key: cord-0227566-zdhimtwn authors: Liu, Chien-Hao; Yau, Shing-Tung title: Soft noncommutative schemes via toric geometry and morphisms from an Azumaya scheme with a fundamental module thereto -- (Dynamical, complex algebraic) D-branes on a soft noncommutative space date: 2021-08-11 journal: nan DOI: nan sha: 2aa55d382a83bdae084e205d447e7cc4732ffeb0 doc_id: 227566 cord_uid: zdhimtwn A class of noncommutative spaces, named `soft noncommutative schemes via toric geometry', are constructed and the mathematical model for (dynamical/nonsolitonic, complex algebraic) D-branes on such a noncommutative space, following arXiv:0709.1515 [math.AG] (D(1)), is given. Any algebraic Calabi-Yau space that arises from a complete intersection in a smooth toric variety can embed as a commutative closed subscheme of some soft noncommutative scheme. Along the study, the notion of `soft noncommutative toric schemes' associated to a (simplicial, maximal cone of index $1$) fan, `invertible sheaves' on such a noncommutative space, and `twisted sections' of an invertible sheaf are developed and Azumaya schemes with a fundamental module as the world-volumes of D-branes are reviewed. Two guiding questions, Question 3.12 (soft noncommutative Calabi-Yau spaces and their mirror) and Question 4.2.14 (generalized matrix models), are presented. The realization of the noncommutative feature of D-brane world-volumes ([H-W] , [Po1] , [Wi] ; [L-Y1] , [Liu] ) makes D-branes a good candidate as a probe to noncommutative geometry. Unfortunately, all the three building blocks of modern Commutative Algebraic Geometry -the notion of localizations of a ring, the method of associating a topology to a ring via Spec, and the approach to understand a ring by studying its category of modules -have their limitation when extending to noncommutative rings and Noncommutative Algebraic Geometry. (See, e.g., [Ro1] , [Ro2] , [B-R-S-S-W] to get a feel.) Unlike a (fundamental) superstring world-sheet, which can have a very abundant class of geometry as its target-spaces -most notably Calabi-Yau spacesit is not immediate clear what class of noncommutative spaces can serve as the target-space of a dynamical super D-string world-sheet and how to construct them, let alone the even deeper issue of Mirror Symmetry phenomenon when two different target-spaces can give rise to isomorphic 2-dimensional supersymmetric quantum field theories on the superstring world-sheet. In this work we introduce a class of noncommutative spaces, named 'soft noncommutative schemes' via toric geometry (Sec. 2 and Sec. 3), to partially take care of the persistent difficulty of gluing in general Noncommutative Algebraic Geometry. This by construction is only a very limited class of noncommutative spaces, yet abundant enough that all the algebraic Calabi-Yau spaces that arise from complete intersections in a smooth toric variety can show up as a commutative subscheme of some soft noncommutative scheme (Corollary 3.11). A mathematical model for (dynamical, complex algebraic) D-brane on such a noncommutative space is then given (Sec. 4) as morphisms from Azumaya schemes with a fundamental module thereto. From this, one is certainly very curious as to how Mirror Symmetry stands when these soft noncommutative spaces are taken to be the target-spaces of D-string world-sheets. Convention. References for standard notations, terminology, operations and facts are (1) Azumaya/matrix algebra: [Ar] , [Az] , [A-N-T]; (2) commutative monoid: [Og] ; (3) toric geometry: [Fu] ; (4) aspects of noncommutative algebraic geometry: [B-R-S-S-W]; (5) commutative algebra and (commutative) algebraic geometry: [Ei] , [E-H] , [Ha] ; (6) string theory and D-branes: [Ba] , [Jo] , [Po1] , [Po2] , [Sz] . · All commutative schemes are over C and Noetherian. · For a sheaf F on a gluing system of charts, the notation 's ∈ F' means a local section s ∈ F(U ) for some chart U . Basic definitions of monoids and monoid algebras we need for the current work are collected in this section to fix terminology and notation. Readers are referred to [De] , [Ge] , [G-H-V] and [Og] , from where these definitions are adapted, for more details. Given a set S, the free monoid F (S) associated to S is the monoid that consists of an element e and all formal words s 1 · · · s k of finitely many elements of S, with the product rules s 1 · · · s k s 1 · · · s l = s 1 · · · s k s 1 · · · s l and e s 1 · · · s k = s 1 · · · s k e = s 1 · · · s k . Let S ⊂ M be a subset of a monoid. We say that M is generated by S if the monoid homomorphism F (S) → M specified by the inclusion S → M , with e → e M and s 1 · · · s k → s 1 · · · s k , is surjective. We say that a monoid M is finitely generated if M is generated by a finite subset S ⊂ M . We say that a generating set S of M is inverse-closed if s ∈ S and s is invertible in M , then s −1 ∈ S. By convention, when a monoid is written multiplicatively and the monoid operation is clear from the content, m 1 m will be denoted simply m · m or mm . Commutative monoids are often written additively, with denoted + and e M denoted 0. Definition 1.2. [product and central extension] Let (C, ·, 1) and (M, , e M ) be monoids. The product of (C, ·, 1) and (M, , e M ) is the monoid with elements (c, m) ∈ C × M with (c 1 , m 1 ) * (c 2 , m 2 ) = (c 1 · c 2 , m 1 m 2 ) and identity (1, e M ). When C is commutative, we'll denote the product C · M or CM and elements cm . In this case, C → C · M → → M , with c → c e M and cm → m, is a central extension of M . of (M, S) is a directed graph with the set of vertices M and, for every pair (m 1 , m 2 ) of vertices a directed edge labelled by s ∈ S from m 1 to m 2 if m 2 = m 1 s. We take the convention that if both s and s −1 are in S, then the edge associated to s and that associated to s −1 coincide with opposite directions; (i.e. the corresponding edge is bi-directed). Since every m ∈ M can be expressed as a finite word in letters from S, Γ Cayley (M, S) is a connected graph. Definition 1.4. [edge-path monoid] (Continuing Definition 1.3.) Endow the Cayley graph Γ := Γ Cayley (M, S) of (M, S) with the topology from the geometric realization of Γ as a 1dimensional simplicial complex. An edge-path at the vertex m 0 on Γ, with the terminal vertex m t , is a direction-preserving continuous map γ : [0, 1] → Γ, with γ(0) = m 0 and γ(1) = m t , that is piecewise linear after a subdivision of the interval [0, 1] as a 1-dimensional simplicial complex. Thus, associated to each edge-path γ at m 0 is a word w γ = s 1 · · · s k in letters from S, where s 1 is the initial edge γ takes from the vertex m 0 and s k is the final edge γ takes to the terminal vertex m t . Given an edge-path γ at m 0 and a vertex m ∈ M , there is a translation T m : γ → m γ, where m γ is the edge-path at m that takes the same word w γ as γ and is piecewise linear under the same subdivision of [0, 1]. The constant edge-path γ : [0, 1] → Γ such that γ([0, 1]) = m ∈ M will be denoted by the vertex m. There is an operation on the set of edge-paths: · For edge-paths γ 1 and γ 2 on Γ such that γ 1 (1) = γ 2 (0), define · For general edge-paths γ 1 and γ 