Introduction
For a smooth, irreducible, complete, algebraic curve $X$ , we denote by $F$ the field of rational functions, by $\mathbb{O}$ the product $\prod _{x\in X_{\text{cl}}}\widehat{{\mathcal{O}}}_{x}$ ranging over closed points, and
This object is called the ring of adèles. André Weil was probably the first to appreciate the close connection between adèles and the geometry of curves (see the letter [Reference WeilWei38b] to Hasse where the case of line bundles is discussed, and [Reference WeilWei38a] for the closely related notion of matrix divisors).
Theorem 0.1 (Weil).
Let $X$ be an algebraic curve, defined over an algebraically closed field $k$ , and let $G$ be a reductive algebraic group. We then have an equivalence between the groupoid of $G$ torsors on $X$ , $BG(X)$ , and the groupoid defined by the double quotient $[G(F)\setminus G(\mathbb{A})/G(\mathbb{O})]$ .
Weil’s theorem is central to the geometric Langlands programme, as it connects the arithmetic conjectures to their geometric counterpart. For a survey of this connection see [Reference FrenkelFre07]. The interplay of Weil’s result with conformal field theory is discussed by Witten [Reference WittenWit88, § V].
In this article we present a generalisation of Weil’s theorem to arbitrary Noetherian schemes. We will deduce it from an adelic descent result for the perfect complexes. The cosimplicial ring $\mathbb{A}_{X}^{\bullet }$ was introduced by Beilinson in [Reference BeilinsonBei80], as a generalisation of the theory of adèles for curves. A similar construction has also been obtained by Parshin for algebraic surfaces. If $X$ is a curve, the cosimplicial ring is given by the diagram
which captures the adelic rings $F$ , $\mathbb{O}_{X}$ and $\mathbb{A}_{X}$ , and the various maps between them, used to formulate Weil’s Theorem 0.1.
Theorem 0.2 (Adelic descent).
Let $X$ be a Noetherian scheme. We denote by $\mathbb{A}_{X}^{\bullet }$ Beilinson’s cosimplicial ring of adèles (see Definition 1.4). We have an equivalence of symmetric monoidal $\infty$ categories $\mathsf{Perf}(X)_{\otimes }\simeq \mathsf{Perf}(\mathbb{A}_{X}^{\bullet })_{\otimes }$ , where the righthand side denotes the $\infty$ category of cartesian $\mathbb{A}_{X}^{\bullet }$ modules.
This theorem also holds for almost perfect complexes, as we show in Corollary 3.39. According to Lieblich, the study of perfect complexes is the mother of all moduli problems (see the abstract of [Reference LieblichLie06]). The Tannakian formalism enables us to make this philosophical principle precise. Using the results of Bhatt [Reference BhattBha16] and Bhatt and HalpernLeistner [Reference Bhatt and HalpernLeistnerBHL15], we obtain a descent result for $G$ torsors (we may replace $BG$ by more general algebraic stacks).
Theorem 0.3. Let $X$ be a Noetherian scheme. The geometric realisation of the simplicial affine scheme $\operatorname{Spec}\mathbb{A}_{X}^{\bullet }$ in the category of Noetherian algebraic stacks with quasiaffine diagonal is canonically equivalent to $X$ . In particular, we have $BG(X)\simeq BG(\operatorname{Spec}\mathbb{A}_{X}^{\bullet })$ , if $G$ is a Noetherian affine algebraic group scheme. Let $G$ be a special group scheme (for example, $G=\operatorname{GL}_{n}$ ). We denote by $G(\mathbb{A}_{X}^{1})^{\text{cocycle}}$ the subset consisting of $\unicode[STIX]{x1D719}\in \mathbb{G}(\mathbb{A}_{X}^{1})$ satisfying the cocycle condition $\unicode[STIX]{x1D719}_{02}=\unicode[STIX]{x1D719}_{01}\circ \unicode[STIX]{x1D719}_{12}$ in $G(\mathbb{A}_{X}^{2})$ . There is an equivalence of groupoids $BG(X)\simeq [G(\mathbb{A}_{X}^{1})^{\text{cocycle}}/G(\mathbb{A}_{X}^{0})]$ .
In characteristic 0, the assumption that $G$ be Noetherian can often be dropped. We refer the reader to Corollary 3.35. We refer the reader to § 3.2.4 for a more detailed discussion of the adelic description of $G$ bundles on Noetherian schemes $X$ . The case of punctured surfaces has also been considered by Garland and Patnaik in [Reference Garland and PatnaikGP]. In [Reference ParshinPar83], Parshin used adelic cocycles for $G$ bundles as above to obtain formulae for Chern classes in adelic terms.
As a further consequence of the adelic descent formalism, we obtain an analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces [Reference Gelfand and NaimarkGN43]. Recall that [Reference Gelfand and NaimarkGN43] shows that a locally compact topological space can be reconstructed from the ring of continuous functions. It is wellknown that a similar result cannot hold for nonaffine schemes. However, our result implies that a Noetherian scheme $X$ can be reconstructed from the cosimplicial ring of adèles.
Theorem 0.4. The functor $\mathbb{A}^{\bullet }:\mathsf{Sch}^{N}{\hookrightarrow}(\mathsf{Rng}^{\unicode[STIX]{x1D6E5}})^{\mathsf{op}}$ from the category of Noetherian schemes to the dual category of cosimplicial commutative rings, has an explicit leftinverse, sending $R^{\bullet }$ to $\operatorname{Spec}\,R^{\bullet }$ .
It is instructive to meditate on the differences and similarities with Gelfand–Naimark’s theorem. While their result copes with plain rings, our Theorem 0.4 requires a diagram of rings (see Corollary 3.33 for a precise statement to which extent the cosimplicial structure is needed). However, the necessary condition of local compactness for topological spaces is not unlike the restriction that the scheme be Noetherian.
For a quasicompact and quasiseparated scheme $X$ we may choose a finite cover by affine open subschemes $\{U_{i}\}_{i=1,\ldots ,n}$ . The coproduct $U=\coprod _{i=1}^{n}U_{i}$ is then still an affine scheme, and we have a map $U\rightarrow X$ . Choosing a finite affine covering for $U\times _{X}U$ , and iterating this procedure, we arrive at a simplicial affine scheme $U_{\bullet }\rightarrow X$ , which yields a hypercovering of $X$ . The coordinate ring yields a cosimplicial ring $\unicode[STIX]{x1D6E4}(U_{\bullet })$ associated to $X$ . However, this construction is a priori not functorial, since it depends on the chosen coverings. Nonetheless, using the construction $X\mapsto X^{Z}$ introduced by Bhatt and Scholze [Reference Bhatt and ScholzeBS15], one obtains another functor as in Theorem 0.4 (the author thanks Bhatt for bringing this to his attention).
Our Theorem 0.2 relies heavily on Beilinson’s [Reference BeilinsonBei80], which constructs a functor, sending a quasicoherent sheaf ${\mathcal{F}}$ on $X$ to an $\mathbb{A}_{X}^{\bullet }$ module $\mathbb{A}_{X}^{\bullet }({\mathcal{F}})$ . Beilinson observes that the latter cosimplicial module gives rise to a chain complex, computing the cohomology of ${\mathcal{F}}$ . This chain complex can be obtained by applying the Dold–Kan correspondence, or taking the alternating sum of the face maps in each degree: $[\mathbb{A}^{0}({\mathcal{F}})\xrightarrow[{}]{\unicode[STIX]{x2202}_{0}\unicode[STIX]{x2202}_{1}}\mathbb{A}^{1}({\mathcal{F}})\cdots \,]$ . Beilinson’s result can be stated as
The reason is that the sheaves $\mathsf{A}_{X}^{k}:U\mapsto \mathbb{A}_{U}^{k}({\mathcal{F}})$ are flasque, and hence it remains to show that the corresponding complex of sheaves defines a flasque resolution of ${\mathcal{F}}$ . The details are explained in [Reference HuberHub91]. Since morphisms in $\mathsf{Perf}(X)$ are closely related to sheaf cohomology, it is not difficult to deduce from Beilinson’s observation the existence of a fully faithful functor $\mathsf{Perf}(X){\hookrightarrow}\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })$ , amounting to a sort of a cohomological descent result. Our proof of the adelic descent theorem is, hence, mainly concerned with establishing that this functor is essentially surjective, that is, establishing effectivity of descent for objects in those $\infty$ categories. In heuristic terms, our theorem asserts that a perfect complex $M$ on $X$ can be described by an iterative formal glueing procedure from the adelic parts $\mathbb{A}_{X}^{\bullet }(M)$ .
