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Geometria Superiore Reference Cards Push out (Sommes amalgam´ees). Given a diagram i : P → Qi (i = 1, 2) we can define 1  c 2000 M. Cailotto, Permissions on last. v0.0  u  −→2 Send comments and corrections to [email protected] Q1 ⊕P Q2 := coker P −→ Q1 ⊕ Q2 u1 The of Monoids. 1 and it is a standard fact that the natural arrows ji : Qi → A (M, ·, 1) (commutative with unit) is a M with a Q1 ⊕P Q2 define a cocartesian square: u1 composition law · : M × M → M and an element 1 ∈ M such P −−−→ Q1 that: the composition is associative: (a · b) · c = a · (b · c) (for   all a, b, c ∈ M), commutative: a · b = b · a (for all a, b ∈ M), u2 y y j1 and 1 is a neutral element for the composition: 1 · a = a (for Q2 −→ Q1 ⊕P Q2 . all a ∈ M). j2 ϕ ϕ More explicitly we have Q1 ⊕P Q2 = (Q1 ⊕ Q2)/R with R the A (M, ·, 1M ) −→(N, ·, 1N ) of monoids is a M → N smallest equivalence stable under product (in Q1 ⊕P of sets commuting with · and 1: ϕ(ab) = ϕ(a)ϕ(b) for all Q2) and making (u1(p), 1) ∼R (1, u2(p)) for all p ∈ P . a, b ∈ M and ϕ(1 ) = 1 . Thus we have defined the cat- M N In general it is not easy to understand the relation involved in egory Mon of monoids. the description of Q1 ⊕P Q2, but in the case in which one of {1} is a monoid, initial and final object of the category Mon. the Qi is a we have a nice description: Q1 ⊕P Q2 = (Q ⊕ Q )/∼ where (x , x ) ∼ (y , y ) if and only if: (N, +, 0) and (Z, ·, 1) are monoids. 1 2 1 2 1 2 ( Free monoids. For any set S there is a free monoid with x1u1(d) = y1u1(c) S there exist c, d ∈ P such that generators the elements of S: the free monoid on S: N . The S S x u (c) = y u (d) natural morphism S → N given by s 7→ es (element of N 2 2 2 2 sending s to 1 and the other elements to zero) realizes a (bi- i.e. if and only if there exist c, d ∈ P such that (x1, x2)(u1(d), u2(c)) = ∼ S functorial) HomSet(S, M) = HomMon(N ,M) for (y1, y2)(u1(c), u2(d)) in Q1 ⊕ Q2. We see in fact that ∼ is an any monoid M. This construction gives a left adjoint to the equivalent relation (without hypothesis) and, by the given hy- forgetful from Mon to Set. pothesis, it is just the relation described in the previous num- Associated Group. There exists a left adjoint to the natural ber. u forgetful functor from Ab to Mon, given by M 7→ Mgp where: Cokernel. Given a morphism P → Q of monoids, we can con- gp  u 1 M := M ×M ∼ and ∼ is the equivalence relation on M ×M sider the diagram P −→ Q ←−{1} and put coker(u) = coker(u, 1) := defined by (x, y) ∼ (m, n) if and only if there exists a ∈ M coker(u, 1) that is: Q ⊕ {1}/∼ = Q/∼ where such that a · x · n = a · y · m. We have the natural morphism 1 1 M −→ Mgp, x 7→ (x, 1). x ∼ y if and only if there exist c, d ∈ P such that xu(c) = yu(d) Projective limits. In the category Mon all projective limits (note that {1} is an abelian group!). exist, and the projective limit functor commutes with the forget It is useful to observe that if we have a cocartesian diagram functor from Mon to Set. In fact this functor is a right ad- u1 P −→ Q1 joint, and so commutes with all projective limits, and we have   a canonical monoid structure (component by component) on u2 y y j1 the (set theoretical) projective limits of a system of monoids. Q2 −→ Q In particular we have (arbitrary) products and fiber products j2 of monoids. the induced morphism between the cokernel: coker(u1) → coker(j2) is always an . f Warning: . Given a morphism M → N of monoids, we Warning: Cokernel. The notion of (i.e. right have also the kernel kerf which is a particular projective limit; erasable map) in Mon is weaker