2 on Γ, define In particular, m γ = T m (γ), γ(0) γ = γ, and γ m = γ, after a subdivision of [0, 1] and a simplicial map on the subdivided [0, 1] that fixes {0, 1}, for all vertex m and edge-path γ. Consider the spaceM of homotopy classes of edge-paths at the vertex e M of Γ relative to the boundary {0, 1} of the interval [0, 1]. The operation on edge-paths on Γ descends to an associative binary operation, still denoted by , onM with the identity the constant path e M . The monoid (M , , e M ) will be called the edge-path monoid associated to (M, S). Definition 1.5. [monoid algebra] Let (M, , e M ) be a monoid. The C-vector space m∈M C · m with the binary operation * generated by C-linear expansion of is called the monoid algebra over C associated to M , in notation C M and the identity 1·e M =: 1. When M is commutative, we will denote C M by C[M ]. Lemma 1.6. [natural homomorphism] There is a built-in monoid epimorphism π :M → M , which extends to an epimorphism π : C M → C M of monoid algebras over C. Proof. This follows from the surjective map π : {edge-paths γ at e M } → M , with γ → γ(1). Example 1.7. [lattice of rank n] Let M be the lattice of rank n (i.e. Z n as a Z-module). As a commutative monoid written multiplicatively, M × n i=1 z Z i and generated by the inverseclosed subset S = {z 1 , z −1 1 , · · · , z n , z −1 n }. The associated C-algebra is a polynomial ring C[z 1 , z 1 , · · · , z n , z n ] (z 1 z 1 − 1, · · · , z n z n − 1) . The Cayley diagram Γ of (M, S) is a homogeneous graph of constant valence 2n for all vertices, with the base vertex 1 ∈ M . Under the above presentation of M , the edge-path monoidM is z 1 , z −1 1 , · · · , z n , z −1 Here, C z 1 , z 1 , · · · , z n , z n is the free associative C-algebra with 2n generators and (z 1 z 1 − 1, z 1 z 1 − 1, · · · , z n z n − 1, z n z n − 1) is the two-sided ideal generated by the elements indicated. The monoid homomorphism π :M → M from the terminal-vertex-of-edge-path-at-1 map and the associated monoid-algebra homomorphism π : The complex torus T := T n := (C × ) n action on the generators induces a T n -action on C · M and C ·M , leaving each C-factor of C · M and C ·M invariant, that render π : C ·M → C · M T n -equivariant. This extends to a T n -action on C M and C[M ] under which π : The lattice M of rank n in Example 1.7 is the starting point of constructions of the current work. As a commutative monoid, there are occasions when it is more convenient for M to be written additively, rather than multiplicatively. We shall switch between these two pictures freely and use whichever suits better. Soft noncommutative toric schemes are introduced in this section. The setting and the terminology used indicate our focus on the function ring, rather than the point-set and topology, of a noncommutative space while striving to retain enough underlying geometric picture for the purpose of studying (dynamical) D-branes on a noncommutative target-space in string theory within the realm of Algebraic Geometry. (Cf. [L-Y1] (D(1)), [L-L-S-Y] (D(2)), and Sec. 4.2 of the current work.) 2.1 The noncommutative affine n-space ncA n C and its 0-dimensional subschemes Definition 2.1.1. [affine scheme, function ring, noncommutative affine n-space over C] Let Ring be the category of rings. An object X in the opposite category Ring op of Ring is called an affine scheme. The ring that underlies X is called the function ring of X. If the function ring of X is noncommutative (resp. commutative), then we say that X is a noncommutative affine scheme (resp. commutative affine scheme). In particular, we define a noncommutative affine nspace over C to be an object in Ring op that corresponds to a free noncommutative associative C-algebra generated by n letters, in notation C z 1 , · · · , z n . Since any two free noncommutative associative C-algebras generated by n letters are isomorphic for a fixed n, we shall denote a noncommutative affine n-space over C commonly by ncA n C or simply ncA n . ncA n is smooth in the sense of the following extension/lifting property: · For any C-algebra homomorphism C z 1 , · · · , z n → A and any C-algebra epimorphism B → → A, there exists a C-algebra homomorphism C z 1 , · · · , z n → B that makes the following diagram commute , defines an embedding A n → ncA n , called the master commutative subscheme of ncA n . Definition 2.1.3. [closed subscheme of ncA n ] A closed subscheme of ncA n is a C-algebra quotient C z 1 , · · · , z n −→ −→ A. When A is commutative, then the closed subscheme is called a commutative closed subscheme of ncA n . Since when A is commutative, such a quotient always factors as C z 1 , · · · , z n , a commutative closed subscheme of ncA n is always contained in the master commutative subscheme A n ⊂ ncA n . When A is a finite-dimensional C-algebra, the closed subscheme is called a 0-dimensional subscheme of ncA n . In particular, a C-point on ncA n is a C-algebra quotient C z 1 , · · · , z n −→ −→ C. Since C is commutative, the set of C-points of ncA n coincide naturally with the set of C-points of A n via the built-in embedding A n → ncA n . Definition 2.1.4. [punctual versus nonpunctual 0-dimensional subscheme] A 0-dimensional subscheme C z 1 , · · · , z n → → A of ncA n is called punctual if A admits a C-algebra quotient Here, Center (A) ⊂ A is the center of A and Nil Center (A) is the ideal of nilpotent elements of Center (A). Else, the 0-dimensional subscheme is called nonpunctual (i.e. "fuzzy without core"). Example 2.1.5. [Grassmann points, Azumaya/matrix points, other special points on ncA n ] (1) All commutative 0-dimensional subschemes of ncA n are punctual. (2) A Grassmann point (resp. Azumaya or matrix point) on ncA n is a C-algebra quotient C z 1 , · · · , z n → → A such that A is isomorphic to a Grassmann algebra over C C[θ 1 , · · · , θ r ] anti-c := C θ 1 , · · · , θ r /(θ α θ β + θ β θ α | 1 ≤ α, β ≤ r) for some r (resp. a matrix algebra M r (C) over C for some rank r ≥ 2). Both are noncommutative points on ncA n . However, Grassmann points are punctual while Azumaya/matrix points are nonpunctual since there exists no C-algebra homomorphism from M r (C) to C for r ≥ 2. (3) Other special points on ncA n include points C z 1 , · · · , z n → → A where A is isomorphic to the C-algebra T upper r (C) of upper triangular matrix for some rank r ≥ 2 (resp. the C-algebra T lower r (C) of lower triangular matrix for some rank r ≥ 2.) They are called upper-triangular matrix points (resp. lower-triangular matrix points). Unlike Azumaya/matrix points on ncA n , both are punctual under the correspondence that sends an upper-or-lower triangular matrix to its diagonal. Definition 2.1.6. [nested master l-commutative closed subscheme of ncA n ] Given two two-sided ideals I, J in C z 1 , · · · , z n . Denote by [I, J] the two-sided ideal in C z 1 , · · · , z n generated by elements of the form [f, g] := f g − gf , where f ∈ I and g ∈ J. Now let m 0 be the two-sided ideal (z 1 , · · · , z n ) of C z 1 , · · · , z n . Then note that [m 0 , m 0 ] and (z i z j − z j z i | 1 ≤ i < j ≤ n) coincide. Consider the closed scheme A n (l) of ncA n , l = 1, 2, · · · , associated the C-algebra one has a nested sequence of closed subschemes in ncA n A n = A n (1) → A n (2) → A n (3) → · · · . Any function on ncA n of the form f 1 · · · f l , where f i ∈ m 0 for all i, becomes commutative when restricted to A n (l ) for l ≤ l. Furthermore, any closed subscheme of ncA n with this property must be a closed subscheme of A n (l) . Thus, we name A n (l) the master l-commutative subscheme of ncA n (with respect to the choice of coordinate functions z 1 , · · · , z n on ncA n ). The restriction of any function f on ncA n of degree d ≥ 1 to any A n (l) with l ≤ d is commutative on A n (l) . Example 2.1.7. [0-dimensional subscheme and A n (l) ] How a 0-dimensional subscheme in ncA n intersects the nested sequence A n (l) , l = 1, 2, . . ., gives one a sense of a depth of noncommutativity the 0-dimensional subscheme of ncA n is located at. (1) A commutative 0-dimensional subscheme of ncA n lies in A n = A (1) and hence in A n (l) for all l. (2) A punctual 0-dimensional subscheme of ncA n contains a commutative subscheme and hence must have nonempty intersection with A n and hence has nonempty intersection with A n (l) for all l. They thus lie in an infinitesimal neighborhood of A n in ncA n . (3) Consider the Azumaya/matrix point on ncA r 2 where e ij is the r × r matrix with ij-entry 1 and elsewhere 0. The kernel of this C-algebra epimorphism is the two-sided ideal I generated by In particular, z ij − z ij 1 z j 1 j 2 · · · z j l−1 j l z j l j ∈ I for all l = 1, 2, . . .. Recall the two-sided ideal Then the two-sided ideal generated by This shows that this Azumaya point on ncA r 2 has no intersection with A r 2 (l) for all l. In a sense, this Azumaya point is located deep in the noncommutative part of ncA n . Definition 2.1.8. [connectivity and hidden disconnectivity of 0-dimensional subscheme] Given a 0-dimensional subscheme C z 1 , · · · , z n → → A on ncA n . We say that the 0-dimensional subscheme is connected if A is not a product A 1 × A 2 of two C-algebras. Else, we say that the 0-dimensional subscheme is disconnected. If A is not a product, yet it admits a decomposition of 1 by non-zero orthogonal idempotents 1 = e 1 + · · · + e k , for some k, where e i = 0 and e 2 i = e i for all i and e i e j = e j e i = 0 for all i = j, then we say that the connected 0-dimensional subscheme has hidden disconnectivity. The phenomenon of hidden disconnectivity can occur for noncommutative points -for example, Azumaya/matrix points, upper-triangular matrix points, lower-triangular matrix pointson ncA n whether they are punctual or not. (Cf. [L-Y1] (D(1)) for hidden disconnectivity of Azumaya schemes via the notion of surrogates.) Any C-algebra homomorphism u : C[t] −→ C z 1 , · · · , z n with u (t) ∈ C is a monomorphism and, by definition/tautology, induces a dominant morphism u : ncA n −→ A 1 . In this way, a 0-dimensional subscheme C z 1 , · · · , z n → → A on ncA n is mapped to a 0dimensional, now commutative, subscheme on the affine line A 1 . One may use such projections to get a sense as to how the 0-dimensional subscheme sits in ncA n and some of its properties. (Cf. Radom transformation in analysis.) Example 2.1.9. [hidden disconnectivity of Azumaya/matrix point on ncA n manifested via projection to A 1 ] Recall the matrix point on ncA r 2 in Example 2.1.7 (3). Consider the projection u : ncA r 2 → A 1 specified by the C-algebra monomorphism and letû be a C-algebra homomorphism from the composition induced by the given matrix point on ncA r 2 . Then the matrix point, though connected, is projected under u to two distinct closed points {p = 0 , p = 1} on A 1 , described by the ideal (t(t − 1)), i.e. the kernel Ker (û ) ofû , of C[t]. In this way, u reveals the hidden disconnectivity of the given matrix point on ncA r 2 . As a (left) C[t]-module throughû , M r (C) is pushed forward under u to a torsion O A 1 -module on A 1 , with fiber-dimension r 2 − r at p = 0 and r at p = 1. For reference, if one instead considers the projection v : ncA r 2 → A 1 specified by the Calgebra monomorphism v : Then the given matrix point is projected to a nonreduced point on A 1 associated to the ideal (t 2 ) of C[t]. ncA n as a smooth noncommutative affine toric scheme Let N be a lattice isomorphic to Z n , with a fixed basis (e 1 , · · · , e n ), and M = Hom Z (N, Z) be the dual lattice, with the evaluation pairing denoted by , : M × N → Z and the dual basis (e * 1 , · · · , e * n ). Denote the associated R-vector spaces N R := N ⊗ Z R and M R := M ⊗ Z R. Let σ be the cone in N R generated by e 1 , · · · e n . Then the commutative monoid M σ := σ ∨ ∩M is generated by {e * 1 , · · · , e * n }. LetM be the associated noncommutative monoid from edge-paths at 0, cf. Definition 1.4. Then the correspondence e * i → z i , i = 1, . . . , n, specifies a C -algebra isomorphism from the monoid algebra C M σ to the function ring C z 1 , · · · , z n of ncA n . Under the built-in monoid homomorphism π :M σ → M σ , this isomorphism descends to an isomorphism C[M σ ] → C[z 1 , · · · , z n ] that gives the standard realization of A n as a toric variety. Furthermore, since the multiplicative group Definition 2.1.10. [ncA n as noncommutative affine toric scheme] We shall call the above construction the realization of ncA n as a smooth noncommutative affine toric scheme over C, associated to the cone σ ⊂ N R . By construction, the restriction of the toric scheme structure on ncA n to the master commutative subscheme A n → ncA n recovers the ordinary realization of A n as a toric variety. Recall the lattice N Z n , with a fixed basis (e 1 , · · · , e n ), the dual lattice M = Hom Z (N, Z), the evaluation pairing denoted by , : M × N → Z, the dual basis (e * 1 , · · · , e * n ) of M , and the associated R-vector spaces N R := N ⊗ Z R and M R := M ⊗ Z R. Let ∆ be a fan in N . (That is, ∆ is a set of rational strongly convex polyhedral cones σ in N R such that (1) Each face of a cone in ∆ is also a cone in ∆; (2) The intersection of two cones in ∆ is a face of each; cf. [Fu: Sec. 1.4 ].) For σ, τ ∈ ∆, denote τ σ or σ τ if τ is a face of σ; and denote τ ≺ σ or σ τ if τ is a face of σ and τ = σ. Definition 2.2.1. [∆-system, inverse ∆-system of objects in a category] Let C be a category, with the set of object denoted Object C and the set of morphisms of objects denoted Morphism C. Then a ∆-system (resp. inverse ∆-system) of objects in C is a correspondence F : ∆ → Object C and a choice of morphisms h τ σ : Recall how a variety Y (∆) can be associated to ∆, (e.g. [Fu] ) as an inverse ∆-system of monoid algebras or a ∆-system of affine schemes: Associated to each cone σ ∈ ∆ is a commutative monoid By construction, there is a built-in monoid inclusion M σ → M τ for τ ≺ σ. This gives rise to a ∆-system of affine schemes where C[M σ ] is the monoid algebra over C determined by M σ , that glue to a toric variety Y (∆) through the system of inclusions of Zariski open sets induced from the monoid inclusion M σ → M τ , that by construction satisfy the gluing conditions: The built-in (C × ) n -action on U σ and the built-in torus embedding for