{F}_{S}$ of degree 1. These are the same as the sections of $\mathscr{F}_{S}$ over $S$ . This proves that $\operatorname{Hom̲}_{\mathbb{G}_{m}}(\mathscr{F}^{\vee },\mathbb{A}^{1})=\mathscr{F}$ .

On the other hand, the sections of $\mathscr{H}om(\mathscr{F}^{\vee },\mathscr{O})$ over $S\rightarrow X$ are the $\mathscr{O}$ -linear homomorphisms $\operatorname{Spec}_{S}\operatorname{Sym}_{\mathscr{O}_{S}}\mathscr{F}_{S}\rightarrow \mathbb{A}_{S}^{1}$ , which are, in particular, $\mathbb{G}_{m}$ -equivariant. Thus we have a natural map $\mathscr{H}om(F^{\vee },\mathscr{O})\rightarrow \operatorname{Hom̲}_{\mathbb{G}_{m}}(\mathscr{F}^{\vee },\mathbb{A}^{1})$ . In total, we have a canonical map

$$\begin{eqnarray}\mathscr{F}^{\vee \vee }=\mathscr{H}om(\mathscr{F}^{\vee },\mathscr{O})\longrightarrow \operatorname{Hom̲}_{\mathbb{ G}_{m}}(\mathscr{F}^{\vee },\mathbb{A}^{1})=\mathscr{F},\end{eqnarray}$$

which is inverse to the tautological map $\mathscr{F}\rightarrow \mathscr{F}^{\vee \vee }$ .

The fifth claim is clear.

The ‘moreover’ follows from the fact that we can reconstruct $\mathscr{F}$ from $\mathscr{F}^{\vee }=\operatorname{Spec}_{X}\operatorname{Sym}_{\mathscr{O}_{X}}\mathscr{F}$ , as the degree 1 part of the graded sheaf of $\mathscr{O}_{X}$ -modules $\unicode[STIX]{x1D70B}_{\ast }(\mathscr{O}_{\mathscr{F}^{\vee }})$ , where $\unicode[STIX]{x1D70B}:\mathscr{F}^{\vee }\rightarrow X$ is the projection morphism.◻

Representable sheaves of modules

If $\unicode[STIX]{x1D711}:E\rightarrow F$ is a homomorphism of vector bundles over $X$ , then $\ker \unicode[STIX]{x1D711}$ , constructed in the category of big sheaves, is a representable sheaf of $\mathscr{O}_{X}$ -modules. In fact, $\ker \unicode[STIX]{x1D711}$ is equal to the fibred product of stacks as shown in the following diagram.

Sheaves such as $\ker \unicode[STIX]{x1D711}$ belong to a class of $\mathscr{O}_{X}$ -modules dual to coherent sheaves.

Proposition 1.2. Let $\mathscr{F}$ be a sheaf of $\mathscr{O}_{X}$ -modules. The following are equivalent.

1. (i) There exists a coherent sheaf $\mathscr{N}$ such that $\mathscr{F}$ is isomorphic to $\mathscr{N}^{\vee }$ .

2. (ii) There exists a smooth cover $X_{i}\rightarrow X$ of $X$ by finite type affine schemes $X_{i}$ such that, for every $i$ , the restriction $\mathscr{F}_{i}$ of $\mathscr{F}$ to the big étale site of $X_{i}$ is isomorphic to the kernel of a homomorphism of vector bundles over $X_{i}$ .

Proof. The fact that (i) implies (ii) follows from the results proved above. So let us indicate how to prove that (ii) implies (i).

Let us first assume that $\mathscr{F}$ is isomorphic to the kernel of a homomorphism of vector bundles $E_{0}\rightarrow E_{1}$ . One checks that $\mathscr{F}$ is then represented by $\operatorname{Spec}_{X}\operatorname{Sym}_{\mathscr{O}_{X}}\operatorname{cok}(E_{1}^{\vee }\rightarrow E_{0}^{\vee })$ . Thus, $\mathscr{F}$ is isomorphic to the dual of the coherent sheaf $\operatorname{cok}(E_{1}^{\vee }\rightarrow E_{0}^{\vee })$ .

Now suppose that $\mathscr{F}$ is locally isomorphic to the kernel of a homomorphism of vector bundles. It suffices to prove that $\mathscr{F}^{\vee }$ is coherent and that $\mathscr{F}\rightarrow \mathscr{F}^{\vee \vee }$ is an isomorphism. Both claims can be checked locally and are true for duals of coherent sheaves.◻

Definition 1.3. We say a sheaf of $\mathscr{O}_{X}$ -modules is locally coherent representable if any of the two equivalent conditions of Proposition 1.2 is satisfied. The terminology is justified by Proposition 1.4 below.

In other words, the category of locally coherent representable sheaves over $X$ is the essential image of the fully faithful functor mentioned in Proposition 1.1. We therefore have an equivalence of categories

(5) $$\begin{eqnarray}\displaystyle (\text{coh. sheaves over }X) & \longrightarrow & \displaystyle (\text{loc. coh. repr. sheaves over }X)\nonumber\\ \displaystyle \mathscr{F} & \longmapsto & \displaystyle \mathscr{F}^{\vee }.\end{eqnarray}$$

The following proposition summarizes facts about locally coherent representable sheaves, which all follow easily from facts mentioned above.