than the notion of epimor- being the kernel of the double map f and the constant 1 map, −1 phism in Set (i.e. surjectivity): moreover an epimorphism has it is realized as for abelian groups by kerf = f (1). But a trivial cokernel, while a map with a trivial cokernel is not the notion of monomorphism (i.e. left eraseable map) in Mon necessarily an epimorphism. coindices with the notion of monomorphism in Set (i.e. injec- tivity): so a monomorphism has a trivial kernel, while a map Consider for example N := (N, +, 0) and the map N×N → N×N with a trivial kernel is not necessarily a monomorphism (con- defined by (x, y) 7→ (x, x + y); the map is not surjective, but s sider for example N := (N, +, 0) and the map N×N → N defined its cokernel is trivial. Composing with N × N →(N, ·, 1) defined by the sum (x, y) 7→ x + y). by s(e1) = 1 and s(e2) = 0, i.e. sending (x, y) to 1 if y = 0 Inductive limits. In the category Mon all inductive limits and 0 if y 6= 0, we find the same map as composing with the exist, but in general it is difficult to calculate them. map r defined by r(e1) = 0 and r(e2) = 0, i.e. sending all (x, y) 6= (0, 0) to 0. Direct sums. Given a family {Mi} the direct sum of the i∈IL Q We can see however that in the category of integral monoids family is the submonoid of the product Mi ⊆ Mi i∈I i∈I the notion of epimorphism coincides with that of “map with of I-ple (s ) where s = 1 for almost all components. i i Mi trivial cokernel” while the notion of surjectivity is stronger. Cokernel of double arrows. We consider a double arrow u, v : Example: the canonical morphism P → P gp is epi, because P Q. The cokernel of u and v is the coker(u, v) := ⇒ P gp/P = 0 (remark however that the morphism is epi also in Q/R were R is the minimal equivalent relation stable by prod- n o the category Mon). uct and containing u(P ) × v(P ) := (u(p), v(p)) : p ∈ P . P . In particular, if we have a submonoid Q of a monoid The canonical morphism Q → Q/R has the attended universal P , we can define the quotient Q/P := coker(Q,→ P ) = Q/∼ property. where x ∼ y if and only if there exist c, d ∈ P such that xc = yd.

1 GeoSupRC - Monoids c 2000 by MC Presentation of monoids by generators and relations. If f : P → Q is a morphism of finitely generated saturated We note that there is a bijection monoids, K a prime of Q with ht(K) = 1 and J = f←(K) (I)  ∼= (I)  with ht(J) = 1, then the ramification index of f at K is the HomMon ,M −→ M ϕ 7−→ ϕ(ei) gp N i∈I n ≥ 0 such that vK ◦ f = nvJ . (I) where (ei)i∈I is the canonical base of N : ei(j) = δi,j. Submonoids and Faces. We say that a R of a Monoid P is a submonoid if 1 ∈ R and one of the following, equivalent, A family (xi)i∈I of elements of a monoid M generates M if (I) conditions is satisfied: the morphism N → M sending ei 7→ xi is surjective. (s ) R · R ⊆ R This means that every element x of M can be written as a 0 product x = xn1 xn2 ··· xn` with i ∈ I and n ∈ . (s) for all x ∈ R and y ∈ R one has xy ∈ R i1 i2 i` j j N (I) (I) We say that Q ⊆ P is a saturated subset if the converse of Given two sets I and J, and elements rj ∈ N ×N for every j ∈ J, we can define the monoid M with generators (x ) and condition (s) holds, i.e. if xy ∈ Q implies that x and y belong i i∈I to Q. relations (rj)j∈J as the cokernel of the double arrow: p1 (J) r (I) (I) −−−→ (I) A face of P is a submonoid and a saturated subset, i.e. is a N −−−→ N × N −−−→ N subset F of P such that 1 ∈ F and xy ∈ F if and only if x ∈ F p 2 and y ∈ F . ej 7−−−→ rj We see immediately that taking the complement gives a bijec- and we will write the “relations” rj also as “equalities” p1(rj) = tion between the set of ideals (resp. submonoids) and the set p2(rj). of saturated (resp. prime subsets) of a monoid. The Algebra of monoids over a . Let R be a commutative same is true for the set of prime ideals and the set of faces: a ring with unit. The forgetful functor from the category of R- face is the complement of a prime . algebra to Mon given by (A, +, ·) 7→ (A, ·) admits a left adjoint M 7→ R[M] where R[M] is the free R-module with bases ex for Let P be a monoid: x ∈ M and multiplication defined by exey = ex·y and R- (i) If an ideal contains 1 it is the monoid P . linearity. (ii) The invertible elements P × form a submonoid and a face. In particular R[·] commutes with inductive limits. It is the smallest face of the monoid. ∼ R[N] = R[T ] by 1 ↔ T . (iii) The subset P \ P × is the maximal prime ideal. ` ∼ R[N ] = R[T1,...,T`] by ei ↔ Ti. Localization. Let S be a subset of M; then there exists a ` r r ` commutative monoid S−1M endowed with a canonical mor- If P = N /N for a presentation N ⇒ N sending ej to P P phism M −→ S−1M such that the of any element of S nj,iei and mj,iei for j = 1, . . . , r, then i i is invertible, and universal for this property in the category of . ∼ Q nj,i Q mj,i  commutative monoids. We can construct S−1M as the quo- R[P ] = R[T1,...,T`] T − T i i i i j=1,...,r tient (S × M)/ ∼ where (s, m) ∼ (s0, m0) if there exists t ∈ S 0 0 Subsets of monoids. such that tsm = ts m. −1 ∼ −1 Ideals and Prime Ideals. We say that a subset I of a Monoid Remark that S M = F M if F is the face of M generated P is an ideal if one of the following, equivalent, conditions is by S. If J is a prime ideal of M, then we use the notation −1 satisfied: MJ = (MrJ) M. (i ) I · P ⊆ I 0 The category of M-Set. (i1) for all x ∈ I and a ∈ P one has ax ∈ I (i) for all x, y ∈ P : x ∈ I or y ∈ I implies xy ∈ I Let M be a monoid. An M-Set is a set X with an action of We say that J ⊆ P is a prime subset if 1 ∈/ J and the converse M, i.e. a morphism of monoids M → (X). The category of condition (i) holds, i.e. if xy ∈ J implies that x or y belongs M-Set admits arbitrary projective limits, and they commute to J. with the forgetful functor from M-Set to Set. So a prime ideal (i.e. an ideal and a prime subset) is a subset The category M-Set admits arbitrary inductive limits, but the K of P such that 1 ∈/ K (i.e. K 6= P ) and xy ∈ K if and only construction does not commute in general with the forgetful if x ∈ K or y ∈ K. functor. The direct sum of M-Sets is given by the disjoint (as for Sets) and we can make the quotient of an M-Set Height and valuations. Define the dimension dim(P ) of a monoid by a congruent relation (i.e. an equivalent relation stable under P as the maximal r such that there exists a chain J0 ( J1 ( the action of M). ··· ( Jr of prime ideals of P . Let J a prime ideal of P ; the −1 . The universal problem Bil (X × Y,Z) =∼ height is defined by ht(J) = dim(PJ ), where PJ = (P rJ) P , M i.e. ht(J) is the maximal lenght r of a chain of prime ideals HomM (X ⊗M Y,Z) for any Z ∈ M-Set define an M-set X ⊗M ∼ starting with J. Y = X × Y/ ∼ where ∼ is the generated by (mx, y) ∼ (x, my) for x ∈ X, y ∈ Y and m ∈ M. The Notice that a finitely generated integral monoid P is the inter- bifunctor ⊗ is commutative and associative in the obvious section in P gp of all P where J varies in the ideals of P with M J sense; M ⊗ X =∼ X for any X. ht(J) = 1. Moreover we have dim(P ) = rank (P gp/P ×) and M Z If M → N is a morphism of monoids, and X in an M-set, then dim(P ) = ht(J) + dim(P r J). N ⊗ X (resp. Hom (N,X)) has a natural structure of N- Let P be a