Our main theorem uses the language of stable $\infty$ categories. Replacing $\mathsf{Perf}(X)$ and $\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })$ by their homotopy categories would render the result incorrect. However, the theorem could be formulated in more classical language. The $\infty$ category of cartesian perfect modules over a cosimplicial ring, such as $\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })$ , has a model, as is discussed in [Reference Toën and VezzosiTV08, 1.2.12] by Toën and Vezzosi.
Since the construction of $\mathbb{A}_{X}^{\bullet }$ involves iterative completion and localisation procedures, the formal descent result of Beauville and Laszlo [Reference Beauville and LaszloBL95] and BenBassat and Temkin [Reference BenBassat and TemkinBBT13] (for quasicoherent sheaves in each case) are closely related. These theorems allow one to glue sheaves on a scheme $X$ with respect to the formal neighbourhood of a closed subvariety $Y$ , and its open complement $X\setminus Y$ . Beauville–Laszlo developed such a descent theory for an affine scheme $X$ , and a closed subvariety $Y$ given by a principal ideal. This result was motivated by the study of conformal blocks [Reference Beauville and LaszloBL94]. The second article does not require the restrictions of $X$ to be affine and $Y$ to be principal, however it utilises the theory of Berkovich spaces to formulate the glueing result. Recently their theory has been extended to treat flags of subvarieties by Hennion, Porta and Vezzosi [Reference Hennion, Porta and VezzosiHPV16]. Our Theorem 0.2 gives a very similar descent theory, but uses all closed subvarieties at once and avoids rigid geometry.
One of the key properties of the adèles allowing one to establish effectivity of adelic descent data is a strengthening of the theory of flasque sheaves of algebras.
Theorem 0.5. Let $X$ be a quasicompact topological space and $\mathsf{A}$ a lâche sheaf of algebras (see Definition 2.9), with ring of global sections $R=\unicode[STIX]{x1D6E4}(\mathsf{A})$ . The global section functor $\unicode[STIX]{x1D6E4}:\mathsf{Mod}(\mathsf{A})\rightarrow \mathsf{Mod}(\mathsf{A})$ restricts to a symmetric monoidal equivalence $\mathsf{Perf}(\mathsf{A})_{\otimes }\simeq \mathsf{Perf}(R)_{\otimes }$ .
We show in Lemma 1.14 that the adèles $\mathbb{A}_{X}^{k}$ are lâche sheaves of algebras. The derived equivalence underlying our theorems decomposes into two parts:
The second equivalence is deduced from Theorem 0.5. The first equivalence can be established by local verifications.
1 A reminder of Beilinson–Parshin adèles
In [Reference BeilinsonBei80] Beilinson generalised the notion of adèles to arbitrary Noetherian schemes, and studied the connection adèles bear with coherent cohomology. We briefly review his definition and the main properties of relevance to us. Except for the assertion that adèles are flasque sheaves (Corollary 1.15), we will not provide a proof for those statements and refer the reader instead to Huber [Reference HuberHub91]. Examples can be found in Morrow’s survey article about adèles and their relation to higher local fields [Reference MorrowMor12].
1.1 Recollection
Henceforth we denote by $X$ a Noetherian scheme, with underlying topological space $X$ and structure sheaf ${\mathcal{O}}_{X}$ .
Definition 1.1. Let $X$ be a scheme with underlying topological space $X$ . For $x,y\in X$ we write $x\leqslant y$ for $x\in \overline{\{y\}}$ ; this defines a partially ordered set $X_{{\leqslant}}$ . We denote the set $\{(x_{0},\ldots ,x_{k})\in X^{\times k+1}x_{0}\leqslant \cdots \leqslant x_{k}\}$ by $X_{k}$ .
One sees that $X_{k}$ is in fact the set of $k$ simplices of a simplicial set $X_{\bullet }$ . This simplicial structure will be reflected in a cosimplicial structure for Beilinson–Parshin adèles.
Definition 1.2. The simplicial set $X_{\bullet }:\unicode[STIX]{x1D6E5}^{\mathsf{op}}\rightarrow \operatorname{Set}$ is defined to be the functor, sending $[n]\in \unicode[STIX]{x1D6E5}^{\mathsf{op}}$ to the set of ordered maps $[n]\rightarrow X_{{\leqslant}}$ , where $X_{{\leqslant}}$ refers to the partially ordered set defined in Definition 1.1.
Following [Reference BeilinsonBei80] we define adèles with respect to a subset $T\subset X_{k}$ . The case of interest to us will be $T=X_{k}$ , but the recursive nature of the definition necessitates a definition for general subsets $T\subset X_{k}$ . We begin with the following preliminary definitions.
Definition 1.3. Let $X$ be a Noetherian scheme and $k\in \mathbb{N}$ a nonnegative integer:

(a) for $x\in X$ and $T\subset X_{k}$ we define $\text{}_{x}T=\{(x_{0}\leqslant \cdots \leqslant x_{k1})\in X^{\times k}(x_{0}\leqslant \cdots \leqslant x_{k1}\leqslant x)\in T\}$ ;

(b) for $x\in X$ we denote by ${\mathcal{O}}_{x}$ the local ring at $x$ with maximal ideal $\mathfrak{m}_{x}$ ; there is a canonical morphism $j_{rx}:\operatorname{Spec}{\mathcal{O}}_{x}/\mathfrak{m}_{x}^{r}\rightarrow X$ .
It is convenient to define adèles in a higherdimensional situation as sheaves of ${\mathcal{O}}_{X}$ modules.
Definition 1.4. Let $X$ be a Noetherian scheme. The adèles are the unique family of exact functors $\mathsf{A}_{X,T}():\mathsf{QCoh}(X)\rightarrow \mathsf{Mod}({\mathcal{O}}_{X})$ , indexed by subsets $T\subset X_{k}$ , satisfying the following conditions:

(a) the functor $\mathsf{A}_{X,T}()$ commutes with directed colimits;

(b) for ${\mathcal{F}}\in \mathsf{Coh}(X)$ and $k=0$ we have $\mathsf{A}_{X,T}({\mathcal{F}})=\prod _{x\in T}\mathop{\varprojlim }\nolimits_{r\geqslant 0}(j_{rx})_{\ast }(j_{rx})^{\ast }{\mathcal{F}}$ ;

(c) for ${\mathcal{F}}\in \mathsf{Coh}(X)$ and $k\geqslant 1$ we have $\mathsf{A}_{X,T}({\mathcal{F}})=\prod _{x\in X}\mathop{\varprojlim }\nolimits_{r\geqslant 0}\mathsf{A}_{X,\text{}_{x}T}((j_{rx})_{\ast }(j_{rx})^{\ast }{\mathcal{F}})$ .
We refer the reader to [Reference HuberHub91] for a detailed verification that the above family of functors is welldefined and exact. The ring of adéles with respect to $T\subset X_{n}$ is defined by taking global sections of the sheaf of rings $\mathsf{A}_{X,T}({\mathcal{O}}_{X})$ . Moreover, it is important to emphasise that the sheaves of ${\mathcal{O}}_{X}$ modules $\mathsf{A}_{X,T}({\mathcal{F}})$ are in general not quasicoherent.
Definition 1.5. We denote the abelian group $\unicode[STIX]{x1D6E4}(X,\mathsf{A}_{X,T}({\mathcal{F}}))$ by $\mathbb{A}_{X,T}({\mathcal{F}})$ ; and reserve the notation $\mathbb{A}_{X,T}$ for $\mathbb{A}_{X,T}({\mathcal{O}}_{X})$ . By construction $\mathbb{A}_{X,T}({\mathcal{F}})$ is an $\mathbb{A}_{X,T}$ module.
As we already alluded to, the most interesting case for us is when $T=X_{k}$ . We reserve a particular notation for this situation.
Definition 1.6. We denote the sheaf $\mathsf{A}_{X,X_{k}}({\mathcal{F}})$ by $\mathsf{A}_{X}^{k}({\mathcal{F}})$ . The abelian group $\unicode[STIX]{x1D6E4}(X,\mathsf{A}_{X}^{k}({\mathcal{F}}))$ will be denoted by $\mathbb{A}_{X}^{k}({\mathcal{F}})$ .