Proposition 1.4. Let $\mathscr{F}$ be a locally coherent representable sheaf over the algebraic stack $X$ . Then:

1. (i) the sheaf $\mathscr{F}$ is locally coherent;

2. (ii) the sheaf $\mathscr{F}^{\vee }$ is coherent;

3. (iii) the canonical homomorphism $\mathscr{F}\rightarrow \mathscr{F}^{\vee \vee }$ is an isomorphism of sheaves of $\mathscr{O}_{X}$ -modules;

4. (iv) the sheaf $\mathscr{F}$ is representable by an algebraic stack $F\rightarrow X$ , which is of finite type and affine over $X$ ;

5. (v) in fact, $\mathscr{F}=\operatorname{Spec}_{X}\operatorname{Sym}_{\mathscr{O}_{X}}\mathscr{F}^{\vee }$ .

Moreover, the functor $\mathscr{F}\mapsto \mathscr{F}^{\vee }$ is an essential inverse to the functor (5). It maps left exact sequences of locally coherent representable sheaves to right exact sequences of coherent sheaves.

Proposition 1.5. Let $\mathscr{F}$ be a locally coherent representable sheaf over the finite type algebraic stack $X$ . There is a unique stratification $X^{\prime }\rightarrow X$ with the property that an $X$ -scheme $S$ factors through $X^{\prime }$ if and only if $\mathscr{F}|_{S}$ is a vector bundle. More precisely, $X^{\prime }=\coprod _{n\geqslant 0}X_{n}$ , and $X_{n}\rightarrow X$ is a locally closed immersion of algebraic stacks with the property that $S\rightarrow X$ factors through $X_{n}$ if and only if $\mathscr{F}|_{S}$ is a vector bundle of rank $n$ .

Proof. The sought-after stratification is the flattening stratification of the coherent sheaf $\mathscr{F}^{\vee }$ .◻

Example 1.6. Consider $X=\mathbb{A}^{1}$ , with coordinate $t$ , and let $\mathscr{C}$ be the cokernel of the homomorphism of vector bundles $t:\mathbb{A}_{X}^{1}\rightarrow \mathbb{A}_{X}^{1}$ . It is the skyscraper sheaf of the origin, considered as a coherent sheaf on $X$ and extended to a big sheaf over $X$ in the usual way. The sheaf $\mathscr{C}$ is an example of a coherent sheaf which is not representable.

Let $\mathscr{K}$ be the kernel of $t:\mathbb{A}_{X}^{1}\rightarrow \mathbb{A}_{X}^{1}$ . This is locally coherent representable, but not cartesian, and hence not coherent.

Note that $\mathscr{C}^{\vee }=\mathscr{K}$ . This shows that $\mathscr{F}^{\vee }$ may not be coherent, even if $\mathscr{F}$ is.

Note also that $\mathscr{K}^{\vee }=\mathscr{C}$ , which shows that $\mathscr{F}^{\vee }$ may not be representable, even if $\mathscr{F}$ is.

Finally, note that $(\mathscr{K}_{X})^{\vee }=0^{\vee }=0$ , but $(\mathscr{K}^{\vee })_{X}=\mathscr{C}_{X}$ is the structure sheaf of the origin in $X$ , considered as a skyscraper sheaf on $X_{\acute{\text{e}}\text{t}}$ , which is not zero. This gives an example where $(\mathscr{F}^{\vee })_{U}\not =(\mathscr{F}_{U})^{\vee }$ .

Remark 1.7. Of course, the category of coherent sheaves on an algebraic stack $X$ has kernels and internal homs, but they do not agree with those in the category of big sheaves, which we considered above. It is therefore important to specify the context when dealing with kernels or duals in the category of coherent sheaves. Unless specified otherwise, we will always consider sheaves of $\mathscr{O}_{X}$ -modules as big sheaves.

1.2 Linear algebraic stacks

We will review the definition of linear algebraic stacks and some basic constructions. For definitions and basic properties of fibred categories we refer the reader to [Reference GrothendieckSGA1, Exposé VI]. The material here is presumably known, but we could not find a suitable reference.

Suppose $\mathfrak{X}\rightarrow \mathscr{S}$ is a category over $\mathscr{S}$ . We write $\mathfrak{X}(S)$ for the fibre of $\mathfrak{X}$ over the object $S$ of $\mathscr{S}$ . If $f:S^{\prime }\rightarrow S$ is a morphism in $\mathscr{S}$ , and $x^{\prime }\in \mathfrak{X}(S^{\prime })$ and $x\in \mathfrak{X}(S)$ are $\mathfrak{X}$ -objects lying over $S^{\prime }$ and $S$ , respectively, we write $\operatorname{Hom}_{f}(x^{\prime },x)$ for the set of morphisms from $x^{\prime }$ to $x$ in $\mathfrak{X}$ , lying over $f$ . For $S^{\prime }=S$ and $f=\operatorname{id}_{S}$ , we write $\operatorname{Hom}_{S}(x^{\prime },x)$ .

Recall that a morphism $\unicode[STIX]{x1D6FC}:x^{\prime }\rightarrow x$ lying over $f:S^{\prime }\rightarrow S$ is cartesian if for every object $x^{\prime \prime }$ of $\mathfrak{X}(S)$ , composition with $\unicode[STIX]{x1D6FC}$ induces a bijection $\operatorname{Hom}_{S}(x^{\prime \prime },x^{\prime })\stackrel{\simeq }{\longrightarrow }\operatorname{Hom}_{f}(x^{\prime \prime },x)$ . Recall further that $\mathfrak{X}\rightarrow \mathscr{S}$ is a fibred category if every composition of cartesian morphisms is cartesian and if for every $f:S^{\prime }\rightarrow S$ in $\mathscr{S}$ and every $x$ over $S$ , there exists a cartesian morphism over $f$ with target $x$ . A cartesian functor between categories over $\mathscr{S}$ is one that preserves cartesian morphisms.