each σ ∈ ∆ glues to a (C × ) n -action on Y (∆) and a torus embedding T n → Y (∆). In this section, we shall construct a class of noncommutative spaces, named soft noncommutative toric schemes, associated to ∆ by constructing an inverse ∆-system {M σ } σ∈∆ of submonoids inM := e * 1 , e * 1 −1 , · · · , e * n , e * n −1 via lifting the above construction to the edge-path monoids of commutative monoids with generators and propose soft gluings to bypass the generally unresolvable issue of localizations of noncommutative rings when trying to gluing. Assumption 2.2.2. [on fan ∆] Recall Sec. 2.1, theme: ncA n as a smooth noncommutative affine toric scheme. To ensure that we have at least one uncomplicated chart in the atlas to begin with for the noncommutative toric scheme to be constructed, we shall assume that · The cone generated by e 1 , · · · , e n is in ∆. The corresponding chart ncA n is called the reference toric chart, cf. Definition 2.2.13. Furthermore, to ensure that we have enough good fundamental charts in the atlas before gluings, we shall assume in addition that · All the maximal cones in ∆ are n-dimensional, simplicial, and of index 1. In particular, Y (∆) is smooth. Inverse ∆-systems of ∆-admissible submonoids ofM Recall from Example 1.7 the edge-path monoidM := e * 1 , e * 1 −1 , · · · , e * n , e * n −1 (cf. slightly different notation here) and the built-in monoid epimorphisms π :M → M , π : C ·M → C · M and monoid-algebra epimorphism π : C M → C[M ] via commutatization. Theorem 2.2.5. [existence of inverse ∆-system of admissible submonoids ofM ] There exists an inverse ∆-system {M σ } σ∈∆ of submonoids ofM such thatM σ is admissible to σ for all σ ∈ ∆. We shall call such a system an inverse ∆-system of admissible submonoids ofM . Proof. We proceed in three steps. Step (a) : Submonoids ofM admissible to maximal cones in ∆ Let σ be a maximal cone in ∆. By assumption, σ is n-dimensional, simplicial, and of index 1. Thus, σ ∨ is n-dimensional, simplicial, of index 1 in M R and the generators u 1 , · · · , u n of the monoid σ ∨ ∩ M generates M as well. Lemma 2.2.6. [submonoid admissible to maximal cone] Letȗ i ∈ π −1 (u i ) ⊂M , for i = 1, . . . , n. Then the submonoid ȗ, · · · ,ȗ n ⊂M is monoid-isomorphic to the free associative monoid of n letters z 1 , · · · , z n . Proof. Recall the basis (e * 1 , · · · , e * n ) for M . Since (u 1 , · · · , u n ) generates M as well, up to a relabelling of indices one may assume that u i = n j=1 a ij e * j with the coefficient a ii = 0, for i = 1 . . . , n. Since (ȗ 1 , · · · ,ȗ n ) projects to (u 1 , · · · , u n ) under π, as a word in 2n letters By construction, ȗ 1 , · · · ,ȗ n is admissible to the maximal cone σ and we may takȇ M σ = ȗ 1 , · · · ,ȗ n . Step (b) : Submonoids ofM admissible to lower-dimensional cones in ∆ For each maximal cone σ ∈ ∆, fix a submonoidM σ ⊂M admissible to σ as constructed in Step (a). For a cone τ ∈ ∆ of dimension < n, let ∆ τ := {σ | τ ≺ σ, σ maximal cone} ⊂ ∆ and consider the submonoid ofM generated by allM σ , where σ is a maximal cone in ∆ that contains τ :M It follows that the built-in monoid epimorphism restricts to a monoid epimorphism Proof. For σ a maximal cone in ∆, let u σ,1 , · · · , u σ,n be the generators of M σ andȗ σ,1 , · · · ,ȗ σ,n be the generators of the monoidM σ with π(ȗ σ,i ) = u σ,i . Then, by construction, {ȗ σ,i | σ ∈ ∆ τ , i = 1, . . . , n} ⊂M generates the submonoidM τ ofM . Since τ ⊥ ⊂ ∂τ ∨ := ∪ ρ≺τ ρ and τ ∨ = σ∈∆τ σ ∨ is convex with τ ⊥ a linear stratum of ∂τ ∨ , if m 1 , m 2 ∈ M τ satisfy the condition m 1 + m 2 ∈ τ ⊥ ∩ M , then both m 1 and m 2 must be in τ ⊥ ∩ M . If follows that the submonoid π −1 (τ ⊥ ∩ M ) ∩M τ ofM τ is generated by and augmentM τ toM τ := M τ ,S τ ⊂M . Then,M τ is now a submonoid ofM admissible to τ ∈ ∆. By construction, note that the submonoidM * τ ofM τ that consists of all the invertible elements of the monoidM τ coincides withS τ ; and thatM 0 =M . Step (c). The inverse ∆-system {M σ } σ∈∆ Consider the category of submonoids ofM with morphisms of objects given by inclusions of submonoids ofM and the collection {M σ } σ∈∆ of submonoids ofM that are admissible to cones in ∆ constructed in Steps (a) and (b). For τ ≺ σ, ∆ τ ⊃ ∆ σ and henceM σ ⊂M τ . For ρ ≺ τ ≺ σ, M σ ⊂M τ ⊂M ρ naturally. This shows that the collection {M σ } σ∈∆ is an inverse ∆-system of submonoids ofM and concludes the proof of the theorem. Similar arguments to the proof of Theorem 2.2.5 give the following: Proposition 2.2.9. [completion to inverse ∆-system] Let ∆(n) be the set of n-dimensional cones in ∆. Given {M σ } σ∈∆(n) such thatM σ is a submonoid ofM admissible to σ, there exists an inverse ∆-system {M τ } τ ∈∆ of admissible submonoids ofM such thatM σ =M σ for σ ∈ ∆(n). We shall call {M τ } τ ∈∆ a completion of {M σ } σ∈∆(n) to an inverse ∆-system of admissible submonoids ofM . Proof. The same proof as the proof of Theorem 2.2.5 goes through as long asM σ , σ ∈ ∆(n), in Step (a) is admissible to σ. Proposition 2.2.10. [augmentation by ∆-indexed submonoids] Given an inverse ∆system {M σ } σ∈∆ of admissible submonoids ofM and a collection {S σ } σ∈∆ of finitely generated submonoids ofM such that π(S σ ) ⊂ M σ , there exists an augmentation Proof. We begin with maximal cones σ ∈ ∆(n) and defineM σ := M σ ,S σ . Since σ ∨ in this case is strongly convex in M R ,M σ must already be admissible to σ. We then proceed by induction by the condimension of cones in ∆ and assume thatM σ for σ ∈ ∆(i), ThenM τ ∩ π −1 (τ ⊥ ) is finitely generated and hence so is the submonoid Corollary 2.2.11. [from ∆-indexed to inverse ∆-system] Let {S σ } σ∈∆ be a collection of finitely generated submonoids ofM such that π(S σ ) ⊂ M σ . Then there exists an inverse ∆-system {M σ } σ∈∆ of admissible submonoids ofM such thatS σ ⊂M σ for all σ ∈ ∆. Proof. Choose any inverse ∆-system of admissible submonoids ofM from Theorem 2.2.5 and apply Proposition 2.2.10. Soft nocommutative toric schemes associated to ∆ Definition 2.2.12. [soft noncommutative toric schemes associated to ∆] (Continuing the previous notations.) Let {M σ } σ∈∆ be an inverse ∆-system of admissible submonoids ofM such thatM σ z 1 , · · · , z n for all maximal cones σ ∈ ∆(n) andM 0 =M and {Ȓ σ } σ∈∆ := {C M σ } σ∈∆ be the associated inverse ∆-system of monoid algebras over C. Then the corresponding ∆-system of noncommutative affine schemes is called a soft noncommutative toric scheme with smooth fundamental charts associated to ∆. Here,Ȗ σ is the noncommutative affine scheme associated toȒ σ (cf. Definition 2.1.1) for σ ∈ ∆, the notationȖ(∆) emphasizes that this is a gluing system of some particular charts, and the notation Y (∆) emphasizes that this is treated as a noncommutative space as a whole. 1 With an abuse of language, we shall callY (∆) n-dimensional since the function ring of each fundamental chart is freely generated by n coordinate functions. For σ ∈ ∆, the monoid algebra C M σ is called interchangeably the function ring or the local coordinate rings of the chartȖ σ . The assignment is called the structure sheaf ofY (∆) and is denoted OY (∆) as in ordinary Commutative Algebraic Geometry. By construction, for cones τ ≺ σ in ∆, one has a dominant morphism of noncommutative affine schemesȖ τ →Ȗ σ and a monoid-algebra monomorphism 2 1 Here, we use the notation Y (∆), rather than X(∆) following [Fu] , due to that such a space will be used as a target space for D-branes, whose world-volume is generally denoted X Az . 