As the superscript indicates, these sheaves can be assembled into a cosimplicial object. The proof of this can be found in [Reference HuberHub91, Theorem 2.4.1].
Proposition 1.7. Let $X$ be a Noetherian scheme, and $T_{\bullet }\subset X_{\bullet }$ a simplicial subset. There is an exact functor $\mathsf{A}_{X,T_{\bullet }}^{\bullet }:\mathsf{QCoh}(X)\rightarrow \mathsf{Fun}(\unicode[STIX]{x1D6E5},\mathsf{Mod}({\mathcal{O}}_{X}))$ , which commutes with directed colimits, and maps $([k],{\mathcal{F}})$ to $\mathsf{A}_{X,T_{k}}({\mathcal{F}})$ . We denote the functor $\unicode[STIX]{x1D6E4}(X,\mathsf{A}_{X,T_{\bullet }}^{\bullet }()):\mathsf{QCoh}(X)\rightarrow \mathsf{Fun}(\unicode[STIX]{x1D6E5},\mathsf{AbGrp})$ by $\mathbb{A}_{X,T_{\bullet }}^{\bullet }()$ ; it is exact and commutes with directed colimits. The notation $\mathbb{A}_{X}^{\bullet }()$ is reserved for the functor corresponding to the case $T_{\bullet }=X_{\bullet }$ . We shall write $\mathbb{A}_{X}^{\bullet }$ to denote the cosimplicial ring obtained by applying this functor to the structure sheaf ${\mathcal{O}}_{X}$ .
Let $X$ be an irreducible Noetherian scheme of dimension 1, and ${\mathcal{F}}$ a coherent sheaf on $X$ . We will discuss how the definitions above recover the classical theory of adèles for algebraic curves. Following classical conventions, we denote by
the restricted product ranging over all closed points $x\in X_{\text{cl}}$ . We denote by
and by $F_{X}({\mathcal{F}})$ the ${\mathcal{O}}$ module ${\mathcal{F}}_{\unicode[STIX]{x1D702}}$ , where $\unicode[STIX]{x1D702}$ is the generic point of $X$ . With respect to this notation we may identify the cosimplicial ${\mathcal{O}}$ module $\mathbb{A}_{X}^{\bullet }({\mathcal{F}})$ with
where $F_{X}({\mathcal{F}})\rightarrow \mathbb{A}_{X}({\mathcal{F}})$ is the diagonal inclusion, and $\mathbb{O}_{X}({\mathcal{F}})\rightarrow \mathbb{A}_{X}({\mathcal{F}})$ the canonical map. Embracing the usual redundancies in cosimplicial objects, that is the continual reappearance of factors already seen at a lower degree level, we observe that Beilinson’s $\mathbb{A}_{X}^{\bullet }$ captures the classical theory of adèles.
It is also helpful to understand the cosimplicial structure in the local case. Let $\unicode[STIX]{x1D70E}:[n]\rightarrow X_{{\leqslant}}$ be an element of $X_{n}$ . We denote the ring of adèles $\mathbb{A}_{X,\{\unicode[STIX]{x1D70E}\}}$ corresponding to $T=\{\unicode[STIX]{x1D70E}\}\subset X_{n}$ by $\mathbb{A}_{X,\unicode[STIX]{x1D70E}}$ . Proposition 1.7 implies that for every map $f:[m]\rightarrow [n]$ in $\unicode[STIX]{x1D6E5}$ we have a ring homomorphism $\mathbb{A}_{X,\unicode[STIX]{x1D70E}\circ f}\rightarrow \mathbb{A}_{X,\unicode[STIX]{x1D70E}}$ . The following assertion is also proven in [Reference HuberHub91, Theorem 2.4.1].
Lemma 1.8. Let $X$ and $T_{\bullet }$ be as in Proposition 1.7. The cosimplicial ring $\mathbb{A}_{X,T_{\bullet }}^{\bullet }$ injects into the cosimplicial ring
We also need the following observation, which is a consequence of the definitions of adèles.
Remark 1.9. If ${\mathcal{F}}$ is a quasicoherent sheaf on $X$ , settheoretically supported on a finite union of closed points $Z\subset X$ , then we have ${\mathcal{F}}\simeq \mathsf{A}_{X}^{k}({\mathcal{F}})$ .
Another observation which we will need is that for an affine Noetherian scheme $X$ , the functor
can be expressed as $\otimes _{\unicode[STIX]{x1D6E4}({\mathcal{O}}_{X})}\mathbb{A}_{X,T}$ . This is the case, since $\mathbb{A}_{X,T}()$ , and $\otimes $ commute with filtered colimits. Since $\mathbb{A}_{X,T}()$ takes values in flasque sheaves by [Reference HuberHub91, Proposition 2.1.5] (see also Corollary 1.15), we see that $\mathbb{A}_{X,T}()$ is an exact functor. Therefore, $\mathbb{A}_{X,T}$ is a flat algebra over $\unicode[STIX]{x1D6E4}({\mathcal{O}}_{X})$ . We record this for later use.
Lemma 1.10. Let $X=\operatorname{Spec}R$ be an affine Noetherian scheme. Then $\mathbb{A}_{X,T}$ is a flat $R$ algebra.
1.2 Functoriality
If $f:X\rightarrow Y$ is a morphism of Noetherian schemes, we have an induced map of partially ordered sets $X_{{\leqslant}}\rightarrow Y_{{\leqslant}}$ . Indeed, $x\in \overline{\{y\}}$ implies $f(x)\in \overline{\{f(y)\}}$ . In addition, we have an induced morphism of local rings ${\mathcal{O}}_{Y,f(x)}\rightarrow {\mathcal{O}}_{X,x}$ . These observations are the building blocks of a functoriality property satisfied by adèles. To the best of the author’s knowledge, this property has not yet been recorded in the literature.
Lemma 1.11. Let $f:X\rightarrow Y$ be a morphism of Noetherian schemes, and ${\mathcal{F}}\in \mathsf{QCoh}(Y)$ a quasicoherent sheaf on $Y$ . For $T\subset X_{n}$ and $f(T)\subset T^{\prime }\subset Y_{n}$ we have a morphism $\mathsf{A}_{Y,T^{\prime }}({\mathcal{F}})\rightarrow f_{\ast }\mathsf{A}_{X,T}(f^{\ast }{\mathcal{F}})$ in $\mathsf{Mod}({\mathcal{O}}_{Y})$ , fitting into the following commutative diagram.
Commutativity of this diagram amounts to the construction inducing a map of augmented cosimplicial objects in $\mathsf{Mod}({\mathcal{O}}_{Y})$ from ${\mathcal{F}}\rightarrow \mathsf{A}_{Y}^{\bullet }({\mathcal{F}})$ to $f_{\ast }f^{\ast }{\mathcal{F}}\rightarrow f_{\ast }\mathsf{A}_{X}^{\bullet }(f^{\ast }{\mathcal{F}})$ .
Proof of Lemma 1.11.
The morphism $\mathsf{A}_{Y,T^{\prime }}({\mathcal{F}})\rightarrow f_{\ast }\mathsf{A}_{X,T}(f^{\ast }{\mathcal{F}})$ is constructed by induction on $n$ (where $T\subset X_{n}$ ). For $n=0$ and ${\mathcal{F}}\in \mathsf{Coh}(Y)$ , we have
and $\mathsf{A}_{Y,T^{\prime }}{\twoheadrightarrow}\mathsf{A}_{Y,f(T)}({\mathcal{F}})=\prod _{x\in f(T)}\mathsf{lim}_{r\geqslant 0}(j_{rx})_{\ast }j_{rx}^{\ast }{\mathcal{F}}$ . We have a map
for each $x\in T$ , which defines the required map for $T\subset X_{0}$ .
Assume that the morphism $\mathsf{A}_{Y,T^{\prime }}({\mathcal{F}})\rightarrow f_{\ast }\mathsf{A}_{X,T}(f^{\ast }{\mathcal{F}})$ has been constructed for all $T\subset X_{k}$ and $f(T)\subset T^{\prime }\subset Y_{k}$ , where $k\leqslant n$ , such that we have the following commutative diagram.