If $\mathfrak{X}$ is a fibred category over $\mathscr{S}$ , the subcategory of $\mathfrak{X}$ consisting of the same objects and all cartesian morphisms is a category fibred in groupoids over $\mathscr{S}$ . We denote it by $\mathfrak{X}_{\text{cfg}}$ and call it the underlying category fibred in groupoids.

Definition 1.8. A category $\mathfrak{X}$ over $\mathscr{S}$ is an $\mathscr{O}$ -linear category over  $\mathscr{S}$ if for every $f:S^{\prime }\rightarrow S$ in $\mathscr{S}$ and all $x^{\prime }\in \mathfrak{X}(S^{\prime })$ , $x\in \mathfrak{X}(S)$ , the set $\operatorname{Hom}_{f}(x^{\prime },x)$ is endowed with the structure of an $\mathscr{O}(S^{\prime })$ -module in such a way that for every pair of morphisms $g:S^{\prime \prime }\rightarrow S^{\prime }$ , $f:S^{\prime }\rightarrow S$ and every triple of objects $x^{\prime \prime }\in \mathfrak{X}(S^{\prime \prime })$ , $x^{\prime }\in \mathfrak{X}(S^{\prime })$ , $x\in \mathfrak{X}(S)$ , the composition

$$\begin{eqnarray}\operatorname{Hom}_{f}(x^{\prime },x)\times \operatorname{Hom}_{g}(x^{\prime \prime },x^{\prime })\longrightarrow \operatorname{Hom}_{f\circ g}(x^{\prime \prime },x)\end{eqnarray}$$

is $\mathscr{O}(S^{\prime })$ -bilinear.

An $\mathscr{O}$ -linear functor $F:\mathfrak{X}\rightarrow \mathfrak{Y}$ between $\mathscr{O}$ -linear categories is a functor of categories over $\mathscr{S}$ such that for every $f:S^{\prime }\rightarrow S$ and all $x^{\prime }\in \mathfrak{X}(S^{\prime })$ , $x\in \mathfrak{X}(S)$ , the map $\operatorname{Hom}_{f}(x^{\prime },x)\rightarrow \operatorname{Hom}_{f}(F(x^{\prime }),F(x))$ is $\mathscr{O}(S^{\prime })$ -linear.

Assume given an $\mathscr{O}$ -linear fibred category $\mathfrak{X}$ over $\mathscr{S}$ . Pullback in $\mathfrak{X}$ is $\mathscr{O}$ -linear, i.e., if $f:S^{\prime }\rightarrow S$ is a morphism in $\mathscr{S}$ and $x,y\in \mathfrak{X}(S)$ are objects with pullbacks $x^{\prime },y^{\prime }\in \mathfrak{X}(S^{\prime })$ , the pullback map $f^{\ast }:\operatorname{Hom}_{S}(x,y)\rightarrow \operatorname{Hom}_{S^{\prime }}(x^{\prime },y^{\prime })$ is $\mathscr{O}(S)$ -linear.

So, if we fix objects $x,y\in \mathfrak{X}(S)$ , the presheaf $\operatorname{Hom̲}_{S}(x,y)$ over the usual small étale site of $S$ , defined by $\operatorname{Hom̲}_{S}(x,y)(T)=\operatorname{Hom}_{T}(x|_{T},y|_{T})$ for every étale $T\rightarrow S$ , is a presheaf of $\mathscr{O}_{S_{\acute{\text{e}}\text{t}}}$ -modules. Moreover, for any morphism $f:S^{\prime }\rightarrow S$ in $\mathscr{S}$ , we have a natural homomorphism of presheaves of $\mathscr{O}_{S}$ -modules $\operatorname{Hom̲}_{S}(x,y)\rightarrow f_{\ast }\operatorname{Hom̲}_{S^{\prime }}(x^{\prime },y^{\prime })$ . Put together, the small presheaves $\operatorname{Hom̲}_{T}(x,y)$ , as $T\rightarrow S$ varies over the big étale site of the scheme $S$ , form a big presheaf, which we denote by $\operatorname{Hom̲}(x,y)$ .

Examples

Example 1.11. Let $X$ be a projective $R$ -scheme. The linear stack $\mathfrak{Coh}_{X}$ has as objects lying over the $R$ -scheme $S$ the coherent sheaves on $X\times S$ , which are flat over $S$ . For a morphism of $R$ -schemes $f:S^{\prime }\rightarrow S$ , and $\mathscr{F}^{\prime }\in \mathfrak{Coh}_{X}(S^{\prime })$ and $\mathscr{F}\in \mathfrak{Coh}_{X}(S)$ , we set $\operatorname{Hom}_{f}(\mathscr{F}^{\prime },\mathscr{F})=\operatorname{Hom}_{\mathscr{O}_{X\times S^{\prime }}}(\mathscr{F}^{\prime },f^{\ast }\mathscr{F})$ . A morphism $\mathscr{F}^{\prime }\rightarrow \mathscr{F}$ in $\mathfrak{Coh}_{X}$ over $f$ in $\mathscr{S}$ is cartesian if it induces an isomorphism $\mathscr{F}^{\prime }\cong f^{\ast }\mathscr{F}$ .

The linear stack $\mathfrak{Coh}_{X}$ is algebraic.