2 Though our focus in this work is on the rings and their homomorphisms for Noncommutative Algebraic Geometry, we try to keep the contravariant underlying geometry as in Commutative Algebraic Geometry in mind whenever possible. This is why we use the notation ι τ σ here and reserve ιτσ for the corresponding morphism of underlying "spaces". Here, there is no indication that ιτσ :Ȗτ →Ȗσ in any sense is an inclusion, though it is true that, when restricted to Y (∆), ιτσ : Uτ → Uσ is an open set inclusion in the sense of affine schemes in Commutative Algebraic Geometry. These replace the role of 'open sets' and 'restriction to open sets' respectively in the usual definition of the 'structure sheaf ' of a scheme in Commutative Algebraic Geometry. An ideal sheafȊ onY is an inverse ∆-system {Ȋ σ } σ∈∆ , whereȊ σ is a two-sided ideal of R σ , such that ι τ σ (I σ ) = I τ . Given an ideal sheafȊ = {Ȋ σ } σ∈∆ onY (∆), one can form an inverse ∆-system of C-algebras {Ȓ σ /Ȋ σ } σ∈∆ , with the inclusion ι τ σ :Ȓ σ /Ȋ σ →Ȓ τ /Ȋ τ , for τ ≺ σ ∈ ∆, naturally induced from ι τ σ . We will think of this inverse ∆-system as defining a soft noncommutative closed subschemeZ ofY (∆) and denoted its structure sheaf OZ and write the quotient as OY (∆) → OZ. RedenoteȊ by IZ. Then one has a short exact sequence Let N be another lattice (isomorphic to Z n for some n via a fixed basis (e 1 , · · · , e n ) of N ), ∆ be a fan in N R that satisfies Assumption 2.2.2, andY (∆ ) be a soft noncommutative toric scheme associated to ∆ , with the underlying inverse ∆ -system of submonoids ofM denoted is a homomorphism ϕ : N → N of lattices that satisfies the conditions: (1) ϕ induces a map between fans (in notation, ϕ : ∆ → ∆ ); i.e., for each σ ∈ ∆ there exists a σ ∈ ∆ such that ϕ(σ) ⊂ σ . (2) The induced monoid-homomorphismφ :M →M restricts to a monoid homomorphism Definition 2.2.13. [reference toric chart and fundamental toric charts ofY (∆)] (Continuing Definition 2.2.12.) A chartȖ σ ofY (∆) that is associated to a maximal cone σ ∈ ∆(n) is called a fundamental toric chart ofY (∆). Among them, the one associated to the cone generated by the given basis (e 1 , · · · , e n ) of the lattiice N is called the reference toric chart ofY (∆). Fundamental toric charts are all isomorphic to the smooth noncommutative affine n-space ncA n . These charts should be thought of as the basic, good pieces to be glued via the subcollection {Ȗ τ } τ ∈∆−∆(n) . The reference chart ensures that we have at least one fundamental toric chartȖ σ such that the morphismȖ 0 →Ȗ σ of noncommutative affine schemes resembles the inclusion of an Zariski open set. Together, this allows one to think ofY (∆) as a "partial compactification" of ncA n via soft gluings of additional copies of ncA n 's. Definition 2.2.14. [softening of noncommutative toric scheme] LetY (∆) andY (∆) be two soft noncommutative toric schemes associated to a fan ∆, with the underlying inverse ∆-system of monoid algebras {C M σ } σ∈∆ and {C M σ } σ∈∆ respectively. ThenY (∆) is called a softening ofY (∆) ifM σ =M σ for maximal cones σ ∈ ∆ (i.e.Y (∆) andY (∆) have identical fundamental toric charts) andM σ ⊂M σ for nonmaximal cones σ ∈ ∆ (i.e. the gluings inY (∆) is softer than those inY (∆)). By construction, there is a built-in toric morphism ϕ :Y (∆) →Y (∆), called the softening morphism. Remark 2.2.15. [meaning of softening: comparison to Isom -functor for moduli stack] Introducing the notion of 'softening' in the play is meant to remedy the fact that we don't have a good theory of localizations of a noncommutative ring to apply to our construction of noncommutative schemes. Rather we keep a collection of charts as principal ones and the remaining charts serve for the purpose of gluing these principal charts. These 'secondary charts' are then allowed to be 'dynamically adjusted' to fulfill the purpose of serving as a medium between principal charts. One may compare this to the gluing in the construction of a moduli stack. There one cannot directly glue two charts for the moduli stack as schemes but rather has to create the medium chart via the Isom -functor, cf. [Mu] . See Sec. 3 how softening is used to construct invertible sheaves and twisted sections on a soft noncommutative toric scheme. (c) The T n -action The T n -action on C[M ] leaves each C · m, m ∈ M , invariant. As C = Center (C M ), the T n -action on C[M ] naturally lifts to a T n -action on C M : that leaves each C ·m,m ∈M , invariant. It follows that T n leaves each C M σ , σ ∈ ∆, invariant and that ι τ σ : C M σ → C M τ , τ ≺ σ, are T n -equivariant. This defines the T n -action onY (∆). The monoid epimorphism π :M → M restricts to monoid epimorphisms π :M σ → M σ , for all σ ∈ ∆. Which in turn gives a ∆-collection of T n -equivariant monoid-algebra epimorphisms {C M σ → → C[M σ ]} σ∈∆ that commute with inclusions: This defines a built-in T n -equivariant embedding Y (∆) →Y (∆). (f ) The master commutative subscheme ofY (∆) This defines another inverse ∆-system of commutative C-algebras and hence a (generally singular) commutative schemeY (∆) ♣ over C. By construction any morphism from a commutative scheme X toY factors through a morphism X →Y (∆) ♣ →Y (∆). In particular, one has commutative diagrams and hence Y (∆) ⊂Y (∆) ♣ ⊂Y (∆). Note that, by construction, However, such isomorphism on local charts may not holds for τ ∈ ∆(k), 1 ≤ k ≤ n − 1. We construct in this section a class of noncommutative spaces, named 'soft noncommutative schemes' from invertible sheaves and their twisted sections onY (∆). Definition 3.1. [sheaf onY (∆) and OY (∆) -module] (1) An inverse ∆-system of objects in a category C (cf. Definition 2.2.1) is also called a sheaf (of objects in C) onY (∆). For example, OY (∆) is a sheaf of C-algebras onY (∆). (Which justifies its name: the structure sheaf ofY (∆)). (2) LetY (∆) = {Ȗ σ } σ∈∆ be a soft noncommutative toric scheme with the underlying inverse ∆-system of monoid algebras ({Ȓ σ whereF σ is a leftȒ σ -module and h στ : ι στFσ :=Ȓ τ ⊗Ȓ σF σ →F τ onȖ τ such that the following diagram commutes ι σρFσ An element ofF σ is called a local section ofF overȖ σ . A section (or global section)ofF is a collection {s σ } σ∈∆ of local sectionss σ ∈F σ , σ ∈ ∆, ofF such that h στ (s σ ) =s τ , τ ≺ σ ∈ ∆. In the last expression,s σ is identified as 1 ⊗s σ ∈Ȓ τ ⊗Ȓ σF σ . The C-vector space of sections of F onY (∆) is denoted H 0 (Y (∆),F) or simply H 0 (F). A homomorphism f :F := ({F σ } σ∈∆ , {h στ } τ, σ∈∆, τ ≺σ ) −→F := ({F σ } σ∈∆ , {h στ } τ, σ∈∆, τ ≺σ ) of left OY (∆) -modules is a collection {f σ } σ∈∆ , where f σ :F σ →F σ is a homomorphism of left R σ -modules, such that the following diagrams commute . f a monomorphism) if in addition f σ is injective for all σ ∈ ∆; surjective (i.e. f an epimorphism) if in addition