Let $T\subset X_{n+1}$ . For every $x\in X$ , we then have a welldefined map $\mathsf{A}_{Y,\text{}_{f(x)}f(T)}({\mathcal{F}})\rightarrow f_{\ast }\mathsf{A}_{X,\text{}_{x}T}(f^{\ast }{\mathcal{F}})$ , since $f(\text{}_{x}T)\subset \text{}_{f(x)}f(T)\subset \text{}_{f(x)}T^{\prime }$ . This induces a morphism
We have a commutative diagram
which commutes levelwise (before taking the inverse limits and products) by the induction hypothesis.
We precompose the map (1) with
to obtain the required morphism $\mathsf{A}_{Y,T^{\prime }}({\mathcal{F}})\rightarrow f_{\ast }\mathsf{A}_{X,T}(f^{\ast }{\mathcal{F}})$ . The required commutativity assumption holds by commutativity of (2).◻
Setting ${\mathcal{F}}={\mathcal{O}}_{Y}$ in the assertion above, we obtain the following result as a consequence.
Corollary 1.12. For a morphism of Noetherian schemes, we obtain a map of augmented cosimplicial objects
in sheaves of algebras on the topological space $Y$ .
Taking global sections, we obtain a functor from Noetherian schemes to cosimplicial rings.
Definition 1.13. We denote the functor $(\mathsf{Sch}^{\mathsf{N}})^{\mathsf{op}}\rightarrow (\mathsf{Rng}^{\unicode[STIX]{x1D6E5}})$ , sending a Noetherian scheme $X$ to the cosimplicial ring $\mathbb{A}_{X}^{\bullet }({\mathcal{O}}_{X})$ by $\mathbb{A}^{\bullet }$ .
As we have alluded to in Theorem 0.4, and will prove as Corollary 3.32, this functor has a leftinverse.
1.3 Taking a closer look at the flasque sheaf of adèles
In this subsection we give a close analysis of flasqueness of the sheaf $\mathsf{A}_{X,T}({\mathcal{F}})$ . We show that the restriction map $\mathbb{A}_{X,T}({\mathcal{F}})\rightarrow \mathbb{A}_{U,T\cap U_{n}}({\mathcal{F}})$ is not only surjective, but admits an $\mathbb{A}_{X,T}({\mathcal{O}}_{X})$ linear section. As a consequence we obtain that $\mathsf{A}_{X,T}({\mathcal{O}}_{X})$ is what we call a lâche sheaf of algebras in Definition 2.9 (see also Corollary 2.17). A similar statement is contained in [Reference HuberHub91, Proposition 2.1.5], however the $\mathsf{A}_{X,T}({\mathcal{O}}_{X})$ linearity is not investigated in [Reference HuberHub91].
Lemma 1.14. Let $X$ be a Noetherian scheme, $T\subset X_{n}$ and ${\mathcal{F}}$ a quasicoherent sheaf on $X$ . For every open subset $U\subset X$ the restriction map $\mathbb{A}_{X,T}({\mathcal{F}})\rightarrow \mathbb{A}_{U,T}({\mathcal{F}})$ has a section, which is moreover $\mathbb{A}_{X,T}({\mathcal{O}}_{X})$ linear and functorial in ${\mathcal{F}}$ .
Proof. We denote the inclusion $U{\hookrightarrow}X$ by $j$ , and will construct a section to the map of sheaves $\mathsf{A}_{X,T}({\mathcal{F}})\rightarrow j_{\ast }\mathsf{A}_{U,T}({\mathcal{F}})$ . Recall that for a coherent sheaf ${\mathcal{F}}$ we have an equivalence $\mathsf{A}_{X,T}({\mathcal{F}})\simeq \prod _{x\in X}\mathop{\varprojlim }\nolimits_{r}\mathsf{A}_{X,\text{}_{x}T}(j_{rx}^{\ast }{\mathcal{F}}),$ and $j_{\ast }\mathsf{A}_{U,T\cap U_{n}}({\mathcal{F}})\simeq \prod _{x\in U}\mathop{\varprojlim }\nolimits_{r}\mathsf{A}_{U,\text{}_{x}T\cap U_{n1}}(j_{rx}^{\ast }{\mathcal{F}}).$ Suppose that we have already constructed a section $\unicode[STIX]{x1D719}_{{\mathcal{F}}}$ for $\mathsf{A}_{X,T^{\prime }}({\mathcal{F}})\rightarrow j_{\ast }\mathsf{A}_{U,T^{\prime }}({\mathcal{F}})$ for $T^{\prime }\subset X_{n1}$ , such that for each map ${\mathcal{F}}\rightarrow {\mathcal{G}}$ we obtain the following commutative square.
We can then map the limit $\prod _{x\in U}\mathop{\varprojlim }\nolimits_{r}\mathsf{A}_{U}(\text{}_{x}T;j_{rx}^{\ast }{\mathcal{F}})$ to $\prod _{x\in X}\mathop{\varprojlim }\nolimits_{r}\mathsf{A}_{U}(\text{}_{x}T;j_{rx}^{\ast }{\mathcal{F}})$ , by defining the components of the map to be $0$ for $x\in X\setminus U$ , and given by the section $\unicode[STIX]{x1D719}$ otherwise.
By induction we see that we may assume that $T\subset X_{0}$ . We may also assume that ${\mathcal{F}}$ is coherent. Hence, $\mathsf{A}_{X,T}({\mathcal{F}})$ is equal to the product $\prod _{x\in T}\mathop{\varprojlim }\nolimits_{r}j_{rx}^{\ast }{\mathcal{F}}$ , and $\mathsf{A}_{U}(T,{\mathcal{F}})$ to $\prod _{x\in T\cap U}\mathop{\varprojlim }\nolimits_{r}j_{rx}^{\ast }{\mathcal{F}}$ . The natural restriction map is given by the canonical projection. A canonical section with the required properties is given by the identity map for components corresponding to $x\in U\cap T$ , and the map 0 for $x\in T\setminus U$ .◻
As a corollary one obtains the following observation of Beilinson.
Corollary 1.15. The sheaves $\mathbb{A}_{X,T}({\mathcal{F}})$ are flasque.
In [Reference BeilinsonBei80] Beilinson continues to observe that for any quasicoherent sheaf ${\mathcal{F}}$ the canonical augmentation ${\mathcal{F}}\rightarrow \mathsf{A}_{X}^{\bullet }({\mathcal{F}})$ induces an equivalence ${\mathcal{F}}\simeq \mathsf{A}_{X}^{\bullet }({\mathcal{F}})$ . A detailed proof is given by Huber [Reference HuberHub91, Theorem 4.1.1].
Theorem 1.16 (Beilinson).
Let $X$ be a Noetherian scheme and ${\mathcal{F}}$ a quasicoherent sheaf on $X$ . The augmentation ${\mathcal{F}}\rightarrow \mathsf{A}_{X}^{\bullet }({\mathcal{F}})$ defines a cosimplicial resolution of ${\mathcal{F}}$ by flasque ${\mathcal{O}}_{X}$ modules. Applying the global sections functor $\unicode[STIX]{x1D6E4}(X,)$ we obtain $R\unicode[STIX]{x1D6E4}(X,{\mathcal{F}})=\mathbb{A}_{X}^{\bullet }({\mathcal{F}})$ , where the cosimplicial realisation $\cdot $ is taken in the derived $\infty$ category $\mathsf{D}(\mathsf{AbGrp})$ of abelian groups.
It is instructive to test the general considerations above on the special case of algebraic curves. For the rest of this subsection we will thus assume that $X$ is an algebraic curve. We denote by $\mathsf{A}_{X}$ the sheaf, assigning to an open subset $U\subset X$ the ring of adèles $\mathbb{A}_{U}$ . Similarly we have sheaves $\mathsf{F}_{X}$ and $\mathsf{O}_{X}$ of rational functions and integral adèles.
The sheaves $\mathsf{A}_{X}$ and $\mathsf{O}_{X}$ satisfy the conclusion of Lemma 1.14, because a section over $U\subset X$ can be extended by 0, outside of $U$ . Since $\mathsf{F}_{X}(U)=\mathsf{F}_{X}(X)$ , as long as $U\neq \emptyset$ , we see that the conclusion of Lemma 1.14 is trivially satisfied for $\mathsf{F}$ . Beilinson’s Theorem 1.16 is in the present situation tantamount to the assertion that the complex
is exact. In other words, we observe that a rational function without any poles on $U\subset X$ , defines a regular function on $U$ . While this is a tautology in the onedimensional case, the general setting of Noetherian schemes requires more subtle arguments from commutative algebra. We refer the reader to the proof of [Reference HuberHub91, Theorem 4.1.1] for more details.
