The ring of regular functions of an algebraic monoid

Let M be an irreducible normal algebraic monoid with unit group G. It is known that G admits a Rosenlicht decomposition, G=G_antG_aff, where G_ant is the maximal anti-affine subgroup of G, and G_aff the maximal normal connected affine subgroup of G. In this paper we show that this decomposition extends to a decomposition M=G_antM_aff, where M_aff is the affine submonoid M_aff=\bar{G_aff}. We then use this decomposition to calculate $\mathcal{O}(M)$ in terms of $\mathcal{O}(M_aff)$ and G_aff, G_ant\subset G. In particular, we determine when M is an anti-affine monoid, that is when $\mathcal{O}(M)=K$.


Introduction
The theory of affine algebraic monoids has been investigated extensively over the last thirty years. See [11,12,18] for different accounts of these developments. More recently there has been some important progress on the structure of non-affine algebraic monoids. By generalizing a classical theorem of Chevalley, the authors of [6] prove that any normal algebraic monoid is an extension of an affine algebraic monoid by an abelian variety. This allows one to analyze the structure of such monoids in terms of more basic objects: affine monoids, abelian varieties and anti-affine algebraic groups.
To state our results we first introduce some notation. Let k be an algebraically closed field. We work with algebraic varieties X over k, that is, integral, separated schemes over k. An algebraic group is assumed to be a smooth group scheme of finite type over k. If X is an algebraic variety we denote by O(X) the ring of regular functions on X. If X is an affine variety and I ⊂ O(X) is an ideal, we denote by V(I) = x ∈ X : f (x) = 0 ∀ f ∈ I ; if Y ⊂ X is a subset, we denote by I(Z) = f ∈ O(X) : f (y) = 0 ∀ y ∈ Y . If X is irreducible we denote by k(X) the field of rational functions on X. If A is any integral domain we denote by [A] its quotient field. Hence, if X is an irreducible affine variety then k(X) = O(X) .
The first named author was partially supported by a grant from NSERC. The second named author was partially supported by grants from IMU/CDE, NSERC and PDT/54-02 research project.
Let M be a connected, normal, algebraic monoid with unit group G (see Definition 2.1 below). The original motivation for this paper was to investigate the following basic question, first posed by M. Brion.
"How does one describe O(M), and when is it finitely generated?" Although we do not answer this question completely, we obtain many remarkable results about O(M).
Let M and G be as above. By the results of [6], if α G : G → A is the unique Albanese morphism of G such that α G (1 G ) = 0 A (note the additive notation for A), then there exists a unique morphism α M : M → A such that α M | G = α G . Furthermore, α is an affine morphism, and the scheme-theoretic fibers of α are normal varieties. The fiber at 1 ∈ A is M aff , the unique irreducible, affine submonoid of M with unit group G aff , the kernel of α. See Theorem 2.2 below.
The purpose of this paper is three-fold. Finally, we determine the conditions under which O(M) = k (that is when M is an anti-affine algebraic monoid, Theorem 3.19). In order to establish our results we define the notion of a stable algebraic monoid (see Definition 3.9).
To obtain our main results we make use of the generalized Chevalley decomposition presented in [6], which states that, if M is an irreducible monoid then where G aff is the smallest affine algebraic group such that G/G aff is an abelian variety (see Theorem 2.2 below). This structural result allows us to present a Rosenlicht decomposition M = G ant * G aff ∩Gant M aff that generalizes the corresponding decomposition G = (G ant × G aff )/(G aff ∩ G ant ) of G, where G ant is the largest anti-affine subgroup of G. See [5] and Proposition 2.12 below. We then use this decomposition (of M) in Theorem 3.1 to calculate O(M) in terms of O(M aff ) and H = G aff ∩ G ant .
Next we identify a set of central idempotents e of M such that O(M) = O(eM). But such an idempotent can be chosen so that eM is a stable monoid. Consequently, we reduce ourselves to the study of stable monoids, thereby obtaining a characterization of the algebraic monoids M such that O(M) = k, the anti-affine algebraic monoids. See Theorem 3.19.