f σ is surjective for all σ ∈ ∆; an isomorphism if in addition f σ is an isomorphism of leftȒ σ -modules for all σ ∈ ∆. (3) Recall the built-in T n -action g Explicitly, note that the T n -action onY (∆) leaves each chartȖ σ , σ ∈ ∆ invariant and φ t restricts to anȒ σ -module homomorphism φ t :F t σ −→F σ , withs −→ φ t (s) =: g t ·s on each chartȖ σ , σ ∈ ∆. HereF t σ := g * tF σ , which is the same C-vector space asF σ but with thȇ R σ -module structure defined byȓ t ·s := g t (ȓ) ·s. This defines a T n -action onF σ that satisfies φ t (g t (ȓ) ·s) =ȓ · (g t ·s) by tautology, for t ∈ T n ,ȓ ∈Ȓ σ , ands ∈F σ . The T n -action onF σ , σ ∈ ∆, is equivariant under gluings: fors ∈F σ , t ∈ T n , and τ ≺ σ ∈ ∆. (4) Similarly, for right and two-sided OY (∆) -modules. a rightȂ-module, andG = ({G σ } σ∈∆ , {hG στ } τ ≺σ∈∆ ) a leftȂ-module, all three onY (∆). Then define the tensor productF ⊗ȂG ofF andG overȂ to be the following (right-on-F, left-on-G) For each σ ∈ ∆ , ∆ σ := {σ ∈ ∆ | ϕ(σ) ⊂ σ } is a subfan of ∆ and specifies a subsystem OY (∆ σ ) := {Ȓ σ } σ∈∆ σ of OY (∆) . Through the restriction ofφ : rendered a leftS σ -module. The gluing data {hF στ } τ ≺σ∈∆ ofF gives rise to a gluing data for the ∆ -collection {H 0 (Y (∆ σ ), F|Y (∆ σ ) )} σ ∈∆ and turn the ∆ -collection into a left OY (∆ ) -module, denotedφ * F and named the direct image sheaf or pushforward ofF underφ. For each σ ∈ ∆, there exists a unique ρ σ ∈ ∆ such that ϕ(σ) ⊂ ρ σ and that ρ σ σ for all σ ∈ ∆ with ϕ(σ) ⊂ σ . The property ρ τ ρ σ for τ ≺ σ ∈ ∆ and the gluing data {hG σ τ } τ ≺σ ∈∆ ofG render the ∆-collection {G ρ σ } σ∈∆ a sheaf of C-vector spaces onY (∆), denotedφ −1G and named the inverse image sheaf ofG underφ. In particular,φ −1 OY (∆ ) is a sheaf of C-algebras onY (∆) and, by construction,φ −1G is a leftφ −1 OY (∆ ) -module. Similarly, for right modules and two-sded modules onY (∆) andY (∆ ). such thatF σ Ȓ σ as left (resp. right, two-sided)Ȓ σ -modules for σ ∈ ∆ and h στ is an isomorphism of left (resp. right, two-sided)Ȓ τ -modules for τ, σ ∈ ∆, τ ≺ σ. Here,m σσ acts onF σσ Ȓ σσ as a leftȒ σσ -module homomorphism given by 'the multiplication bym σσ from the right'. In particular,m τ ρ •m στ =m στmτ ρ . Two such collections {c στmστ } τ ≺σ∈∆, • and {c στm στ } τ ≺σ∈∆, • define isomorphic invertible OY (∆) -modules if and only if there exists a collection Proof. Observe that the multiplicative monoid of invertible elements of the monoid algebra C M is given by C ×M and that the multiplicative monoid of invertible elements of the monoid algebraȒ σ , σ ∈ ∆, is contained in C ×M and is given by C ×M σ ∩ π −1 (σ ⊥ ∩ M ). It follows from Definition 3.3 that an invertible OY (∆) -module is determined by gluings of the collection {Ȓ σ } σ∈∆ for each pair τ ≺ σ asȒ τ -modules. This gives the gluing data {c στmστ } τ ≺σ∈∆,• in the Statement. Different choices of isomorphismsF σ Ȓ σ (i.e. local trivializations of L) define isomorphic invertible OY (∆) -modules. Proof. Under Assumption 2.2.2, Y (∆) is smooth and thus L O Y (∆) (D) for some T n -invariant Cartier divisor D on Y (∆) and can be specified by a collection {m σ ∈ M } σ∈∆(n) , where m σ ∈ M and ∆(n) is the collection of maximal cones in ∆, that satisfies m −1 σ m τ ∈ (σ ∩ τ ) ⊥ ∩ M , for σ, τ ∈ ∆(n). (Cf. [Fu: Sec. 3.4] , with the monoid M here presented multiplicatively for convenience.) For each τ ∈ ∆(k), k ≤ n − 1, fix an σ ∈ ∆(n) such that τ ≺ σ and set m τ = m σ . This extend {m σ } σ∈∆(n) to a collection {m σ } σ∈∆ such that m −1 σ m τ ∈ (τ ∩ σ) ⊥ ∩ M . Let {M σ } σ∈∆ be the underlying inverse ∆-system of monoids associated toY (∆) and, for each σ ∈ ∆, fix anm σ ∈M σ such that π(m σ ) = m σ . Set Then, This is almost the gluing data in Lemma 3.4 for an invertible OY (∆) -module except that in generalm στ / ∈M τ . To remedy this, let S σ = the empty set for σ ∈ ∆(n) , This defines a collection of ∆-indexed finitely generated submonoids ofM . It follows from Proposition 2.2.10 that one can augment {M σ } σ∈∆ to an inverse ∆-system {M σ } σ∈∆ of submonoids ofM such thatS σ ⊂M σ . Furthermore, sinceS σ = empty set for σ ∈ ∆(n), it follows from the proof of Proposition 2.2.10 that it can be made thatM σ =M σ for σ ∈ ∆(n). Thus, {M } σ∈∆ defines a softeningY (∆) ofY (∆) and L extends to an invertible OY (∆) -moduleL on Y (∆). Since a soft noncommutative toric scheme associated to ∆ always exists (cf. Theorem 2.2.5, Proposition 2.2.9, Definition 2.2.12), Proposition 3.5 can be rephrased as: Proposition 3.5 . [existence of invertible sheaf ] Let L be an invertible sheaf on the smooth toric variety Y (∆). Then there exists a soft noncommutative toric schemeY (∆) ⊃ Y (∆) associated to ∆ such that L extends to an invertible sheafL onY (∆). Note that the pullback of an invertible sheaf under a morphism of noncommutative toric schemes is an invertible sheaf. In particular, any invertible sheaf onY (∆) pulls back to invertible sheaves on softenings ofY (∆). Let L be an invertible sheaf on the n-dimensional smooth toric variety Y (∆). Then L is isomor- (1), and a τ ∈ Z. The divisor D determines an m σ ∈ M for each maximal cone σ ∈ ∆(n): Here v τ ∈ τ ∩ N is the first lattice point on the ray τ . Writing the commutative monoid M multiplicatively as a multiplicatively closed subset of R 0 = C[M ] and passing to the isomorphism (Cf. [Fu: Sec. 3.3 & Sec. 3.4 Definition 3.6. [twisted section of invertible sheaf ] LetL = ({F σ } σ∈∆ , {h στ } τ ≺σ∈∆ ) be a (left) invertible sheaf on a soft noncommutative toric schemeY (∆) = {Ȓ σ } σ∈∆ . Two local sectionss 1 ,s 2 ∈F σ , σ ∈ ∆, are said to be equivalent, in notation,s 1 ∼s 2 , if there exists an invertible elementȓ ∈Ȓ * σ such thats 1 =ȓs 2 . This is clearly an equivalence relation onF σ , σ ∈ ∆. A twisted section ofL is a collection Fix a local trivializing isomorphismF σ Ȓ σ of leftȒ σ -modules, σ ∈ ∆. Then,s is specifoed uniquely by {ȓ σ } σ∈∆ , withȓ ∈Ȓ σ . We will called {ȓ σ } σ∈∆ the presentation ofs with respect to the local trivialization ofL. Different choices of local trivializations ofL give presentations ofs that differ by a right multiplication of unit (i.e. invertible element) chart by chart. Lemma 3.7. [twisted section under pullback] Let φ :Y (∆) →Y (∆) be a morphism of soft noncommutative toric schemes,L be an invertible sheaf onY (∆), ands be a twisted section ofL. Then, φ * s is a twisted section of the pullback φ * L ofL toY (∆). In particular, a twisted section pulls back to a twisted section under softening. Proof. This follows from the fact that φ takes an invertible element to an invertible element chart by chart. Proposition 3.8. [extension to twisted section after passing to softening] LetL be an invertible sheaf onY (∆), L be the restriction ofL to Y (∆) ⊂Y (∆), and s 1 , · · · , s k ∈ H 0 (Y (∆), L) be sections of L on Y (∆). Then, there exists a softeningY (∆) →Y (∆) such that all s 1 · · · , s k of L extend to twisted sections of the pullbackL ofL toY (∆). Proof. Present L as a T-Cartier divisor O Y (∆) (D), recall the polytope P D in M R associated to D, and denote the section of L corresponding to m ∈ P D ∩ M by s m . Since each s i is a