The text explicitly recalls that EGA defines a formal scheme as a pair (X, O_X) where X is a topological space and O_X is a sheaf of topological rings such that every point has an open neighbourhood isomorphic to an affine formal scheme. It also notes that the associated functor h_X satisfies the sheaf property for the Zariski topology, illustrating the same sheaf-theoretic style of local representability used for schemes.

In the section 'Schemes as functors', the Stacks Project describes the Yoneda embedding h_X: (Sch)ᵒᵖ → Sets sending a scheme X to the functor h_X(T) = Mor(T, X). It then states that schemes can be viewed as certain sheaves on the big Zariski site (Sch)_{Zar}, i.e. those functors h_X which are sheaves for the Zariski topology and are locally isomorphic to representable functors by affine schemes. This gives a characterization of schemes as Zariski sheaves on an appropriate site that are locally representable by affine schemes.

The Stacks Project defines the big affine Zariski site (Aff/S)_{Zar} as the full subcategory of (Sch/S)_{Zar} consisting of objects U/S such that U is an affine scheme, with coverings given by standard Zariski coverings of affine schemes.[4] In the general discussion of sheaves on these sites, it is explained that representable functors h_U for affine schemes U are sheaves for the Zariski topology, and that many geometric objects can be studied as sheaves on the big (or affine) Zariski site which are locally given by such representable functors.

In this preprint, the author studies when a functor F from Rings^{op} to Sets is represented by a scheme. The introduction observes that any scheme X gives a functor of points h_X, and that one can ask for an *intrinsic* characterization of those functors that arise in this way. The paper proves that under suitable conditions, a functor that is a sheaf in the fpqc or Zariski topology and is locally isomorphic to representable functors corresponds to a scheme. This formalizes the intuition that schemes are exactly those sheaves on the affine site that are locally representable by affine schemes.

The page states: "One can characterize the essential image of the category of schemes in the category of presheaves over the category of affine schemes, as the full subcategory spanned by those small presheaves that are Zariski-locally representable and are Zariski-sheaves, where Zariski is the Zariski topology on Aff." It then clarifies: "A presheaf is **locally representable** if there is a set-indexed covering family {U_i → F} where each U_i is the representable sheaf of an affine scheme and the morphisms are open immersions." This directly formulates the equivalence between schemes and Zariski sheaves on Aff that are locally representable by affines.

This section explains the fully faithful Yoneda-type embedding of schemes into presheaves on the category of affine schemes. It states that for a scheme X, the functor h_X: Aff^{op} → Sets given by h_X(Spec A) = Hom(Spec A, X) is a sheaf for the Zariski topology on Aff, and that the functor X ↦ h_X is fully faithful. Later sections (referenced here) identify which sheaves on Aff arise this way, describing them as those sheaves that are Zariski-locally isomorphic to representable sheaves of affine schemes via open immersions.

The lecture notes on 'Schemes as functors of points' describe the construction which associates to any scheme X the functor h_X(T) = Hom(T, X) on the category of schemes. The notes emphasize that h_X is in fact a sheaf for the Zariski topology on the category of schemes, and that this process embeds schemes fully faithfully into the category of Zariski sheaves. Moreover, schemes are characterized among Zariski sheaves by the property that they admit a cover by representable open subfunctors h_{Spec A}, i.e. they are Zariski sheaves locally representable by affine schemes.

Vakil defines representable presheaves and representable morphisms in a general site G, and then: "14.1. Definition. We define locally representable sheaves LRepSh_G. I might want to call them locally representables for short, to deliberately elide the word 'sheaves', so I can think of them as 'spaces'." Earlier he gives the motivating example: "Things that are locally schematic in the Zariski topology are just schemes" and explains that a locally representable sheaf whose local models are schemes corresponds to what algebraic geometers call a scheme. This provides an explicit link between schemes and Zariski sheaves that are locally representable by (affine) schemes.