Let M be an anti-affine algebraic monoid and let e ∈ E(M) be the minimum idempotent of M. In Theorem 3.21 we show that the retraction ℓ e : M → eM, ℓ e (m) = em, is Serre's universal morphism from M to a commutative algebraic group (see [17,Thm. 8]).
We conclude the paper with Theorem 3.22. Here we show that there is an analogue of the Rosenlicht decomposition for a large class of normal, algebraic monoids. In particular the fibre ϕ −1 (1), of the canonical map ϕ : M → Spec(O(M)), is an anti-affine monoid which we identify explicitly in terms of the internal structure of M.
Acknowledgements: This paper was written during a stay of the second author at the University of Western Ontario. He would like to thank them for the kind hospitality he received during his stay.

Preliminaries
In this section we assemble some of what is known about algebraic monoids with nonlinear unit groups. These results are due to M. Brion and the second named author [3,6,4,15].
We denote the set of idempotent elements by E(M) = {e ∈ M : e 2 = e}.
It has been proved that G(M) is an algebraic group, open in M (see for example [14]). The structure of M is significantly influenced by the structure of G(M). Recall that if G is algebraic group, then the Albanese morphism p : G → A(G) fits into an exact sequence where G aff is a normal connected affine algebraic group (since the group A(G) is commutative, its law will be denoted additively). Moreover, G aff is the smallest affine algebraic subgroup such that G/G aff is an abelian variety. This structure theorem is originally due to Chevalley, but now there is a modern proof in [7]. Recently it has been generalized from groups to monoids. The following theorem is a summary of this development.
Theorem 2.2 (Brion, Rittatore [3,6,4,15]). Let M be a normal irreducible algebraic monoid with unit group G. Then M admits a Chevalley decomposition: are the Albanese morphisms of M and G respectively, and M aff is an affine algebraic monoid.
Definition 2.3. Let G be an algebraic group, and let a closed subgroup H ⊂ G act on an algebraic variety X.
Under mild conditions on X (e.g. X normal and covered by quasiprojective H-stable open subsets), this quotient exists. Clearly, G * H X is a G-variety, for the action induced by a · (g, x) = (ag, x). We will denote the class of (g, x) in G×X by [g, x] ∈ G * H X. The fundamental properties of G * H X were established by Bialynicki-Birula in [1]. See also [19].
Remark 2.4. Let G be an algebraic group, and H ⊂ G a closed subgroup acting over an algebraic variety X. Then π : G * H X → G/H, which is induced by (g, x) → gH, is a fiber bundle over G/H, with fiber isomorphic to X. Theorem 2.5 (Brion, Rittatore, [3,6]). Let M be a normal algebraic monoid and let Z 0 be the connected center of G. Then A(G) ∼ = Z 0 /(Z 0 ∩ G aff ) and Definition 2.6. If M is an algebraic monoid with unit group, we define the center of M be the set of central elements.
It is clear that Z(M) is a closed submonoid of M, with unit group G Z(M) = Z(G), the center of G. However, one should be aware that this monoid is not necessarily connected. Moreover, the following example shows that the Z(G) is not necessarily dense in Z(M).
Example 2.7. Let r, s ∈ N, r = s, and consider the affine algebraic monoid

In particular the zero matrix is a central idempotent which does not belong to Z(G).
In what follows we collect some results about the Rosenlicht decomposition of an algebraic group G. This decomposition depicts G as the product of two subgroups, one affine and the other anti-affine. We refer the reader to [16], [8, Sec. III.3.8] and [5] for proofs and further results about this decomposition.
, and G aff ∩G ant contains (G ant ) aff as an algebraic group of finite index.