finitely generated saturated monoid, J a prime ideal M M Set. The functor N ⊗ - (resp. Hom (N, -)) is the right (resp. of heght 1; then P and P /P × are saturated, P /P × ∼ , M M J J J J J = N left) adjoint of the “restriction of monoid” from N to M. P gp/P × =∼ . We define the at J as the J Z Tensor and Symmetric monoids. The tensor monoid over v : P gp −→ P gp/P × =∼ . J J Z the M-set X is the (non commutative) monoid TM (X) := gp `∞ i 0 1 i+1 Then we have PJ = {x ∈ P |vJ (x) ≥ 0}. i=0 TM (X) where TM (X) := M, TM (X) := X, and TM (X) :=

2 GeoSupRC - Monoids c 2000 by MC i TM (X)⊗M X. It is endowed in a natural way of a (non commu- is cartesian. tative) monoid structure, with a canonical monoid morphism A monoid M is saturated if it is integral and for any x ∈ Mgp, M → TM (X). if xn ∈ M for n ≥ 1 then x ∈ M, i.e. if the previous diagram is The symmetric monoid SM (X) over the M-set X is the quo- cartesian for any p, or for any prime p, so that M is saturated tient of TM (X) by the congruent relation generated by the if and only if it is p-saturated for all primes p. symmetric group of order i acting on T i (X) for any i. M For an integral monoid M define the saturation of M as Msat := Modules and Algebras. Let M → A be a log ring, i.e. a {x ∈ Mgp : ∃n ≥ 1 s.t. xn ∈ M}; for a generic monoid M we morphism between a monoid M and the moltiplicative monoid sat put Msat := Mint . of a ring A. Then the forgetful functor from A-Mod to M-Set admits the right adjoint sending X to A(X), and the forgetful The correspondence M 7→ Msat defines a functor which is a left functor from A-Alg to M-Mon admits the right adjoint sending adjoint of the (inclusion) forgetful functor from the subcategory N (M-monoid by M → N) to A ⊗ [L]. of saturated monoids to Mon. Z[M] Z (X) Valuatif monoids. An integral monoid M is valuatif if for We have the canonical A⊗ [M]Z[TM (X)] −→ TA(A ) Z any a ∈ Mgp we have that either a ∈ M or a−1 ∈ M.A and A ⊗ [S (X)] −→ S (A(X)). Z[M] Z M A valuatif monoid is saturated. Consider the class of all submonoid N ⊆ Mgp endowed with Properties of monoids. the order relation defined by N ≤ N0 iff N ⊆ N0 and N× = 0 × Invertible elements. An element x ∈ M is invertible if there N ∩ (N ) . Then M is valuatif if and only if it is maximal in exists y ∈ M with xy = 1 = yx. The set of invertible elements that class. of a monoid M is a submonoid (in fact a and a face, If M is an integral monoid, then there exists a valuative monoid × see below) and will be indicated by M . The correspondence V ⊆ Mgp such that M ⊆ V and M = M ∩ V ×; moreover × M → M gives a functor Mon → Ab. Msat is characterized as the intersection of all such valuative A monoid is sharp if M× is trivial, i.e. the only invertible submonoids of Mgp. element of M is the identity. For any monoid M, we put M = Remark the analogy between monoids and local rings, where in- × M/M , which is sharp. tegral monoids corresponds to local domains, saturated monoids We have the obvious relations: {1} ⊆ M× ⊆ M,(P/P ×)× = to integrally closed local domains, valuatif monoids to valua- {1},(P ×)gp = P ×. tion (local) rings. But in general the composite morphism M× ,−→ M −→ Mgp is Finiteness. A monoid M is of finite type if it admits a finite × gp not injective, as the case (Z, ·) = {1, −1} ,→(Z, ·) →(Z, ·) = family of generators, i.e. if and only if there exists a surjective n {1} shows. morphism: N −→ M. Torsion elements. An element x ∈ M is a torsion element A monoid M is of finite presentation if it admits a finite family n if there exists n ∈ N \{0} such that x = 1. Any torsion of generators, subject to a finite