This paper, "Schemes and sheaves", discusses the relationship between schemes and sheaves on a site. It considers the functor of points perspective and characterizes schemes in terms of sheaves on the category of affine schemes. The authors explain that if a sheaf on the Zariski site of affine schemes is locally isomorphic to representable sheaves (i.e., locally represented by affine schemes), then it can be glued to give a scheme, and conversely the functor of points of a scheme satisfies these local representability conditions. Thus, schemes appear as Zariski sheaves on Aff that are locally representable.

This section defines the big and small affine Zariski sites. It introduces "the big affine Zariski site of S, denoted (Aff/S)_{Zar}," as "the full subcategory of (Sch/S)_{Zar} consisting of objects U/S such that U is an affine scheme," with coverings given by standard Zariski coverings of affines. It also defines the "small affine Zariski site" S_{affine,Zar} similarly. These constructions provide the site (category of affine schemes with Zariski topology) on which schemes can be viewed as Zariski sheaves.

In the Stacks Project, representable functors h_X from schemes X to Sets are discussed, with X affine giving representable presheaves on Aff. The text explains that for schemes, these functors are sheaves for the Zariski topology and that open immersions of schemes correspond to certain monomorphisms of sheaves. This framework is used later to define when a sheaf on the affine site is "locally represented" by affine schemes via open immersions, yielding an equivalence between schemes and such sheaves.

This part of the Stacks Project articulates the philosophy of viewing schemes as sheaves. It notes that a scheme X gives a functor h_X that is a sheaf for the Zariski topology on the big/small site of schemes and that the Yoneda embedding is fully faithful. It then characterizes the essential image in terms of sheaves that locally look like affine schemes via open immersions: such sheaves, when restricted to the category of affine schemes with the Zariski topology, provide a description of schemes as Zariski sheaves locally representable by affines.

This section defines the big Zariski site (Sch)_{Zar} and discusses sheaves on it. It explains that representable functors h_X are sheaves for this site and that open immersions correspond to certain monomorphisms of sheaves. It then outlines how one can glue affine schemes along open immersions in the sheaf-theoretic setting, giving criteria for when a sheaf on the big Zariski site (or its affine subsite) is representable by a scheme, phrased in terms of being Zariski-local and locally affine.

Here the Stacks Project defines the big affine site Aff/S and the small affine site S_{affine} and discusses restrictions of sheaves from the big site of all schemes to the affine subsite. It notes that for a fixed base S, schemes over S correspond to certain sheaves on Aff/S that are Zariski sheaves and that are locally representable by affine S-schemes, meaning that there exists a covering by representable sheaves of affine schemes via open immersions.

The zbMATH review of "Schemes and sheaves" summarizes: the authors investigate "the characterization of schemes among all functors from commutative rings to sets" and show conditions under which such a functor is a scheme. The review notes that the characterizations use sheaf conditions (e.g. Zariski or fpqc) and local representability by affine schemes. Thus, the paper contributes to the idea that a scheme is equivalently a sheaf on the affine site which is locally given by affine schemes.

These notes define an affine scheme as a locally ringed space isomorphic to Spec(R) for some ring R, and a scheme as a locally ringed space in which each point has an open neighborhood isomorphic to an affine scheme.[7] Later in the course, the functor of points perspective is introduced, where each scheme X determines a functor h_X on rings or affine schemes, and this functor satisfies a sheaf condition for the Zariski topology and is locally represented by affine schemes, giving an equivalence between schemes and such locally representable Zariski sheaves.

In a survey-style discussion of functorial methods in algebraic geometry, the article recalls that any scheme X defines its functor of points h_X on the category of affine schemes, and that h_X is a sheaf in the Zariski topology. It further remarks that conversely, "a sheaf on the Zariski site of affine schemes which is Zariski-locally isomorphic to some h_{Spec A} corresponds to a scheme obtained by gluing the Spec A." This encapsulates the equivalence between schemes and Zariski sheaves on Aff that are locally representable by affine schemes.