To see this, suppose $\mathscr{F}$ and $\mathscr{G}$ are coherent sheaves on $X\times S$ , flat over $S$ . The fact that $\operatorname{Hom̲}(\mathscr{F},\mathscr{G})$ is a locally coherent representable sheaf of $\mathscr{O}_{X}$ -modules follows from the fact that there exists a coherent sheaf $\mathscr{N}$ on the big étale site of $S$ such that $\operatorname{Hom̲}(\mathscr{F},\mathscr{G})=\mathscr{N}^{\vee }$ (see [Reference Grothendieck and DieudonnéGD67, EGA III 7.7.8, 7.7.9]). In fact, for a morphism of schemes $T\rightarrow S$ , we have $\operatorname{Hom̲}_{T}(\mathscr{F},\mathscr{G})={\unicode[STIX]{x1D70B}_{T}}_{\ast }\mathscr{H}om(\mathscr{F}_{X\times T},\mathscr{G}_{X\times T})$ . The fact that pushforward does not commute with arbitrary pullbacks means that $\operatorname{Hom̲}(\mathscr{F},\mathscr{G})$ is not in general cartesian and hence not in general coherent. On the other hand, by [Reference Grothendieck and DieudonnéGD67, EGA III 7.7.8, 7.7.9], we have ${\unicode[STIX]{x1D70B}_{T}}_{\ast }\mathscr{H}om(\mathscr{F}_{X\times T},\mathscr{G}_{X\times T})=(\mathscr{N}_{T})^{\vee }$ , which proves that, indeed, $\operatorname{Hom̲}(\mathscr{F},\mathscr{G})=\mathscr{N}^{\vee }$ .

The fact that $(\mathfrak{Coh}_{X})_{\text{cfg}}$ is algebraic and locally of finite type is proved in [Reference Laumon and Moret-BaillyLMB00, 4.6.2.1].

Example 1.12. As a special case of the previous example, consider the case $X=\operatorname{Spec}R$ . Then the linear algebraic stack $\mathfrak{Coh}_{\operatorname{Spec}R}$ is the linear stack of vector bundles, denoted by $\mathfrak{Vect}$ . The underlying algebraic stack $\mathfrak{Vect}_{\text{cfg}}$ is the disjoint union $\coprod _{n\geqslant 0}B\operatorname{GL}_{n}$ . The sheaf $\mathscr{H}$ over

$$\begin{eqnarray}\coprod _{n\geqslant 0}B\operatorname{GL}_{n}\times \coprod _{n\geqslant 0}B\operatorname{GL}_{n}=\coprod _{n,m\geqslant 0}B(\operatorname{GL}_{n}\times \operatorname{GL}_{m})\end{eqnarray}$$

is given by the natural representation $M(m\times n)$ of $\operatorname{GL}_{n}\times \operatorname{GL}_{m}$ over $B(\operatorname{GL}_{n}\times \operatorname{GL}_{m})$ .

Example 1.13. A generalization of $\mathfrak{Vect}$ in a different direction is given by quiver representations.

Let $Q$ be a quiver. The stack of representations of $Q$ , denoted by $\mathfrak{Rep}_{Q}$ , has as $\mathfrak{Rep}_{Q}(S)$ the set of diagrams $(\mathscr{F})$ in the shape of $Q$ of locally free finite rank $\mathscr{O}_{S}$ -modules. For a morphism $f:S^{\prime }\rightarrow S$ of $R$ -schemes, we have that $\operatorname{Hom}_{f}(\mathscr{F}^{\prime },\mathscr{F})$ is the $\mathscr{O}(S^{\prime })$ -module of homomorphisms $\mathscr{F}^{\prime }\rightarrow f^{\ast }\mathscr{F}$ of diagrams of locally free $\mathscr{O}_{S^{\prime }}$ -modules.

Example 1.14. As a toy example, let $A$ be a vector bundle over $\operatorname{Spec}R$ , endowed with the structure of a sheaf of $\mathscr{O}_{\operatorname{Spec}R}$ -modules, with smooth group scheme of units $A^{\times }$ , also of finite type. Then we define the linear stack of $A^{\times }$ -torsors to have as objects over the $R$ -scheme $S$ the right $A^{\times }$ -torsors over $S$ , and for $f:S^{\prime }\rightarrow S$ and $A^{\times }$ -torsors $P^{\prime }$ over $S^{\prime }$ and $P$ over $S$ , we set $\operatorname{Hom}_{f}(P^{\prime },P)=\operatorname{Hom}_{S^{\prime }}(P^{\prime },f^{\ast }P)=P^{\prime }\times _{A^{\times }}A\times _{A^{\times }}f^{\ast }P$ . In this example, the underlying algebraic stack is $BA^{\times }$ and we have $\mathscr{H}=A^{\times }\backslash A/A^{\times }$ .

The case $A=0$ is not excluded. The associated linear stack is $\operatorname{id}:\mathscr{S}\rightarrow \mathscr{S}$ . All $\operatorname{Hom}_{f}(x,y)$ are singletons, endowed with their unique module structure. This stack is represented by $\operatorname{Spec}R$ . It can also be thought of as the stack of zero-dimensional vector bundles.