family of relations, i.e. if and ∼ r n element is invertible. The set of torsion elements of a monoid only if there exists an isomorphism: M = coker (N ⇒ N ). M is a submonoid and will be indicated as M ⊆ M×. The tor (Gabber) If M is of finite type, then it is of finite presenta- correspondence M → M gives a functor Mon → Ab. tor tion. So the two notions are equivalent and a monoid of finite M is says to be without torsion if Mtor = {1}. Obviously M type can be presented with a finite number of generators and × × is without torsion if and only if M is (Mtor = (M )tor). relations. We have that M/Mtor is torsion free, and the functorial cor- Remark that if P is finitely generated, also P sat is. respondence M 7→ M/Mtor gives a left adjoint of the forgetful functor from the (sub)category of torsion-free monoids to Mon, Warning. A submonoid of a monoid of finite type is not neces- sarily of finite type. For example a “line with irrational slope” using the canonical morphism M → M/Mtor. 2 define twos submonoids of N which are not of finite type. Integrality. A monoid M is integral (resp. quasi-integral) × if the M −→ Mgp is injective (resp. has a trivial A monoid M is finitely generated iff M and M are finitely kernel); i.e. if and only if for any a, b, c ∈ M we have ab = generated (as monoids). ac =⇒ b = c (resp. ab = a =⇒ b = 1). A monoid M is fine if it is of finite type and integral; fs (fine A monoid M is pre-integral if the composite of canonical and saturated) if it fine and saturated; toric if it is fs and gp ∼ d morphisms M× −→ Mgp has trivial kernel (i.e. is injective); torsion-free. If M is toric, then M = Z . × the condition is equivalent to the following: for any u ∈ M , If P is a fine and sharp monoid, then it has a unique minimal if ua = a for same a ∈ M, then u = 1. This means that the system of generators, given by the irreducible elements of P , × action of M on M is free. i.e. the non invertible elements p such that if p = q + r, then q The definition Mint := im(M → Mgp) defines a functor from or r is invertible (so the other is p). Mon to the subcategory of integral monoids which is the left adjoint of the forgetful functor. Properties of maps of monoids. The canonical morphism M → Mint, factorization of M → Mgp, × induces an isomorphism Mgp →(Mint)gp. Sharp morphisms. A morphism f : M → N is sharp if f : M× −→ N× is an isomorphism. Saturation. A monoid M is p-saturated, p being a prime, if it is integral and for any x ∈ Mgp, if xp ∈ M then x ∈ M. In Strict morphisms. A morphism f : M → N is strict if f : other words if the diagram M/M× −→ N/N× is an isomorphism. Remark that f is in- p ← × i M −→ M jective iff f (N ) ⊆ M nv, and it is surjective iff N = N× · f(M). can   can y y If f is a sharp and strict morphism, then it is surjective; more- Mgp −→ Mgp over, if N is pre-integral, then f is a bijection. p

3 GeoSupRC - Monoids c 2000 by MC Local morphisms. A morphism f : P −→ Q of monoids is p). It is p-saturated for a prime p (resp. saturated) if it is local if f←(Q×) = P × (the trivial inclusion P × ⊆ f←(Q×) integral and p-quasi-saturated (resp. quasi-saturated). is always true). For M and N p-saturated (resp. saturated) monoids and f : Exact morphisms. Let M and N be integral monoids; M −→ N an integral morphism, the following conditions are we say that a morphism f : M −→ N is exact if the following equivalent: gp ← condition holds: (f ) (N) = M, i.e. if the diagram (i) f is p-saturated (resp. saturated); f M −→ N (ii) for any M → M0 with M0 p-saturated (resp. saturated), ,−→ ,−→ 0 the push-out in the category of monoids M ⊕M N is p- saturated (resp. saturated); gp gp M −→ N  p f  f gp (iii) N ⊕ (M, p) := cocart M ← M → N is p-saturated. is cartesian. A morphism f is exact iff the following condition M holds: for any a, b ∈ M, if f(a)|f(b) then a|b (the converse is Let M and N be p-saturated monoids and f : M −→ N an true for any morphism of monoids). Note that composition of integral morphism; then f is p-saturated if and only if the fol- exact morphisms is exact, and also that for an exact morphism lowing condition holds: for any a ∈ M and b ∈ N such that we have f←(N×) = M×, i.e. it is local. Finally, if a compo- f(a)|bp there exists c ∈ M with a|cp and f(c)|b. sition g ◦ f and g are exact, then f is too. A sharp and exact Composition of p-quasi-saturated morphisms is p-quasi-saturated, morphism is injective. All the notions of p-quasi-saturated, quasi-saturated, p-saturated Let f be an exact morphism of monoids; then the canonical and saturated morphisms are stable for composition, cocarte- morphism from ker(f) to ker(fgp) is a bijection, in particular sian diagrams, localization and quotient. Moreover if a com- f is a monomorphism (i.e. injective) if and only if ker(f) is position g ◦ f is p-saturated (resp. saturated) and g is exact, trivial. then f is p-saturated (resp. saturated). Let Ex be the category of exact morphism of integral monoids, Valuative criterion. Let F be a morphism of finitely generated a (full) subcategory of F`(Mon) (morphism of the category of saturated monoids; then f is saturated iff for any prime ideal K integral monoids). Then the inclusion functor Ex ,−→F`(Mon) of height 1 sach that J = f←(K) has height 1, the ramification f f ex admits a left adjoint M → N 7−→ Mex → N defined by Mex := index of f at K is 1. In particular the only saturated morphism (fgp)←(N) and fex := fgp . of N into itself is the identity. |M ex Kummer morphisms. A morphism f : M → N of integral Integral morphisms. A morphism f : M −→ N of integral monoids is “of Kummer type” if it is injective and for any b ∈ N monoids is integral if the following equivalent conditions hold: there exists an integer n ≥ 1 such that bn ∈ f(M). 0 (i) for any other morphism M −→ M of integral monoids, the If f is Kummer, then coker(f) is a torsion group; in case f is push-out in the category of monoids is integral; exact, then it is Kummer iff coker(f) is a torsion group. 0 (ii) for any other morphism M −→ M of integral monoids, the Vertical morphisms. A morphism f : M → N is vertical if 0 0 push-out M ⊕M N = M ⊕ N/ ∼ in the category of for any b ∈ N there exists a ∈ M such that b|f(a) (i.e. f(a) ∈ monoids is described by the equivalence relation defined bN); if and only if for any b ∈ N we have bN ∩ f(M) 6= ∅. as in the case of “push-out with a group”; If f is vertical, then coker(f) is a group; in case f is exact, then (iii) for any a1, a2 ∈ M, b1, b2 ∈ N with f(a1)b1 = f(a2)b2 it is vertical iff coker(f) is a group. there exist a3, a4 ∈ M and b ∈ N such that ( If M,N are finitely generated integral monoids, then a vertical b1 = f(a3)b and p-saturated morphism is saturated. b2 = f(a4)b a1a3 = a2a4 . of monoids. The following properties are equivalent: ∗ For a monoid P we define its dual P := HomMon(P, N). Us- (i) f : M −→ N is injective and integral; ing the fact that N is integral, sharp and saturated we can see immediately that (P int)∗ =∼ P ∗,(P/P ×)∗ =∼ P ∗ and (ii) Z[f]: Z[M] −→ Z[N] is flat; (P sat)∗ =∼ P ∗. Moreover P ∗ is automatically integral, sharp (iii) for any field k, k[f]: k[M] −→ k[N] is flat. and saturated; it is also finitely generated if P is. The composition of integral morphisms is integral, and