In this lecture note on scheme theory, after developing the functor-of-points perspective, the author remarks that a scheme X defines a functor F from commutative rings (or equivalently, from affine schemes) to sets, which is a sheaf for the Zariski topology and is locally affine. The note states in substance: if F is a sheaf for the Zariski topology and locally affine, then one can write F as a Zariski sheaf with pieces representable by affine schemes, glued along open immersions, thereby recovering a scheme. This matches the description of schemes as Zariski sheaves on Aff that are locally representable by affine schemes.

The lecture notes state: "a (derived) scheme is a Zariski sheaf admitting an open covering by affines." They also explain that when the category of affine schemes is embedded by Yoneda, the universal target for gluing with respect to the Zariski topology is the category of Zariski sheaves, and that the intended notion of scheme is a sheaf with an open affine cover.

In the preliminaries, the author sets notation: "Sch is the category of schemes, and Aff is the category of affine schemes." The paper uses the functor-of-points viewpoint, referring to schemes via their restrictions to affine opens, consistent with viewing them as sheaves on the category Aff equipped with the Zariski topology. Although the main theorem is about morphisms, the setup assumes the standard equivalence between schemes and Zariski sheaves on Aff that are locally representable by affines.

Grothendieck and Dieudonné introduce schemes as locally ringed spaces that can be covered by open subschemes isomorphic to spectra of rings. Later in the Éléments, they promote the 'functor of points' point of view, where a scheme X is understood by the contravariant functor it represents on the category of schemes (or affine schemes). This functor satisfies a sheaf condition for the Zariski topology and is locally representable by affine schemes, foreshadowing the modern characterization of schemes as Zariski sheaves on Aff that are locally affine-representable.

In a discussion of functorial geometry and the functor of points, the post says: "Namely, locally, F must look like the functor of points of a scheme, moreover F must be a sheaf, i.e. F must have a gluing property that allows us to patch..." It then explains that on the category of affine schemes with the Zariski topology, a functor F which is a sheaf and is locally isomorphic to a representable functor h_X (with X an affine scheme) corresponds to a scheme obtained by gluing these local affine pieces. This is an informal restatement that schemes are Zariski sheaves on Aff that are locally representable by affine schemes.

The paper discusses functorially associating a Zariski sheaf of rings or modules to data over an affine scheme, and repeatedly treats the affine scheme as the local base object for sheaf-theoretic constructions. While the paper is about idempotentization rather than the definition of schemes, it confirms the centrality of Zariski sheaves on affine schemes in modern scheme-like formalisms.

In this MathOverflow discussion, users describe the precise statement that "the category of schemes is equivalent to the category of sheaves on the site of affine schemes with the Zariski topology that are locally isomorphic to affine schemes." Answers explain that a scheme X defines a Zariski sheaf on Aff via its functor of points, and conversely, a Zariski sheaf on Aff that is locally representable by affine schemes (through open immersions) comes from a unique scheme.

These lecture notes define a Grothendieck topology on a category C and the notion of a sheaf on a site (C, J) as a functor F: Cᵒᵖ → Set satisfying a gluing condition for all coverings.[1] Although not specific to schemes, the notes explain that when C is the category of affine schemes with the Zariski topology, sheaves on this site correspond to geometric objects whose functor of points is determined locally by affine schemes, motivating the description of schemes as Zariski sheaves on Aff that are locally representable by affine schemes.

Modern expositions of algebraic geometry (following Grothendieck) commonly state: the category Sch of schemes is equivalent to the full subcategory of Sh(Aff_{Zar}) consisting of those sheaves F such that there is a Zariski cover {U_i → F} with each U_i representable by an affine scheme and the maps are open immersions. This is the precise sense in which a scheme is a Zariski sheaf on the category of affine schemes that is locally representable by affine schemes.

The notes say that affine schemes are the local models for schemes in general. They describe schemes as being built from affine pieces and use this perspective to motivate the local nature of scheme theory, though they do not give the full categorical characterization in the way the Stacks project and nLab do.