C-linear combination of finitely many s m 's, m ∈ P D ∩ M , it follows from Lemma 3.7 that we only need to prove the proposition for a section of L in the form s m for some m ∈ P D ∩ M . Write out more explicitly:Y (∆) = {C M σ } σ∈∆ andL = ({C M σ } σ∈∆ , {c ρτmστ } τ ≺σ∈∆ ) under a local trivialization (cf. Lemma 3.4). Recall the collection {m σ } σ∈∆(n) ⊂ M associated to D from the beginning of the theme. For any τ ∈ ∆ − ∆(n), fix an σ ∈ ∆(n) such that τ ≺ σ and assign m τ to be m σ . This enlarges the collection {m σ } σ∈∆(n) to a collection {m σ } σ∈∆ with the property that Here, we write the operation of the commutative monoid M multiplicatively. Choosem σ ,m σ ∈M such that for all σ ∈ ∆. Since π :M → M restricts to a surjectionM σ ·m σ → M σ · m σ and m ∈ M σ · m σ , such a choice always exists. It follows thatm σ =ȓ σ ·m σ for someȓ σ ∈M σ ⊂Ȓ σ , σ ∈ ∆. Consider now the collections := {ȓ σ } σ∈∆ of local sections ofL. By construction, it recovers the section s m of L when restricted to Y (∆) ⊂Y (∆). For τ ≺ σ ∈ ∆, h στ (ȓ σ ) =ȓ σ · c στmστ ∈Ȓ τ . As elements inM ,ȓ σm στ = (ȓ σm στȓ τ −1 ) ·ȓ τ . Since Thus,s is almost a twisted section ofL except that the twisting factorȓ σm στȓ τ −1 may not yet be inM τ . The same argument as in the proof of Proposition 3.5 and applying Proposition 2.2.10 show that there exists a softening φ :Y (∆) →Y (∆) such that the pullback collection φ * s is indeed a twisted section of the pullback invertible sheaf φ * L onY (∆). This proves the proposition. Since a soft noncommutative toric scheme associated to ∆ with an invertible sheaf that extends L on Y (∆) exists (cf. Proposition 3.5 ), Proposition 3.8 is equivalent to: Proposition 3.8 . [extension to twisted section] Let L be an invertible sheaf on Y (∆) and s 1 , · · · , s k ∈ H 0 (Y (∆), L) be sections of L. Then, there exists a soft noncommutative toric schemeY (∆) with an invertible sheafL that restricts to L on Y (∆) ⊂Y (∆) such that s i extends to a twisted sections i ofL, for i = 1, . . . , k. Soft noncommutative closed subschemes ofY (∆) associated to twisted sections of invertible sheaves onY (∆) Continuing the notation and discussion of the previous theme. Lets i = {s i, σ } σ∈∆ be a (nonzero) twisted section of an invertible sheafL i onY (∆), for i = 1, . . . , k. With respect to a local trivialization of eachL i ,s i has a presentation {ȓ i, σ } σ∈∆ , withȓ i, σ ∈Ȓ σ , σ ∈ ∆. A different local trivialization ofL i gives rise to a different presentation ofs i of the form {ȓ i, σȗi, σ } σ∈∆ , whereȗ i, σ is a unit (i.e. invertible element) ofȒ σ . Denote Γ = {s 1 , · · · ,s k }. It follows that the two-sided idealȊ Γ, σ := (ȓ 1, σ , · · · ,ȓ k, σ ) ofȒ σ , σ ∈ ∆, associated to Γ is independent of the local trivialization ofL i , i = 1, . . . , k. This defines an ideal sheaf I Γ := {Ȋ Γ, σ } σ∈∆ onY (∆) and hence a soft noncommutative closed subschemeZ Γ ofY (∆). Definition 3.9. [soft noncommutative closed subscheme associated to twisted sections]Z Γ is called the soft noncommutative closed subscheme ofY (∆) associated to Γ. When Γ = {s},Zs :=Z Γ is called the soft noncommutative hypersurface ofY (∆) associated tos. Definition 3.10. [soft noncommutative scheme/space (via toric geometry)] A soft noncommutative toric schemeY (∆) associated to a fan ∆ or a soft noncommutative closed subschemeZ of aY (∆) for some ∆ will be called a soft noncommutative scheme via toric geometry, or simply a soft noncommutative scheme, or even a soft noncommutative space. The structure sheaf OY of such a noncommutative spaceY is defined to the inverse ∆-system of C-algebras that describesY . It is this class of noncommutative spaces that we will study further in sequels. Definition 2.2.12 and Definition 3.1 can be generalized routinely to soft noncommutative schemes and their modules. A straightforward generalization of Proposition 3.8 to a finite collection of invertible sheaves with sections {(L 1 , s 1 ), · · · , (L k , s k )} on Y (∆) implies that: Corollary 3.11. [commutative complete intersection subscheme] Any complete intersection subscheme Z of a smooth toric variety Y (∆) embeds as a commutative closed subscheme of a soft noncommutative schemeZ in aY (∆) (⊃ Y (∆)) such thatZ ∩Y (∆) = Z. In particular, this applies to complete intersection Calabi-Yau spaces in a smooth toric variety. Indeed it is the attempt to make this corollary holds that the notion of soft noncommutative schemes via toric geometry is discovered. Let us spell out a guiding question behind the NCS spin-off of the D-project before leaving this section: The notion of morphisms from an Azumaya scheme with a fundamental module over C to a soft noncommutative scheme over C is developed in this section. The notion of Azumaya schemes (or equivalently matrix schemes) with a fundamental module and morphisms therefrom were developed in [L-Y1] (D(1)) and [L-L-S-Y] (D(2)) in the realm of Algebraic Geometry and in [L-Y2] (D(11.1)) and [L-Y3] (D(11.2)) in the realm of Differential Geometry to capture the Higgsing/unHiggsing feature of D-branes when they coincide or separate and to address dynamical D-branes moving in a space-time, [L-Y4] (D(13.1))and [L-Y5] (D (13.3) ). The generalization of such notion to cover fermionic D-branes in Superstring Theory has been a major focus in more recent years; cf. the SUSY spin-off of the D-project, e.g. [L-Y6] (SUSY(2.1)). The key setup of Azumaya/matrix schemes in the realm of Algebraic Geometry that is relevant to the current work is reviewed below for the terminology and notation. Readers are referred to the above works for more details and to [Liu] for a review of the first four years (fall 2007 -fall 2011) of the D-project. Definition 4.1.1. [D-brane world volume: Azumaya/matrix scheme over C] Let X be a (Noetherian) scheme over C and E be a locally free O X -module of rank r. Then the sheaf End O X (E) := Hom O X (E, E) of endomorphisms of E is a noncommutative O X -algebra with center O X . The fiber of End O X (E) over each C-point on X is isomorphic to the Azumaya algebra (or interchangeably matrix algebra) of r × r matrices over C. E is in the fundamental representation of End O X (E). The new ringed space from the enhanced structure sheaf, with its fundamental representation encoded, is called an Azumaya scheme (or interchangeably matrix scheme) with a fundamental module. The ringed space (X, O Az X ) serves as the world-volume of a D-brane and the fundamental (left) O Az X -module E serves as the Chan-Paton sheaf on the world-volume. For a complete description of the data on the world-volume of a D-brane, there is also a connection ∇ on E and, in this case, one may introduce a Hermitian structure on E and require that ∇ be unitary. These two additional structures on X Az are irrelevant to the discussion of the current work and hence will be dropped. As the category of (left) O Az X -modules is equivalent to the category of O X -modules (cf. Morita equivalence), the noncommutativity of X Az may look only of a mild kind -yet important for the correct mathematical description of dynamical D-branes in string theory. where M r (C) is the algebra of r × r