In loose terms, an anti-affine algebraic group G is a non-split extension of an abelian variety by an affine, commutative algebraic group. See [5,Thm. 2.7]. But in the case char(k) = p > 0 the situation is considerably less complicated. Indeed we have the following simplifying result. See [5,Prop. 2.2]. Proposition 2.11 (Brion [5]). Let G be an anti-affine, connected algebraic group over an algebraically closed field k. If char(k) = p > 0 then G is a semi-abelian variety. i.e. G is the extension of an abelian variety by an affine torus group.
The following result generalizes Rosenlicht's decomposition to the case of algebraic monoids. It is essential for determining the ring of regular functions on an algebraic monoid.
If a, b ∈ G ant and m, n ∈ M aff are such that am = bn, then b −1 am = n. It follows from Theorem 2.5 that b −1 a ∈ G aff ∩ G ant , and thus [a, m] = [b, n]; that is, ϕ is injective. Since ϕ| G : G = G * G aff G aff → G is an isomorphism and that M is normal, it follows from Zariski's main Theorem that ϕ is an isomorphism. Since 0 is central, it follows that 0M = 0G ant is an algebraic group. Consider the multiplication morphism ℓ : G ant → 0G ant , ℓ(g) = 0g. Then It follows that g ∈ ℓ −1 (0) if and only if [g, 0] = [1, 0] ∈ G ant * G aff ∩Gant M aff ; that is, if and only if g ∈ G aff ∩ G ant . Since ℓ is a separable morphism, it follows that 0G ant ∼ = (G aff ∩ G ant ).

The algebra of regular functions of M
The following Theorem is the key to understanding the ring of regular functions on an algebraic monoid.
where for the last equality we used that O(G ant ) = k.
where eM aff = eG aff = eG aff , and eM = eG. (2) Since eM = {x ∈ M : xe = e}, it is clear that eM is a closed subset. Hence, eM is an algebraic monoid and ℓ e : M → eM, ℓ e (m) = em, is a morphism of algebraic monoids.
(3) Since ℓ e : M → eM is a surjective morphism of algebraic monoids, it follows that G(eM) = eG. In particular, G → eG is a surjective morphism of algebraic groups, and hence, by [5, Lemma 1.5], eG ant ⊂ (eG) ant . It is clear that eG aff ⊂ (eG) aff and (eG aff )(eG ant ) = eG. Hence, eG/eG ant ∼ = eG aff /(eG aff ∩ eG ant ) is an affine algebraic group, since eG aff ∩ eG ant is a central subgroup of eG aff . It follows that (eG) ant ⊂ eG ant . On the other hand, it is clear that eG aff is a normal subgroup of eG, and the morphism G → eG/eG aff , g → eg(eG aff ), induces a surjective morphism G/G aff → eG/eG aff . It follows that eG/eG aff is an abelian variety and hence eG aff = (eG) aff .
Finally, since eG ∼ = G/G e , and G e ⊂ G aff , it follows that A(eG) = eG/(eG aff ) ∼ = G/G aff = A(G), with the Albanese morphism fitting into the following commutative diagram where ϕ : eG/(eG) aff ∼ = (G/G e )/(G/G e ) aff → G/G aff is the canonical isomorphism obtained by observing that (G/G e ) aff = G aff /G e . (4) follows from the description above and Proposition 2.12.  Assume now that e ∈ E G ant ∩ G aff and let f ∈ O(eM aff ) Gant∩G aff . If x ∈ M, then f (x) = f (g·x) for all g ∈ G ant ∩G aff . It follows that f (x) = f (ex), since e ∈ G ant ∩ G aff . In particular, if f ∈ I(eM aff ) Gant∩G aff , then f (x) = f (ex) = 0 for all x ∈ M.