Fibred products

Let $F:\mathfrak{X}\rightarrow \mathfrak{Z}$ and $G:\mathfrak{Y}\rightarrow \mathfrak{Z}$ be cartesian morphisms of $\mathscr{O}$ -linear fibred categories. We define a new $\mathscr{O}$ -linear fibred category $\mathfrak{W}$ as follows: objects of $\mathfrak{W}$ over the object $T$ of $\mathscr{S}$ are triples $(x,\unicode[STIX]{x1D6FC},y)$ , where $x$ is an $\mathfrak{X}$ -object over $T$ , $y$ is a $\mathfrak{Y}$ -object over $T$ , and $\unicode[STIX]{x1D6FC}$ is an isomorphism $\unicode[STIX]{x1D6FC}:F(x)\rightarrow G(y)$ over $T$ . A morphism from $(x^{\prime },\unicode[STIX]{x1D6FC}^{\prime },y^{\prime })$ to $(x,\unicode[STIX]{x1D6FC},y)$ over $T^{\prime }\rightarrow T$ is a pair of morphisms $f:x^{\prime }\rightarrow x$ over $T^{\prime }\rightarrow T$ and $g:y^{\prime }\rightarrow y$ over $T^{\prime }\rightarrow T$ such that $\unicode[STIX]{x1D6FC}\circ F(f)=G(g)\circ \unicode[STIX]{x1D6FC}^{\prime }$ .

In other words, we can write the set of morphisms from $(x^{\prime },\unicode[STIX]{x1D6FC}^{\prime },y^{\prime })$ to $(x,\unicode[STIX]{x1D6FC},y)$ over $\unicode[STIX]{x1D711}:T^{\prime }\rightarrow T$ as the fibred product

$$\begin{eqnarray}\operatorname{Hom}_{\unicode[STIX]{x1D711}}(x^{\prime },x)\times _{\operatorname{ Hom}_{\unicode[STIX]{x1D711}}(F(x^{\prime }),G(y))}\operatorname{Hom}_{\unicode[STIX]{x1D711}}(y^{\prime },y),\end{eqnarray}$$

and as each of the sets in this fibred product is an $\mathscr{O}(T^{\prime })$ -module, and the maps are linear, this fibred product is also an $\mathscr{O}(T^{\prime })$ -module. We leave it to the reader to verify that composition is bilinear.

Let us verify that $\mathfrak{W}$ is a fibred category. Suppose that $(x,\unicode[STIX]{x1D6FC},y)$ is a triple over $T$ and $\unicode[STIX]{x1D711}:T^{\prime }\rightarrow T$ a morphism in $\mathscr{S}$ . We construct a triple $(x^{\prime },\unicode[STIX]{x1D6FC}^{\prime },y^{\prime })$ over $T^{\prime }$ by taking as $x^{\prime }$ a pullback of $x$ via $\unicode[STIX]{x1D711}$ and as $y^{\prime }$ a pullback of $y$ via $\unicode[STIX]{x1D711}$ . Then, as $G$ is cartesian, $G(y^{\prime })$ is a pullback of $G(y)$ via $\unicode[STIX]{x1D711}$ . Hence, there exists a unique morphism $\unicode[STIX]{x1D6FC}^{\prime }:F(x^{\prime })\rightarrow G(y^{\prime })$ covering $T^{\prime }$ such that $\unicode[STIX]{x1D6FC}\circ F(x^{\prime }\rightarrow x)=G(y^{\prime }\rightarrow y)\circ \unicode[STIX]{x1D6FC}^{\prime }$ . Then $\unicode[STIX]{x1D6FC}^{\prime }$ is cartesian, because cartesian morphisms satisfy the necessary ‘two out of three’ property. Then $\unicode[STIX]{x1D6FC}^{\prime }$ is invertible, because cartesian morphisms covering an identity are invertible. The triple $(x^{\prime },\unicode[STIX]{x1D6FC}^{\prime },y^{\prime })$ comes with a given morphism to $(x,\unicode[STIX]{x1D6FC},y)$ that covers $\unicode[STIX]{x1D711}$ . It is easily verified that this morphism is cartesian.

Therefore, $\mathfrak{W}$ is an $\mathscr{O}$ -linear fibred category. By construction, the two projections $\mathfrak{W}\rightarrow \mathfrak{X}$ and $\mathfrak{W}\rightarrow \mathfrak{Y}$ are cartesian. We call $\mathfrak{W}$ the fibred product of $\mathfrak{X}$ and $\mathfrak{Y}$ over $\mathfrak{Z}$ .

Suppose $\mathfrak{X}$ , $\mathfrak{Y}$ , and $\mathfrak{Z}$ are algebraic, with underlying algebraic stacks $X$ , $Y$ , and $Z$ , respectively. For triples $(x^{\prime },\unicode[STIX]{x1D6FC}^{\prime },y^{\prime })$ and $(x,\unicode[STIX]{x1D6FC},y)$ over $S$ , the presheaf $\operatorname{Hom̲}((x^{\prime },\unicode[STIX]{x1D6FC}^{\prime },y^{\prime }),(x,\unicode[STIX]{x1D6FC},y))$ is equal to the fibred product

$$\begin{eqnarray}\operatorname{Hom̲}(x^{\prime },x)\times _{\operatorname{ Hom̲}(Fx^{\prime },Gy)}\operatorname{Hom̲}(y^{\prime },y)\end{eqnarray}$$

and is therefore a locally coherent representable sheaf of $\mathscr{O}_{S}$ -modules. We see that $\mathfrak{W}$ is again a linear algebraic stack. Moreover, the underlying algebraic stack of $W$ is the fibred product $X\times _{Z}Y$ .

Special linear stacks

For a linear algebraic stack $\mathfrak{M}$ , every fibre category $\mathfrak{M}(S)$ is an $R$ -linear category. By putting special requirements on these linear categories, we get stronger notions of linear algebraic stack.

For a linear algebraic stack $\mathfrak{M}$ , we denote the universal sheaf of endomorphisms by $\mathfrak{A}\rightarrow \mathfrak{M}$ .

Definition 1.16. A linear algebraic stack $\mathfrak{M}$ has a zero object if the $R$ -linear category $\mathfrak{M}(R)$ admits a zero object.