2 Perfect complexes and lâche sheaves of algebras
In this section we introduce the notion of lâche sheaves of algebras and prove Theorem 0.5.
2.1 Lâche sheaves of algebras
The main example of a lâche sheaf of algebras $\mathsf{A}$ is Beilinson’s sheaf of adèles. This is the content of Corollary 2.17 below.
2.1.1 Flasque sheaves
In this section we record a few wellknown lemmas on flasque sheaves for the convenience of the reader.
Lemma 2.1. If ${\mathcal{F}}$ is a sheaf on $X$ , such that every point $x\in X$ has an open neighbourhood $U$ with ${\mathcal{F}}_{U}$ flasque, then ${\mathcal{F}}$ is a flasque sheaf.
Proof. Let $V\subset X$ be an open subset and $s\in {\mathcal{F}}(V)$ a section. We claim that there exists $t\in {\mathcal{F}}(X)$ with $t_{V}=s$ . Consider the set $I$ of pairs $(W,t)$ , where $W\subset X$ is an open subset containing $V$ , and $t\in {\mathcal{F}}(W)$ , such that $t_{V}=s$ . Inclusion of open subsets induces a partial ordering on $I$ , where we say that $(W,t)\leqslant (W^{\prime },t^{\prime })$ if $W\subset W^{\prime }$ , and $t^{\prime }_{W}=t$ . Moreover, $I$ is inductively ordered, that is, for every totally ordered subposet $J\subset I$ , there exists a common upper bound $i\in I$ , such that we have $i\geqslant j$ for all $j\in J$ . Indeed, denoting the pair corresponding to $j\in J$ by $(W_{j},t_{j})$ , we have $W_{j}\subset W_{k}$ for $j\leqslant k$ in $J$ , and $t_{k}_{W_{j}}=t_{j}$ . If we define $W=\bigcup _{j\in J}W_{j}$ , the fact that ${\mathcal{F}}$ is a sheaf allows us to define a section $t\in {\mathcal{F}}(W)$ with $t_{W_{j}}=t_{j}$ . In particular, $(W,t)\in I$ is a common upper bound for the elements of $J$ .
Zorn’s lemma implies that the poset $I$ has a maximal element $(W,t)$ . It remains to show that $W=X$ . Assume that there exists $x\in X\setminus W$ . By assumption, $x$ has an open neighbourhood $U$ , such that ${\mathcal{F}}_{U}$ is flasque. In particular, there exists a section $r\in {\mathcal{F}}(U)$ , such that $r_{U\cap W}=t_{U\cap W}$ . By virtue of the sheaf property we obtain a section $t^{\prime }\in {\mathcal{F}}(W\cup U)$ , satisfying $t^{\prime }_{W}=t$ , which contradicts maximality of $(W,t)$ .◻
Lemma 2.2. If $X$ is a quasicompact topological space and $\mathsf{A}$ a sheaf of algebras, then every locally finitely generated $\mathsf{A}$ module $\mathsf{M}$ which is flasque is globally finitely generated, that is there exists a surjection $\mathsf{A}^{n}\rightarrow \mathsf{M}$ .
Proof. For every point $x\in X$ there exists a neighbourhood $U_{x}$ , such that $\mathsf{M}_{U_{x}}$ is finitely generated. Since $X$ is quasicompact, we may choose a finite subcover $X=\bigcup _{i=1}^{n}U_{i}$ , and generating sections $(s_{ij})_{j=1,\ldots n_{i}}$ . Because $\mathsf{M}$ is assumed to be flasque, we may extend each $s_{ij}$ to a global section $t_{ij}$ , and see that this finite subset of $\unicode[STIX]{x1D6E4}(X,\mathsf{M})$ generates $\mathsf{M}$ .◻
Lemma 2.3. Assume that we have a short exact sequence of $\mathsf{A}$ modules with
with $\mathsf{M}_{i}$ flasque for $i>0$ , then $\mathsf{M}_{0}$ is flasque as well.
Proof. Since flasque sheaves have no higher cohomology, we have $H^{1}(X,\mathsf{M}_{2})=0$ , and therefore the following commutative diagram has exact rows
Commutativity of the righthand square, and the fact that $\mathsf{M}_{1}(X){\twoheadrightarrow}\mathsf{M}_{1}(U){\twoheadrightarrow}\mathsf{M}_{0}(U)$ is surjective, implies surjectivity of $\mathsf{M}_{0}(X){\twoheadrightarrow}\mathsf{M}_{0}(U)$ .◻
Lemma 2.4. Let $\mathsf{A}$ be an arbitrary sheaf of algebras on a topological space $X$ . Consider the abelian category of sheaves of $\mathsf{A}$ modules. The full subcategory, given by $\mathsf{A}$ modules $\mathsf{M}$ , such that $\mathsf{M}$ is a flasque sheaf, is extensionclosed.
Proof. Assume that we have a short exact sequence of $\mathsf{A}$ modules $\mathsf{M}_{1}{\hookrightarrow}\mathsf{M}_{2}{\twoheadrightarrow}\mathsf{M}_{3}$ , with $\mathsf{M}_{i}$ flasque for $i=1$ and $i=3$ . Since flasque sheaves do not have higher cohomology, we see that for every open subset $U\subset X$ we have a short exact sequence of abelian groups $\mathsf{M}_{1}(U){\hookrightarrow}\mathsf{M}_{2}(U){\twoheadrightarrow}\mathsf{M}_{3}(U)$ . In particular, we obtain a commutative diagram with exact rows
with the left and right vertical arrows being surjective. The snake lemma or a simple diagram chase reveal that the vertical map in the middle also has to be surjective. This proves that $\mathsf{M}_{2}$ is a flasque sheaf.◻
Definition 2.5. We denote the exact category, given by the extensionclosed full subcategory of $\mathsf{Mod}(\mathsf{A})$ consisting of modules whose underlying sheaf is flasque, by $\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A})$ .
We refer the reader to [Reference KellerKel96] and [Reference BühlerBüh10] for the notion of derived categories of exact categories. We also emphasise that we use the notation $\mathsf{D}()$ to denote the stable $\infty$ category obtained by applying the dgnerve construction of [Reference LurieLur, § 1.3.1] to the dgcategory of [Reference KellerKel96]. We will also consider similarly constructed stable $\infty$ categories $\mathsf{D}^{+}$ , $\mathsf{D}^{}$ and $\mathsf{D}^{b}$ , corresponding to complexes which are bounded below, bounded above and bounded, respectively.
It is important to emphasise that for a substantial part of this text we will not need to delve deeply into the theory of stable $\infty$ categories. The homotopy category of a stable $\infty$ category is naturally triangulated. To check that a functor $F:\mathsf{C}\rightarrow \mathsf{D}$ is fully faithful, essentially surjective, or an equivalence, it suffices to prove the same statement for its homotopy category (that is a classical triangulated category). This is essentially a consequence of the Whitehead lemma. Distinguished triangles $X\rightarrow Y\rightarrow Z\rightarrow \unicode[STIX]{x1D6F4}X$ correspond to socalled bicartesian squares
that is, commutative diagrams which are cartesian and cocartesian. A functor $\mathsf{C}\rightarrow \mathsf{D}$ between stable $\infty$ categories is called exact, if it preserves bicartesian squares. In particular, this is the case if and only if the induced functor $\mathsf{Ho}(\mathsf{C})\rightarrow \mathsf{Ho}(\mathsf{D})$ is exact in the sense of triangulated categories. The embedding $\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}){\hookrightarrow}\mathsf{Mod}(\mathsf{A})$ induces an exact functor of derived categories.
Lemma 2.6. The canonical functor $\mathsf{D}^{+}(\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}))\rightarrow \mathsf{D}^{+}(\mathsf{A})$ , induced by the exact functor $\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}){\hookrightarrow}\mathsf{Mod}(\mathsf{A})$ , is fully faithful.