Remark 3.6. The reader should notice that  Let M be an algebraic monoid such that G(M) is an anti-affine algebraic group. Then M is anti-affine. The converse is not true, as the following example shows. Example 3.8. Let T = k * × k * be an algebraic torus of dimension 2, and consider the affine toric variety T ⊂ A 2 . Then A 2 is an affine algebraic monoid with unit group T . Let A be a non-trivial connected abelian variety and consider an extension The quotient M = (A 2 × H)/T 1 is an algebraic monoid with unit group G and with M aff ∼ = A 2 . Thus Hence, M is an anti-affine algebraic monoid while G(M) is not an anti-affine algebraic group. Definition 3.9. Let G be an algebraic group and X be a G-variety. We say that the action is generically stable (equivalently that X is a generically stable G-variety) if there exists an open subset consisting of closed orbits. We say that an algebraic monoid M is stable if it is generically stable as a (G aff ∩ G ant )-variety. Definition 3.11. Let G be an affine algebraic group acting on an affine variety X. We say that the action is observable if for every non-zero G-stable ideal I ⊂ O(X), I G = (0). Here we consider the induced The concept of observable action is a generalization of the notion of observable subgroup. Observable subgroups were introduced by Bialynicki-Birula, Hochschild and Mostow in [2] and have been researched extensively since then, notably by F. Grosshans (see [10] for a survey on this topic). Given an affine algebraic group G, a closed subgroup H ⊂ G is said to be observable if G/H is a quasi-affine algebraic variety. The equivalent definition of H being observable if every nonzero H-stable ideal I ⊂ O(G) has the property that I G = (0), was first recorded in [9], and then further generalized in [13]. We present here some of the basic results that we need in what follows. We include some of the proofs here for convenience. Proof. We will only prove that if the action is observable then conditions (1) and (2) hold. We refer the reader to [13,Thm. 3.10] for a complete proof. In [13] the reader can find examples showing that both conditions (1) and (2) of Theorem 3.12 are necessary. However, for some families of algebraic groups, generic stability of the action implies observability. This is the case when G is a reductive group (see [13,Thm. 4.7]), or when G = U × L, where L is reductive and U is unipotent. Since L is reductive and that Z is L-stable, it follows that there exists f ∈ O(X) L such that f ∈ I, f | Z = 1. Hence, I L = (0). Since U normalizes L, it follows that I ∩ O(X) L = {0} is an U-submodule, and hence I G = (I L ) U = (0).
Since e ∈ eE G aff ∩ G ant , it follows that eM is stable if and only if ef = e for all f ∈ E G aff ∩ G ant .
Let now G aff ∩G ant = T U be a Levi decomposition. Since G aff ∩G ant is commutative, it follows that E G aff ∩ G ant = E T . Indeed, if follows from [12,Prop. 3.13] that E G aff ∩ G ant = g∈G aff ∩Gant gE(T )g −1 = E(T ). Let f 0 ∈ T the unique minimal idempotent; i.e. f 0 is unique idempotent in the Kernel of the affine toric variety T -the existence of f 0 follows from the fact that there exists a finite number of idempotents elements on T . It is clear that ef 0 = e if and only if ef = e for all f ∈ E G aff ∩ G ant . We now characterize normal, anti-affine, algebraic monoids. Finally, since eM is a group, it follows that E(eM) = e, and hence ef = e for f ∈ E(M).
The following examples indicate why, in Theorem 3.19, one must focus on the idempotents of E G aff ∩ G ant , rather than just any central idempotent.
Proof. Let σ : M → S be Serre's universal morphism from the pointed variety (M, e) into a commutative algebraic group. Since M is antiaffine, it follows that S is necessarily an anti-affine algebraic group (see for example [5, §2.4]). Moreover, it follows from Theorem 3.19 that eM is an anti-affine algebraic group. Thus we have a commutative diagram Since ϕ(0 S ) = e, it follows that ϕ is a morphism of algebraic groups (see for example [5,Lem.1.5]). Consider the associated short exact sequence   is an exact sequence of algebraic monoids, with ϕ a dominant morphism.
Proof. Since O(M) is finitely generated, it follows that N = Spec O(M) is an algebraic monoid. Moreover, the canonical morphism ϕ : M → N is a morphism of algebraic monoids. If we let S = ϕ −1 (1), then we have the following commutative diagram. The top row is exact and the bottom row is left exact.