the no- gp ∼ Obviously we have Hom(P , Z) = Hom(P, Z) (universal prop- tion is stable for cocartesian diagrams, localization and quo- erty of P gp) which is a group, so that we have a canonical tient. Moreover if a composition g ◦ f is integral and g is exact, ∗ gp morphism: (P ) −→ Hom(P, Z) (which factorizes the injec- then f is integral; if f is surjective, then g is integral. ∗ tion P = Hom(P, N) ,→ Hom(P, Z)); it is an isomorphism if We have that f is integral iff f is integral. If M is valuatif, or and only if P is sharp. In particular, we have a canonical iso- ∗ gr M or N is a group, then f is integral. morphism (M ) −→ Hom(M, Z). Saturated morphisms. Let f : M −→ N be a morphism of The canonical morphism P −→(P ∗)∗ sending p to the map integral monoids. It is p-quasi-saturated (resp. quasi-saturated) p∗ 7→ p∗(p) is injective if and only if P is integral and sharp, i.e. if the following condition holds: h : N ⊕M (M, p) −→ N, and surjective if and only if P is saturated. In particular if M defined by the cocartesian diagram is saturated we have a canonical isomorphism M → M∗∗; if M p ∗∗ M −−−→ M is arbitrary, then we have the isomorphism Msat → M . We can recover the invertible elements of P as P × = ker(P →(P ∗)∗) f   −−−−−→f y y and for an integral P we can describe its saturation as P sat = gp ∗ ∗ ∗ N −→ N ⊕M (M, p) −→ N {p ∈ P : p (p) ∈ N ∀p ∈ P }. h (where the lower horizontal composition is the multiplication Suppose M be a fine and sharp monoid; if Mgp is torsion free, r by p) is an exact morphism (resp. is p-quasi-saturated for any then M is isomorphic to a submonoid of N for some r; if M

4 GeoSupRC - Monoids c 2000 by MC r is saturated then it is isomorphic to an exact submonoid of N for some r. Geometry of Monoids. Let Spec(M) be the set of prime ideals of a monoid M; then + × ∅,M = MrM ∈ Spec(M) (the minimal and maximal prime ideals of M). If J ∈ Spec(M) we have the following identifications: Spec(MJ ) = {K ∈ Spec(M)|K ( J} ⊆ Spec(M) and Spec(MrJ) = {K ∈ Spec(M)|K ) J} ⊆ Spec(M). A morphism f : M → N of monoids induces a map Spec(f) = f← : Spec(N) −→ Spec(M). If M is finitely generated, then Spec(M) is finite; moreover we have that the morphism M → M induces a canonical bijection Spec(M) −→ Spec(M) We define on Spec(M) the Zariski topology: the closed sets are the subset of the form V (I) = {K ∈ Spec(M)|I ( K} for I a subset (or an ideal) of M. Remark that V (∅) = Spec(M) + + (i.e. ∅ is the generic point of Spec(M)) and V (M ) = {M } (M+ is the closed point of Spec(M)). Bare for the open sets of the topology are the subsets of the form D(f) = {K ∈ Spec(M)|f∈ / K} arying f ∈ M. Now we define on the topological X = Spec(M) a sheaf M of monoids by setting M (D(f)) = S−1P where S = X X f f {fn|n ≥ 0}. The contravariant functor Spec gives an equivalence of cate- gories between the category of monoids and the essential im- age in the category of monoidal spaces (by definition the “affine monoidal spaces”), with inverse given by the global sections of the monoidal sheaf. The affine monoidal space Spec(M) represents the functor send- ing a monoidal space T to the set Hom(M, Γ(T, MT )); so we ∼ have that Hom(T, Spec(M)) = Hom(M, Γ(T, MT )) and in par- ticular Hom(Spec(N), Spec(M)) =∼ Hom(M,N).

Copyright c 2000 M. Cailotto, January 2000 v0.0 Dip. di Matematica Pura ed Applicata, Univ. Padova (Italy) Thanks to TEX, a trademark of the American Mathematical Society, and DEK. Permission is granted to make and distribute copies of this card provided the copyright notice and this permission notice are preserved on all copies.

5 GeoSupRC - Monoids c 2000 by MC