Proof. According to a theorem of Keller [Reference KellerKel96, Theorem 12.1] it suffices to check that every short exact sequence of $\mathsf{A}$ modules $\mathsf{M}_{1}{\hookrightarrow}\mathsf{M}_{2}{\twoheadrightarrow}\mathsf{M}_{3}$ with $\mathsf{M}_{1}$ flasque fits into a commutative diagram with exact rows
with $\mathsf{M}_{2}^{\prime }$ and $\mathsf{M}_{3}^{\prime }$ flasque. To produce this diagram, we recall that every $\mathsf{A}$ module $\mathsf{M}$ can be embedded into a flasque $\mathsf{A}$ module. Indeed, the sheaf of discontinuous sections, that is, $\mathsf{M}^{\text{dc}}(U)=\prod _{x\in U}\mathsf{M}_{x}$ provides such an embedding. Let $\mathsf{M}_{2}{\hookrightarrow}\mathsf{M}_{2}^{\prime }$ be an embedding of $\mathsf{M}_{2}$ into a flasque $\mathsf{A}$ module. Then, the quotient $\mathsf{M}_{3}^{\prime }=\mathsf{M}_{2}^{\prime }/\mathsf{M}_{1}$ is also flasque by Lemma 2.3.◻
2.1.2 Flasque sheaves of algebras
In this subsection we ponder over what can be said about quasicoherent sheaves of $\mathsf{A}$ modules, if the sheaves of algebras $\mathsf{A}$ itself is known to be flasque. Recall that an $\mathsf{A}$ module $\mathsf{M}$ is quasicoherent, if every point $x\in X$ has a neighbourhood $U\subset X$ , such that the restriction $\mathsf{M}_{U}$ can be represented as a cokernel of a morphism $\mathsf{A}^{\oplus J}_{U}\rightarrow \mathsf{A}^{\oplus I}_{U}$ of free $\mathsf{A}$ modules.
Remark 2.7. For a general sheaf of algebras $\mathsf{A}$ the category $\mathsf{QCoh}(\mathsf{A})$ of quasicoherent $\mathsf{A}$ modules is in general not closed under taking kernels in the abelian category of $\mathsf{A}$ modules $\mathsf{Mod}(\mathsf{A})$ . In particular, one does not expect $\mathsf{QCoh}(\mathsf{A})$ to be abelian in general. If the restriction maps $\mathsf{A}(V)\rightarrow \mathsf{A}(U)$ for $U\subset V$ belonging to a specific subbase for the topology are known to be flat, $\mathsf{QCoh}(\mathsf{A})$ can be shown to be abelian. This assumption is too strong for the sheaves of algebras we care about in this article.
We see from Lemma 2.1 that every locally finite free or locally finite projective sheaf of $\mathsf{A}$ modules is flasque (where $\mathsf{A}$ is itself a flasque sheaf of algebras). In general, one cannot expect every quasicoherent sheaf of $\mathsf{A}$ modules to be flasque. However, we will see in the next subsection that there are certain flasque sheaves of algebras for which this is true.
Lemma 2.8. Let $\mathsf{A}$ be a sheaf of algebras on $X$ , such that every free $\mathsf{A}$ module is flasque. We denote by $\mathsf{P}(\mathsf{A})$ the exact category given by the idempotent completion of free $\mathsf{A}$ modules, and refer to its objects as projective $\mathsf{A}$ modules. The functor $\mathsf{D}^{}(\mathsf{P}(\mathsf{A})){\hookrightarrow}\mathsf{D}^{}(\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}))$ , induced by the inclusion $\mathsf{P}(\mathsf{A}){\hookrightarrow}\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A})$ , is fully faithful.
Proof. We will apply the dual of the result in Keller [Reference KellerKel96, Theorem 12.1], by which it suffices to check that every short exact sequence of flasque $\mathsf{A}$ modules $\mathsf{M}_{1}{\hookrightarrow}\mathsf{M}_{2}{\twoheadrightarrow}\mathsf{M}_{3}$ with $\mathsf{M}_{3}$ projective fits into a commutative diagram with exact rows
with $\mathsf{M}_{1}^{\prime }$ and $\mathsf{M}_{2}^{\prime }$ projective. Since $\mathsf{M}_{2}\in \mathsf{Mod}_{\mathsf{fl}}(\mathsf{A})$ is flasque by assumption, there exists a surjection $\mathsf{A}^{\oplus I}\rightarrow \mathsf{M}_{2}$ for some index set $I$ . Indeed, we can take $I=\{(U,s)U\subset X\text{ open, }s\in \mathsf{M}_{2}(U)\}$ . Choosing an extension $t_{(U,s)}\in \mathsf{M}_{2}(X)$ , satisfying $t_{(U,s)}_{U}=s$ for every element of $I$ , we obtain a surjective morphism of $\mathsf{A}$ module $\mathsf{A}^{\oplus I}\rightarrow \mathsf{M}_{2}$ .
Let $\mathsf{M}_{2}^{\prime }=\mathsf{A}^{\oplus I}$ , and define $M_{1}^{\prime }$ to be the kernel of the composition $\mathsf{M}_{2}^{\prime }\rightarrow \mathsf{M}_{2}\rightarrow \mathsf{M}_{3}$ . Since $\mathsf{M}_{3}$ is a direct summand of a free $\mathsf{A}$ module, and $\mathsf{M}_{2}^{\prime }$ is flasque, there exists a splitting to this surjection. Therefore, $\mathsf{M}_{1}^{\prime }$ belongs to $\mathsf{P}(\mathsf{A})$ , since it is a direct summand of $\mathsf{M}_{2}^{\prime }$ . This concludes the proof.◻
2.1.3 Definition of lâche sheaves of algebras
If $\mathsf{A}$ is a sheaf of algebras on a topological space, then there is a strengthening of the notion of $\mathsf{A}$ being flasque.
Definition 2.9. A sheaf of algebras $\mathsf{A}$ on $X$ is called lâche if for every open subset $U\subset X$ and every map of free $\mathsf{A}_{U}$ modules $\mathsf{A}_{U}^{\oplus J}\xrightarrow[{}]{f}\mathsf{A}_{U}^{\oplus I}$ , the kernel $\ker f$ is a flasque sheaf on $U$ .
To see that there are nontrivial lâche sheaves of algebras, we let $X$ be a topological space where every open subset is also closed in the following example.
Example 2.10. Let $X$ be a topological space, where every open subset is also closed. Then every sheaf of abelian groups ${\mathcal{F}}$ is flasque. If $U\subset X$ is open, and $s\in {\mathcal{F}}(U)$ , then using the sheaf property of ${\mathcal{F}}$ we see that there is a unique section $t\in {\mathcal{F}}(X)$ , such that $t_{U}=s$ and $t_{X\setminus U}=0$ . This is possible because $X\setminus U$ is open by assumption. Hence, every sheaf of algebras on $X$ is lâche.
The lemma below implies that for a lâche sheaf of algebras $\mathsf{A}$ , and a morphism $f:\mathsf{A}^{\oplus J}\rightarrow \mathsf{A}^{\oplus I}$ the sheaves $\operatorname{im}f$ and $\operatorname{coker}f$ are flasque as well.
Lemma 2.11. Let $V_{1}\xrightarrow[{}]{f}V_{2}$ be a morphism of flasque sheaves, such that $\ker f$ is flasque. Then, the sheaves $\operatorname{im}f$ , and $\operatorname{coker}f$ are flasque.
Proof. We have a short exact sequence $\ker f{\hookrightarrow}V_{1}{\twoheadrightarrow}\operatorname{im}f$ , since the first two sheaves are flasque, so is the third (Lemma 2.3). The same argument applies to the short exact sequence $\operatorname{im}f{\hookrightarrow}V_{2}{\twoheadrightarrow}\operatorname{coker}f$ , and implies that $\operatorname{coker}f$ is flasque.◻
We can further generalise the assertion.
Lemma 2.12. Let $V_{1}\xrightarrow[{}]{f}V_{2}$ be a morphism of projective quasicoherent $\mathsf{A}$ modules (that is, direct summands of free modules), where $\mathsf{A}$ is lâche. Then the sheaves $\ker f$ , $\operatorname{im}f$ and $\operatorname{coker}f$ are flasque.
Proof. Since every projective quasicoherent $\mathsf{A}$ module is a direct summand of a free $\mathsf{A}$ module, there exist quasicoherent $\mathsf{A}$ modules $W_{1}$ and $W_{2}$ , such that $V_{i}\oplus W_{i}$ are free $\mathsf{A}$ modules for $i=1,2$ . The induced map
has the same kernel $\ker f\simeq \ker (f\oplus \operatorname{id})$ . However, the Eilenberg swindle
allows us to see that the two sides are, in fact, free $\mathsf{A}$ modules. Therefore, the defining property of lâche sheaf of algebras implies that $\ker f$ is flasque. Lemma 2.11 yields that $\operatorname{im}f$ and $\operatorname{coker}f$ are flasque sheaves.◻
The considerations above imply, in particular, that every quasicoherent $\mathsf{A}$ module of a lâche sheaf of algebras $\mathsf{A}$ is flasque. However, we have to keep in mind that the category of quasicoherent sheaves is not abelian in general, as we pointed out in Remark 2.7. We have the following corollary to Lemma 2.12.