If $\mathfrak{M}$ admits a zero object, then for every $R$ -scheme $S$ , the $R$ -linear category $\mathfrak{M}(S)$ admits a zero object, namely, the pullback of the zero object in $\mathfrak{M}(\operatorname{Spec}R)$ via the unique morphism $S\rightarrow \operatorname{Spec}R$ .

A zero object for $\mathfrak{M}$ defines a section $\operatorname{Spec}R\stackrel{0}{\longrightarrow }\mathfrak{M}$ , which is an isomorphism onto the closed substack of $\mathfrak{M}$ defined by the condition $1=0$ inside $\mathfrak{A}$ .

If $\mathfrak{M}$ admits a zero object, we denote the complement of the zero object in $\mathfrak{M}$ by $\mathfrak{M}_{\ast }$ . It is a linear open substack of $\mathfrak{M}$ .

Definition 1.17. The linear algebraic stack $\mathfrak{M}$ admits direct sums if for every $R$ -scheme $S$ , the $R$ -linear category $\mathfrak{M}(S)$ admits all (finite) direct sums.

The pullback functor $\mathfrak{M}(S)\rightarrow \mathfrak{M}(S^{\prime })$ for a morphism of $R$ -schemes $S^{\prime }\rightarrow S$ commutes with direct sums. Hence, if $\mathfrak{M}$ admits direct sums, there is a canonical morphism of linear stacks

$$\begin{eqnarray}\displaystyle \mathfrak{M}\times \mathfrak{M} & \longrightarrow & \displaystyle \mathfrak{M}\nonumber\\ \displaystyle (x,y) & \longmapsto & \displaystyle x\oplus y.\nonumber\end{eqnarray}$$

See also Remark 1.63.

1.3 Finite type algebras

None of our algebras are assumed to be commutative, but they are all assumed to be unital.

The sheaf of $\mathscr{O}_{X}$ -modules underlying a finite type algebra is locally coherent representable.

For an automorphism $\unicode[STIX]{x1D711}$ of a finite type algebra $A$ , the fixed algebra $A^{\unicode[STIX]{x1D711}}$ is again a finite type algebra, because the underlying sheaf of modules is the kernel of $A\stackrel{1-\unicode[STIX]{x1D711}}{\longrightarrow }A$ .

If $\mathfrak{X}$ is a linear algebraic stack with underlying algebraic stack $X$ , then the universal sheaf of endomorphisms $\mathscr{E}\rightarrow X$ is a finite type algebra.

Note that finite type algebras need not have a coherent underlying sheaf of $\mathscr{O}_{X}$ -modules. For example, let $X=\mathbb{A}^{1}$ , with coordinate $t$ , and let $A\rightarrow X$ be the centralizer in $(M_{2\times 2})_{X}$ of the matrix $(\begin{smallmatrix}t & 0\\ 0 & 0\end{smallmatrix})$ . In this case, the big sheaf underlying $A$ is not cartesian.

Inertia representation

Whenever $A\rightarrow X$ is an algebra over the algebraic stack $X$ , we have a tautological morphism of sheaves of groups over $X$ ,

(6) $$\begin{eqnarray}I_{X}\longrightarrow \operatorname{Aut̲}(A).\end{eqnarray}$$

Here, $I_{X}$ is the inertia stack of $X$ , i.e., the stack of pairs $(x,\unicode[STIX]{x1D711})$ where $x$ is an object of $X$ and $\unicode[STIX]{x1D711}$ an automorphism of $x$ , and $\operatorname{Aut̲}(A)$ is the sheaf of automorphisms of the sheaf of algebras $A$ over $X$ . To construct (6), consider the stack of sheaves of algebras $\mathfrak{Alg}$ over $\mathscr{S}$ , which has as objects over the scheme $S$ the sheaves of $\mathscr{O}_{S}$ -algebras on the usual (small) étale site of $S$ . A morphism from the sheaf of $\mathscr{O}_{S^{\prime }}$ -algebras $A^{\prime }$ over $S^{\prime }$ , covering the morphism of schemes $f:S^{\prime }\rightarrow S$ to the sheaf of $\mathscr{O}_{S}$ -algebras $A$ over $S$ , is, by definition, an isomorphism of sheaves of $\mathscr{O}_{S^{\prime }}$ -algebras $A^{\prime }\rightarrow f^{\ast }A$ . The sheaf of algebras $A\rightarrow X$ gives rise to a morphism of $\mathscr{S}$ -stacks $a:X\rightarrow \mathfrak{Alg}$ . We get an induced morphism on inertia stacks $I_{X}\rightarrow I_{\mathfrak{Alg}}$ and notice that $a^{\ast }I_{\mathfrak{Alg}}=\operatorname{Aut̲}(A)$ .

With this definition, an automorphism $\unicode[STIX]{x1D711}$ of the object $x$ of the stack $X$ is mapped to the inverse of the restriction morphism $\unicode[STIX]{x1D711}^{\ast }:A(x)\rightarrow A(x)$ .

Lemma 1.20. Suppose $X$ is a gerbe over the algebraic space $Y$ and $A\rightarrow X$ is an algebra. Then there exists a sheaf of $\mathscr{O}_{Y}$ -algebras $B$ and an isomorphism $A\cong B|_{X}$ if and only if the inertia representation $I_{X}\rightarrow \operatorname{Aut̲}_{X}(A)$ is trivial.