matrices over C. One may attempt to assign some sensible topology "Space (M r (C))" to the noncommutative C-algebra, in the role of Spec (R) to a commutative ring R. However, all the existing methods only lead to that "Space (M r (C))" = {p}. Thus, underlying topology is the wrong direction to understand the geometry behind the Azumaya point (interchangeably, matrix point) p Az . Now, the D-brane world-volume is only half of the story. The other half is how the worldvolume gets mapped into a target space-time Y (i.e. morphisms from Azumaya schemes to Y ). With this in mind, if A ⊂ M r (C) is a C-subalgebra and we do know how to make sense of Space (A) and morphisms Space (A) → Y -for example, A is commutative C-subalgebra of M r (C) and Y is an ordinary Noetherian commutative scheme in Commutative Algebraic Geometry ( [Ha] ) -then we should expect a morphism p Az → Y from the composition It follows that the correct way to unravel the geometry behind p Az is through C-subalgebras of M r (C). For example, for r ≥ 2, M r (R) contains C-subalgebra in product form A = A 1 ×A 2 . This means p Az has hidden disconnectivity. Indeed, the consideration of commutative C-subalgebras of M r (C) alone is enough to indicate that p Az has a very rich geometry behind. Cf. [Liu: Figure 1 We study in the subsection the notion of morphisms from an Azumaya scheme with a fundamental module to a soft noncommutative schemeY . The design is guided by the requirement to reproduce the Higgsing/unHiggsing phenomenon when D-branes collide/coincide or fall apart/separate in a space-time. Idempotents and gluing of quasi-homomorphisms Definition 4.2.1. [quasi-homomorphism of C-algebras] Given two (generally noncommutative) C-algebras R and A, a quasi-homomorphism from R to A is a C-vector-space homomorphism f : R → A such that f (r 1 r 2 ) = f (r 1 )f (r 2 ). Note that it is not required that f (1 R ) = 1 A , where 1 R and 1 A are the identity of R and A respectively. . A C-algebra quasi-homomorphism f : R → S with f (1 R ) = 1 S is called as usual a C-algebra homomorphism. In the other extreme, the zero-map R → 0 ∈ A is a quasi-homomorphism. Let A, R, S be C-algebras with a fixed C-algebra homomorphism h : R → S. It is standard that two C-algebra homomorphisms f R : R → A and f S : S → A are defined to be glued under h if f R = f S • h. But if, instead, f R and f S are only C-algebra quasi-homomorphisms, it takes some thought as to what the "correct" definition of 'f R and f S glue under h' should be. As a generally noncommutative C-algebra, it can happen that A is not a product of C-algebras and yet still contains idempotents other than 0 and 1. In terms of the objects in the opposite category of the category of C-algebras, the geometry Scheme (A) behind the C-algebra A could have hidden disconnectivity (cf. Definition 2.1.8 and Example 2.1.9) and in terms of geometry it can happen that some connected components in the hidden disconnectivity of Scheme (A) is mapped to the complement of the image of Scheme (S) in Scheme (R) under h. Since such hidden disconnectivity of Scheme (A) is inherited from idempotents of A other than 0 and 1, we need to keep track of f R (1 R ) and f S (1 S ), allowing the situation f R (1 R ) = f S (1 S ), as well when considering gluing of C-algebra quasi-homomorphisms f R and f S under h. A be a C-algebra and e 1 , e 2 ∈ A are idempotents: e 2 1 = e 1 , e 2 2 = e 2 . We say that e 1 is subordinate to e 2 , in notation e 1 ≺ e 2 , or equivalently e 2 is superior to e 1 , in notation e 2 e 1 , if e 1 = e 2 and e 1 e 2 = e 2 e 1 = e 1 . Denote 'e 1 ≺ e 2 or e 1 = e 2 ' by e 1 e 2 or equivalently e 2 e 1 . Note that 0 e 1 for any idempotent e of A. When e 1 ≺ e 2 , e 2 := e 2 − e 1 ∈ A is also an idempotent subordinate to e 2 and it satisfies e 2 e 1 = e 1 e 2 = 0. Thus, e 2 = e 1 + e 2 gives an orthogonal decomposition of e 2 into subordinate idempotents. Or, equivalently in terms of the objects in the opposite category of the category of C-algebras, a diagram of morphisms of (generally noncommutative) schemes: Assume in addition that the idempotent e 3 := 1 − e 1 is also orthogonal to e 2 (i.e. e 2 e 3 = e 3 e 2 = 0 in A) and let A e := {a ∈ A | ae = ea = a} for an idempotent e ∈ A. Note that A e is a C-algebra with a built-in C-algebra quasihomomorphism A e → A. Then, 1 = e 1 + e 2 + e 3 is an orthogonal decomposition of 1 by idempotents in A and is now an honest C-subalgebra inclusion. Since f R (R) ⊂ A e 1 + A e 2 and f S (S) ⊂ A e 3 , one has now a commuting diagram of C-algebra homomorphisms Here π 12 and π 2 are the projection homomorphisms of product C-algebras to the indicated components. The corresponding commuting diagram of morphisms of noncommutative affine schemes in the opposite category is then: ( = disjoint union) Here, ι 12 and ι 2 are the inclusion morphism of the connected components as indicated. Thus, a quasi-homomorphism as defined in Definition 4.2.1 is nothing but a roof of ordinary homomorphisms and, with an additional assumption of the completeness of idempotents involved, a gluing of quasi-homomorphisms in the sense of Definition 4.2.3 is nothing but the ordinary gluing of homomorphisms after the replacement by roofs. Remark 4.2.5. [transitivity of subordination relation of idempotents] Caution that for a general noncommutative C-algebra A, idempotents e 1 ≺ e 2 and e 2 ≺ e 3 do not imply e 1 ≺ e 3 . (E.g. A = C z 1 , z 2 , z 2 /(z 2 1 − z 1 , z 2 2 − z 2 , z 2 3 − z 3 , z 1 z 2 − z 1 , z 2 z 1 − z 1 , z 2 z 3 − z 2 , z 3 z 2 − z 2 ).) However, transitivity of ≺ holds for idempotents of the Azumaya/matrix algebra M r (C) of rank r over C, for all r. (For idempotents e 1 ≺ e 2 , e 2 ≺ e 3 in M r (C), the three e 1 , e 2 , e 3 must be commuting and hence simultaneously diagonizable. Which implies e 1 ≺ e 3 in the end.) Homomorphisms from a ∆-system of C-algebras to a C-algebra A Let ∆ be a fan in N R that satisfies Assumption 2.2.2 and A a C-algebra. Definition 4.2.6. [∆-system of idempotents in A] Let e A ∆ := {e σ } σ∈∆ be a collection of idempotents in A labelled by cones in ∆. e A ∆ is said to be (i) a weak ∆-system of idempotents in A if e τ e σ for all τ ≺ σ ∈ ∆; (ii) a strong ∆-system of idempotents in A if e σ e σ = e σ∩σ for all σ, σ ∈ ∆. Note that since σ ∩ σ = σ ∩ σ, all the idempotents in a strong ∆-system of idempotents in A commute with each other. Furthermore, since τ ∩ σ = τ for τ ≺ σ ∈ ∆, a strong ∆-system of idempotents in A is naturally a weak ∆-system of idempotents in A. Lemma/Definition 4.2.7. [reduced idempotents in strong ∆-system] Let e A ∆ = {e σ } σ∈∆ be a strong ∆-system of idempotents in A. Then, there exists a unique collection {e σ } σ∈∆ of orthogonal (i.e. e σ e σ = 0 for σ = σ ) idempotents in A such that e σ is called the reduced idempotent associated to σ ∈ ∆ from the strong ∆-system e A ∆ . It is the contribution to e σ after being trimmed away all the boundary contributions. Proof. Explicitly, the reduced idempotent e σ , for σ ∈ ∆(k), is given by: Equivalently, let {τ 1 , · · · , τ k } be the set of facets (i.e. codimension-1 faces) of σ ∈ ∆(k), then e σ = e σ − i e τ i + i 1