Corollary 2.13. If $M^{\bullet }\in \mathsf{DMod}(\mathsf{A})$ is a complex of sheaves of $\mathsf{A}$ modules which is locally quasiisomorphic to an object of $\mathsf{D}(\mathsf{P}(\mathsf{A}))$ , then its cohomology sheaves ${\mathcal{H}}^{i}(\mathsf{M}^{\bullet })$ are flasque.
Proof. We have seen in Lemma 2.1 that a sheaf is flasque if and only if it is locally flasque. Therefore, we may assume $M\in \mathsf{D}(\mathsf{P}(\mathsf{A}))$ . Let us choose an explicit presentation by a complex $(V^{\bullet },d)$ , where each $V^{i}$ is a projective $\mathsf{A}$ module. We have ${\mathcal{H}}^{i}(M)\simeq (\ker d^{i})/(\operatorname{im}d^{i1})$ . By Lemma 2.12, $\ker d^{i}$ and $\operatorname{im}d^{i1}$ are flasque. By Lemma 2.3, the quotient ${\mathcal{H}}^{i}(M)$ is flasque.◻
2.1.4 A criterion for being lâche
In this subsection we observe that every sheaf of algebras $\mathsf{A}$ , which admits linear sections to the restriction maps $\mathsf{A}(X)\rightarrow \mathsf{A}(U)$ , is in fact lâche. As a consequence, we obtain that the sheaf of adèles on a Noetherian scheme is lâche (Corollary 2.17).
Definition 2.14. A sheaf of algebras $\mathsf{A}$ is called very flasque if for every open subset $U\subset X$ there exists an $\mathsf{A}(X)$ linear section $\unicode[STIX]{x1D719}_{U}$ of the restriction map $r_{U}:\mathsf{A}(X)\rightarrow \mathsf{A}(U)$ .
Typically the section $\unicode[STIX]{x1D719}_{U}$ is given by a map which extends $s\in \mathsf{A}(U)$ by 0 outside of $U$ , as in the following example.
Example 2.15. Let $X$ be a topological space where every open subset is closed and $\mathsf{A}$ an arbitrary sheaf of algebras, then $\mathsf{A}$ is very flasque.
Proof. For an open subset $U\subset X$ we have a map $\unicode[STIX]{x1D719}:\mathsf{A}(U)\rightarrow \mathsf{A}(X)$ which sends $s\in \mathsf{A}(U)$ to the unique section $\widehat{s}\in \mathsf{A}(X)$ , such that $\widehat{s}_{U}=s$ , and $\widehat{s}_{X\setminus U}=0$ . This definition makes sense because $\mathsf{A}$ is a sheaf, and $X=U\cup X\setminus U$ a disjoint open covering. Since this map is $\mathsf{A}(X)$ linear, we have shown that $\mathsf{A}$ is very flasque.◻
In hindsight we have shown in Lemma 1.14 that for every quasicoherent sheaf of algebras ${\mathcal{F}}$ on a Noetherian scheme $X$ , the sheaves of algebras $\mathbb{A}_{X,T}({\mathcal{F}})$ are very flasque. See also Corollary 2.17 below, where an important consequence of this observation is recorded.
The next lemma is the aforementioned criterion for a sheaf of algebras being lâche.
Lemma 2.16. A very flasque sheaf of algebras $\mathsf{A}$ is lâche.
Proof. Let $f:\mathsf{A}_{V}^{\oplus J}\rightarrow \mathsf{A}_{V}^{\oplus I}$ be an $\mathsf{A}_{V}$ linear map, where $V\subset X$ is open. We have to show that $K=\ker f$ is a flasque sheaf on $V$ . For $U\subset V$ open we have a commutative diagram
with exact rows, because taking global sections is a left exact functor. However, $\mathsf{A}(V)$ linearity of the section $r_{V}\circ \unicode[STIX]{x1D719}_{U}:\mathsf{A}(U)\rightarrow \mathsf{A}(V)$ implies that we have a commutative diagram
where the dashed arrow is provided by the universal property of kernels. The dashed arrow is therefore rightinverse to the restriction map $K(V)\rightarrow K(U)$ , and we conclude that $K=\ker f$ is flasque.◻
Corollary 2.17. For a Noetherian scheme $X$ and a quasicoherent sheaf ${\mathcal{F}}$ of algebras, the sheaves of Beilinson–Parshin adèles $\mathsf{A}_{X,T}({\mathcal{F}})$ are lâche sheaves of algebras.
2.2 Perfect complexes
In this subsection we study the $\infty$ category of perfect complexes of $\mathsf{A}$ modules. This is necessary since the classical category of quasicoherent $\mathsf{A}$ modules is not necessarily abelian (see Remark 2.7).
Definition 2.18. Let $\mathsf{P}(\mathsf{A})$ denote the exact category obtained as the idempotent completion of the exact category of free $\mathsf{A}$ modules. We denote by $\mathsf{D}^{}(\mathsf{A})$ the $\infty$ category corresponding to the full subcategory of $\mathsf{D}(\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}))$ given by complexes of flasque $\mathsf{A}$ modules $\mathsf{M}^{\bullet }$ , which are locally equivalent to objects of $\mathsf{D}^{}(\mathsf{P}(\mathsf{A}))$ . That is, there exists an open covering $X=\bigcup _{i\in I}U_{i}$ and complexes $\mathsf{N}_{i}^{\bullet }\in \mathsf{D}^{}(\mathsf{P}(\mathsf{A}_{U_{i}}))$ such that we have equivalences $\mathsf{N}_{i}^{\bullet }\simeq \mathsf{M}^{\bullet }_{U_{i}}$ .
Recall that every exact functor between exact categories induces a functor between derived $\infty$ categories.
Lemma 2.19. Let $X$ be a quasicompact topological space and $\mathsf{A}$ a lâche sheaf of algebras on $X$ . We denote by $R=\unicode[STIX]{x1D6E4}(\mathsf{A})$ the ring of global sections of $\mathsf{A}$ . The global sections functor $\unicode[STIX]{x1D6E4}:\mathsf{D}^{}(\mathsf{A})\rightarrow \mathsf{D}^{}(R)$ , induced by the exact functor $\unicode[STIX]{x1D6E4}:\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A})\rightarrow \mathsf{Mod}(R)$ , is conservative.
Proof. Pick a complex $\mathsf{M}^{\bullet }=[\cdots \xrightarrow[{}]{d^{i1}}M^{i}\xrightarrow[{}]{d^{i}}M^{i+1}\xrightarrow[{}]{d^{i+1}}\cdots \,]\in \mathsf{D}(\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}))$ representing an object of $\mathsf{D}^{}(\mathsf{A})$ . By definition, we have $\unicode[STIX]{x1D6E4}(\mathsf{M}^{\bullet })=[\cdots \xrightarrow[{}]{\unicode[STIX]{x1D6E4}(d^{i1})}\unicode[STIX]{x1D6E4}(M^{i})\xrightarrow[{}]{\unicode[STIX]{x1D6E4}(d^{i})}\unicode[STIX]{x1D6E4}(M^{i+1})\xrightarrow[{}]{\unicode[STIX]{x1D6E4}(d^{i+1})}\cdots \,]$ . We shall assume that $\unicode[STIX]{x1D6E4}(\mathsf{M}^{\bullet })$ is acyclic, that is quasiisomorphic to the 0complex in $D^{}(R)$ . To establish conservativity of the functor $\unicode[STIX]{x1D6E4}$ , we must show that $\mathsf{M}^{\bullet }$ is acyclic in $D(\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}))$ .