Similarly, if $X$ is a connected gerbe over the Deligne–Mumford stack $Y$ , then an algebra $A$ over $X$ descends to $B$ over $Y$ if and only if the inertia representation restricts to a trivial homomorphism $I_{X}^{\circ }\rightarrow \operatorname{Aut̲}_{X}(A)$ .

In either case, $A$ is representable or of finite type if and only if $B$ is.

We can pull back the sheaf of algebras $A$ over $X$ , via the structure morphism $I_{X}\rightarrow X$ , to obtain the sheaf of algebras $A|_{I_{X}}$ . This sheaf of algebras is endowed with a tautological automorphism, induced from (6). We shall denote the algebra of invariants for this automorphism by $A_{I_{X}}^{\text{fix}}$ .

The following statement is somewhat tautological and holds more generally than for algebras.

Proposition 1.21. Suppose that $A$ is a representable algebra over the algebraic stack $X$ . Then the inertia stack of $A$ is naturally identified with $A_{I_{X}}^{\text{fix}}$ . In particular, $I_{A}$ is a representable algebra over $I_{X}$ .

Proof. We have a commutative diagram of algebraic stacks

which identifies $I_{A}$ with a substack of $A|_{I_{X}}$ . The algebra $A|_{I_{X}}$ is the stack of triples $(x,\unicode[STIX]{x1D711},a)$ , where $x$ is an object of $X$ , $\unicode[STIX]{x1D711}$ is an automorphism of $x$ , and $a\in A(x)$ is an object of $A$ lying over $x$ . Such a triple is in $I_{A}$ if and only if $\unicode[STIX]{x1D711}\in \operatorname{Aut}(x)$ is in the subgroup $\operatorname{Aut}(a)\subset \operatorname{Aut}(x)$ . This is equivalent to $\unicode[STIX]{x1D711}$ fixing $a$ under the action of $\operatorname{Aut}(x)$ on $A(x)$ . This is the claim.◻

In fact, the fibre of $I_{A}$ over the object $x$ of $X$ is equal to

$$\begin{eqnarray}I_{A}(x)=\{(\unicode[STIX]{x1D711},a)\in \operatorname{Aut}(x)\times A(x)\mid \unicode[STIX]{x1D711}^{\ast }(a)=a\}.\end{eqnarray}$$

The fibre of $I_{A}(x)$ over $\unicode[STIX]{x1D711}\in \operatorname{Aut}(x)$ is the subalgebra $A(x)^{\unicode[STIX]{x1D711}}\subset A(x)$ , and the fibre of $I_{A}(x)$ over $a\in A(x)$ is the subgroup $\operatorname{Stab}_{\operatorname{Aut}(x)}(a)\subset \operatorname{Aut}(x)$ .

The degree stratification

Let $k$ be a field and $A$ a finite-dimensional $k$ -algebra. The rank $r$ of $A$ is the dimension of $A$ as a $k$ -vector space. For an element $a\in A$ , we define its degree to be the dimension of the commutative subalgebra $k[a]\subset A$ . It is equal to the degree of the minimal polynomial of $a$ , i.e., the monic generator of the kernel of the algebra map $k[x]\rightarrow A$ , defined by $x\mapsto a$ .

Now let $A$ be an algebra bundle of rank $r$ over the algebraic stack $X$ , and $a\in A(S)$ a local section of $A$ over an $X$ -scheme $S$ .

Definition 1.31. If the cokernel (as a homomorphism of $\mathscr{O}_{S}$ -modules) of the morphism of $\mathscr{O}_{S}$ -algebras $\mathscr{O}_{S}[x]\rightarrow A_{S}$ defined by $x\mapsto a$ is flat over $\mathscr{O}_{S}$ , we say that $a$ is strict, and we call the rank of the image of $\mathscr{O}_{S}[x]\rightarrow A$ the degree of $a$ .

If $f(x)\in \mathscr{O}_{S}[x]$ is the characteristic polynomial of $a$ , the morphism $\mathscr{O}_{S}[x]\rightarrow A$ factors through $\mathscr{O}_{S}[x]/(f)$ , by the theorem of Caley–Hamilton. Hence, the cokernel of $\mathscr{O}_{S}[x]\rightarrow A$ is actually a cokernel of a homomorphism of vector bundles and hence coherent. The condition that this cokernel be flat is equivalent to it being locally free. It implies that forming the image of $\mathscr{O}_{S}[x]\rightarrow A$ commutes with base change, and that this image, denoted by $\mathscr{O}_{S}[a]$ , is also locally free.

Proposition 1.32. For every $n=1,\ldots ,r$ , there exists a locally closed substack $A_{n}\subset A$ with the property that a local section $a\in A(S)$ factors through $A_{n}(S)$ if and only if $a$ is strict of degree $n$ . The $A_{n}$ are pairwise disjoint and their disjoint union

$$\begin{eqnarray}A^{\text{strat}}=\coprod _{n=1}^{r}A_{n}\end{eqnarray}$$

maps surjectively to $A$ . The section $a\in A(S)$ factors through $A^{\text{strat}}\rightarrow A$ if and only if it is strict.

Proof. Consider the tautological section $\unicode[STIX]{x1D6E5}$ of the pullback of $A$ via the structure map $A\rightarrow X$ . It gives rise to a morphism of $\mathscr{O}_{A}$ -algebras $\mathscr{O}_{A}[x]\rightarrow A_{A}$ . Then $A^{\text{strat}}$ is given by the flattening stratification of its cokernel, and $A_{n}\subset A^{\text{strat}}$ is the component where the cokernel has rank $r-n$ .◻

We call the stratification $A^{\text{strat}}\rightarrow A$ the degree stratification of $A$ .