Since $X$ is assumed to be quasicompact, and $\mathsf{M}^{\bullet }$ locally quasiisomorphic to an object of $D^{}(\mathsf{P}(\mathsf{A}))$ , we see that there exists an $i\in \mathbb{Z}$ , such that the cohomology sheaves ${\mathcal{H}}^{j}(\mathsf{M}^{\bullet })=0$ vanish for $j>i$ . We claim that for such an integer $i$ we have that ${\mathcal{Z}}^{i}=\ker d^{i}$ has no higher cohomology. Indeed, by assumption the stupid truncation $\unicode[STIX]{x1D70E}_{i}\mathsf{M}^{\bullet }=[\cdots \rightarrow 0\rightarrow M^{i}\rightarrow M^{i+1}\rightarrow \cdots \,]$ is a flasque resolution of ${\mathcal{Z}}^{i}[i]$ , by assumption on the vanishing of ${\mathcal{H}}^{j}(\mathsf{M}^{\bullet })$ for $j>i$ . However, since $\unicode[STIX]{x1D6E4}(M^{\bullet })$ has no cohomology in all degrees, we see that $\unicode[STIX]{x1D70E}_{i}\mathsf{M}^{\bullet }$ has no cohomology in degrees $j>i$ . This shows that ${\mathcal{Z}}^{i}$ has no higher cohomology.
Let us denote the image sheaf of $d^{i1}$ by ${\mathcal{B}}^{i}$ . It fits into a short exact sequence ${\mathcal{B}}^{i}\rightarrow M^{i}{\twoheadrightarrow}{\mathcal{Z}}^{i}$ . Since $M^{i}$ and ${\mathcal{Z}}^{i}$ are acyclic, one sees from the associated long exact sequence that ${\mathcal{B}}^{i}$ has no higher cohomology if and only if $\unicode[STIX]{x1D6E4}(M^{i})\xrightarrow[{}]{\unicode[STIX]{x1D6E4}(d^{i})}\unicode[STIX]{x1D6E4}({\mathcal{Z}}^{i})$ is surjective. By definition, the cokernel of this map equals $R^{i}\unicode[STIX]{x1D6E4}(M^{\bullet })=0$ . This is the case because $\unicode[STIX]{x1D6E4}(\mathsf{M}^{\bullet })$ is acyclic, and therefore all its cohomology groups vanish.
We have a commutative diagram
where the lower zigzag is a short exact sequence of sheaves without higher cohomology. Applying the functor $\unicode[STIX]{x1D6E4}$ (the short exact sequence is preserved by virtue of the fact that ${\mathcal{B}}^{i}$ has no higher cohomology) we obtain
Using again that $\unicode[STIX]{x1D6E4}(\mathsf{M}^{\bullet })$ has no nonzero cohomology groups, we see that $\unicode[STIX]{x1D6E4}(d^{i1})$ is surjective, and therefore so is the map $\unicode[STIX]{x1D6E4}({\mathcal{B}}^{i})\rightarrow \unicode[STIX]{x1D6E4}({\mathcal{Z}}^{i})$ . This implies $\unicode[STIX]{x1D6E4}({\mathcal{H}}^{i})=0$ . However, we know from Corollary 2.13 that the cohomology sheaves of $\mathsf{M}^{\bullet }$ are locally flasque, and therefore flasque by Lemma 2.1. Since a flasque sheaf without nonzero global sections is the zero sheaf, we obtain ${\mathcal{H}}^{i}(\mathsf{M}^{\bullet })=0$ . Continuing this process by downward induction, we obtain that all cohomology sheaves of $\mathsf{M}^{\bullet }$ vanish.
To conclude the proof we need to show that $\mathsf{M}^{\bullet }$ is acyclic in $\mathsf{D}(\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}))$ . By [Reference BühlerBüh10, Definition 10.1 and Remark 10.19] this is equivalent to the assertion that the sheaves ${\mathcal{Z}}^{i}$ and ${\mathcal{B}}^{i}$ are flasque. Since sheaves are flasque if they are locally flasque by Lemma 2.1, it suffices to show that $\mathsf{M}^{\bullet }$ is locally acyclic.
By assumption we can cover $X$ by open subsets $U$ , such that $\mathsf{M}^{\bullet }_{U}$ is quasiisomorphic to a complex of projective $\mathsf{A}$ modules $P^{\bullet }\in \mathsf{D}^{}(\mathsf{P}(\mathsf{A}_{U}))$ . In particular, we see that $P^{\bullet }$ is a complex of projective sheaves of $\mathsf{A}_{U}$ modules $[\cdots \rightarrow P^{i1}\rightarrow P^{i}\rightarrow 0\rightarrow \cdots \,]$ , such that ${\mathcal{H}}^{k}(P^{\bullet })=0$ for all $k\in \mathbb{Z}$ . This implies the existence of a factorisation
where the lower zigzags are short exact sequences. Since $P^{i}=Q^{i}$ is projective, we see that the first sequence splits, and therefore $Q^{i1}$ is projective too. Continuing by downward induction, we see that $Q^{j}$ is projective for all integers $j$ , and therefore $P^{\bullet }\simeq 0$ in $\mathsf{D}(\mathsf{P}(\mathsf{A}_{U}))$ . We have seen in Lemma 2.8 that the functor $\mathsf{D}^{}(\mathsf{P}(\mathsf{A}_{U})){\hookrightarrow}\mathsf{D}^{}(\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}_{U}))$ is fully faithful. This shows that $\mathsf{M}^{\bullet }_{U}\simeq 0$ is acyclic, and therefore that the restriction of the sheaves ${\mathcal{B}}^{i}$ and ${\mathcal{Z}}^{i}$ to $U$ are flasque. As discussed above this concludes the proof that $\mathsf{M}^{\bullet }$ is acyclic in $\mathsf{D}(\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A}))$ .◻
We also have a localisation functor.
Definition 2.20. For $\mathsf{A}$ a lâche sheaf of algebras on $X$ , we have an exact functor between exact categories $\otimes _{R}\mathsf{A}:\mathsf{P}(R)\rightarrow \mathsf{P}(\mathsf{A}){\hookrightarrow}\mathsf{Mod}_{\mathsf{fl}}(\mathsf{A})$ . The induced exact functor between derived $\infty$ categories will be denoted by
Proposition 2.21. If $X$ is quasicompact and $\mathsf{A}$ is lâche, then $\unicode[STIX]{x1D6E4}:\mathsf{D}^{}(\mathsf{A})\rightarrow \mathsf{D}^{}(R)$ is an equivalence of $\infty$ categories, with inverse equivalence $\mathsf{Loc}$ .
Proof. There is a commutative triangle
of exact functors, inducing a natural equivalence of functors $\operatorname{id}_{\mathsf{D}^{}(\mathsf{A})}\simeq \mathsf{Loc}\circ \unicode[STIX]{x1D6E4}$ . We claim that we also have an equivalence $\unicode[STIX]{x1D6E4}\circ \mathsf{Loc}\simeq \operatorname{id}_{\mathsf{D}^{}(R)}$ . To see this, consider $\mathsf{M}^{\bullet }\in \mathsf{D}^{}(\mathsf{A})$ . We will show that $\mathsf{M}^{\bullet }$ belongs to the essential image of $\mathsf{Loc}$ . Let $g:P^{\bullet }\rightarrow \unicode[STIX]{x1D6E4}(\mathsf{M}^{\bullet })$ be a projective replacement of $\unicode[STIX]{x1D6E4}(\mathsf{M}^{\bullet })$ , given by an actual morphism between chain complexes in $\mathsf{Mod}(R)$ . By the adjunction between $\otimes _{R}\,\mathsf{A}$ and $\unicode[STIX]{x1D6E4}$ , this yields a morphism $f:\mathsf{Loc}(P^{\bullet })\simeq \mathsf{Loc}(\unicode[STIX]{x1D6E4}(\mathsf{M}^{\bullet }))\rightarrow \mathsf{M}^{\bullet }$ in $\mathsf{D}^{}(\mathsf{A})$ . Since $\unicode[STIX]{x1D6E4}(f)=g$ is a quasiisomorphism, and $\unicode[STIX]{x1D6E4}$ is conservative by Lemma 2.19, we see that $f$ is an equivalence. This implies that every $\mathsf{M}^{\bullet }\in \mathsf{D}^{}(\mathsf{A})$ is in fact equivalent to an object of $\mathsf{D}^{}(\mathsf{P}(\mathsf{A}))$ . Therefore, we have a natural equivalence $\mathsf{Loc}\circ \unicode[STIX]{x1D6E4}\simeq \operatorname{id}_{\mathsf{D}^{}(\mathsf{A})}$ as a consequence of the commutative diagram
of exact functors, and Lemma 2.8, which asserted