Semi-simple elements

Let $k$ be an algebraically closed field and $A$ a finite-dimensional $k$ -algebra. Recall that an element $a\in A$ is semi-simple if the following equivalent conditions are satisfied.

1. (i) The map $A\rightarrow A$ given by left multiplication by $a$ is diagonalizable.

2. (ii) The minimal polynomial $f\in k[x]$ of $a$ is separable, i.e., satisfies $(f,f^{\prime })=1$ .

3. (iii) The commutative subalgebra $k[a]\subset A$ is reduced, or, equivalently, étale over $k$ .

Definition 1.33. Let $A\rightarrow X$ be an algebra bundle. A local section $a\in A(S)$ , for an $X$ -scheme $S$ , is called semi-simple if it is strict and for every geometric point $s\in S$ , the element induced by $a$ in $A(s)$ is semi-simple.

For example, an idempotent section $e\in A(S)$ is semi-simple over the open subset of $S$ , where $e$ is neither 0 nor 1 (in a commutative algebra bundle, this subset is also closed).

Assuming $a\in A(S)$ is strict, $a$ is semi-simple if and only if the geometric fibres of the finite flat $S$ -scheme $\operatorname{Spec}_{S}\mathscr{O}_{S}[a]$ are unramified. This condition is equivalent to $\operatorname{Spec}_{S}\mathscr{O}_{S}[a]$ being unramified and hence étale over $S$ .

The semi-simple sections of $A$ form a subsheaf $A^{\text{ss}}\subset A$ .

Proposition 1.34. Let $A\rightarrow X$ be an algebra bundle over the algebraic stack $X$ . Then $A^{\text{ss}}$ is an algebraic stack with a representable structure morphism of finite type $A^{\text{ss}}\rightarrow X$ .

Proof. In fact, $A^{\text{ss}}\subset A^{\text{strat}}$ is the open substack defined by the condition that the finite flat representable morphism $\operatorname{Spec}_{A}\mathscr{O}_{A}[\unicode[STIX]{x1D6E5}]\rightarrow A$ is unramified. (The section $\unicode[STIX]{x1D6E5}$ is the tautological one, as in the proof of Proposition 1.32.) Thus, we have a factorization of the monomorphism $A^{\text{ss}}\rightarrow A$ as

Thus, $A^{\text{ss}}\subset A$ is a constructible substack.◻

The semi-simple centre

For a commutative finite-dimensional algebra over an algebraically closed field, we have the following facts.

1. (i) The primitive idempotents are linearly independent.

2. (ii) An element is semi-simple if and only if it is a linear combination of primitive idempotents.

We need a version of this statement for algebra bundles.

Proposition 1.35. Let $A\rightarrow X$ be a commutative algebra bundle whose stack of idempotents is finite étale, and let $Y\rightarrow X$ be the finite flat cover corresponding to $A$ . There is a canonical finite flat morphism of $X$ -stacks $Y\rightarrow PE(A)$ . Over every geometric point $x$ of $X$ , this morphism maps each point in the fibre $Y_{x}$ to the characteristic function of its connected component in $Y_{x}$ (which is a primitive idempotent in $A|_{x}$ ). Dually, we obtain a strict monomorphism of algebra bundles

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{\ast }\mathscr{O}_{PE(A)}\longrightarrow A,\end{eqnarray}$$

where $\unicode[STIX]{x1D70B}:PE(A)\rightarrow X$ is the structure map.

The induced morphism

$$\begin{eqnarray}(\unicode[STIX]{x1D70B}_{\ast }\mathscr{O}_{PE(A)})^{\text{strat}}\longrightarrow A^{\text{strat}}\end{eqnarray}$$

factors through the open substack $A^{\text{ss}}\subset A^{\text{strat}}$ and induces a surjective closed immersion of algebraic stacks $(\unicode[STIX]{x1D70B}_{\ast }\mathscr{O}_{PE(A)})^{\text{strat}}\rightarrow A^{\text{ss}}$ .

Proof. Consider the finite étale cover of primitive idempotents $\unicode[STIX]{x1D70B}:PE(A)\rightarrow X$ . We have a tautological global section $e$ of $A|_{PE(A)}$ , and $a\mapsto ae$ defines a homomorphism of $\mathscr{O}_{PE(A)}$ -modules $\mathscr{O}_{PE(A)}\rightarrow A|_{PE(A)}$ . Pushing forward with $\unicode[STIX]{x1D70B}$ and composing with the trace map $\unicode[STIX]{x1D70B}_{\ast }(A|_{PE(A)})\rightarrow A$ defines the morphism of algebra bundles over $X$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{\ast }\mathscr{O}_{PE(A)}\longrightarrow A.\end{eqnarray}$$

It is a strict monomorphism of vector bundles, because it is injective over every geometric point, by fact (i), above. Dually, we obtain a morphism of $X$ -stacks $Y\rightarrow PE(A)$ , which is the morphism described in the statement of the proposition. It is flat, because $PE(A)$ is étale over $X$ , and flatness can be checked étale locally.

Passing to the degree stratification commutes with strict monomorphisms of algebra bundles, so we have a cartesian diagram of $X$ -stacks

which shows that $(\unicode[STIX]{x1D70B}_{\ast }\mathscr{O}_{PE(A)})^{\text{strat}}\rightarrow A^{\text{strat}}$ is a closed immersion. That this closed immersion factors through $A^{\text{ss}}\subset A^{\text{strat}}$ and is surjective onto $A^{\text{ss}}$ can be checked over the geometric points of $X$ , where it follows from fact (ii), above.◻