PURITY OF MONOIDS AND CHARACTERISTIC-FREE SPLITTINGS IN SEMIGROUP RINGS

A BSTRACT . Inspired by methods in prime characteristic in commutative algebra, we introduce and study combinatorial invariants of seminormal monoids. We relate such numbers with the singularities and homological invariants of the semigroup ring associated to the monoid. Our results are characteristic independent.

Frobenius splittings have inspired a large number of results in commutative algebra, algebraic geometry, and representation theory.In this manuscript we seek to continue this approach in the context of combinatorics of monoids.Given a monoid M ⊆ Z q ⩾0 for some q ∈ Z >0 , and m ∈ Z >0 , we study the pure M-submodules of 1  m M that are translations of M, which algebraically corresponds to free summands of k[ 1  m M] as k[M]-module.It turns out that the purity of M ⊆ 1 m M detects both normality and seminormality (see Proposition 3.5).The study of pure submodules, or equivalently of free summands, of normal monoids was already initiated by other authors in order to compute the F-signature of normal affine semigroup rings [21,26].Moreover, the structure of 1  m M as M-module was described by Bruns and Gubeladze [4,5] for normal monoids (see [20] for a related result in prime characteristic).
In this manuscript we study combinatorial numerical invariants of a seminormal monoid.Our key motivation is that seminormality for a monoid can be seen as a characteristic-free version of F-purity for affine semigroup rings.For more information and examples on seminormal monoids we direct the interested reader to Li's thesis on this subject [17].
In Definition 3.19 we introduce the notion of pure threshold of a seminormal monoid M, denoted by mpt(M), which is motivated by the F-pure threshold in prime characteristic.This number can be described as the largest degree of a pure translation of M inside the cone R ⩾0 M or, equivalently, of 1  m M for some m.We show that mpt(M) gives an upper bound for the Castelnuovo-Mumford regularity reg(k[M]) defined in terms of local cohomology, and the Castelnuovo-Mumford regularity Reg(k[M]) defined in terms of graded Betti numbers of k[M] (see Section 2 for more details).
Theorem A (Theorem 5.4).Let M be a seminormal monoid with a minimal set of generators {γ 1 , . . ., γ u }.Then, a i (k[M]) ⩽ −mpt(M).As a consequence, Moreover, if we present R as S/I, where S = k[x 1 , . . ., x u ] and each x i has degree d i := deg(x i ) = |γ i | the degree of γ i for i = 1, . . ., u, and I ⊆ S is a homogeneous ideal, then Theorem A allows us to give an upper bound for the degrees of generators of the defining ideal I.We also show that mpt(M) is a rational if M is a normal (see Proposition 4.4).Despite mpt(M) being inspired by F-pure thresholds, these numbers do not always coincide (see Example 3.23 and Remark 3.24).In addition, mpt(M) is defined independently of the field k and so it is a characteristic-free invariant, while the F-pure threshold is only defined when k has prime characteristic.
We introduce the pure prime ideal P(M), and the pure prime face F M , of a seminormal monoid M (see Corollary 3.28, and Definitions 3.26 and 3.29).The former emulates the splitting prime ideal of an F-pure ring, while the latter is related to the quotient of a ring by its splitting prime.In fact, the submonoid F M ∩ M is normal (see Corollary 3.28).We note that the rank of F M ∩ M is a monoid version of the splitting dimension and so we call it the pure dimension and denote it by mpdim(M).It turns out that this rank is equal to the rank of M if and only if M is normal, and it is non-negative if and only if M is seminormal (see Corollary 3.30).Therefore, in some sense, mpdim(M) measures how far a seminormal monoid is from being normal.Furthermore, mpdim(M) is related to the depth of k[M] as the following theorem shows.
We point out that Theorem B recovers Hochster's result that normal semigroup rings are Cohen-Macaulay [14].
Finally, we consider the growth of the number of disjoint pure translations of M in 1 m M as m varies.More specifically, if m ∈ Z >0 is such that 1 m M ∩ ZM = M, we define exists and it is positive for every increasing sequence m t ∈ A (M). Furthermore, if M is normal, then mpr(M) ∈ Q >0 .
We call the limit in Theorem C the pure ratio of M. If the field has prime characteristic, this number coincides with the splitting ratio of k[M] [1].A consequence of Theorem C is that the value of the F-splitting ratio depends only on the structure of M, and so it is independent of the characteristic of the field as long as k[M] is F-pure.Finally, using this result we give a monoid version of a celebrated Theorem of Kunz [16,Theorem 2.1] which characterizes regularity of rings of prime characteristic in terms of Frobenius (see Theorem 5.9).
Throughout this article we adopt the following notation.
Notation 1.1.Let k be a field of any characteristic and q a positive integer.Let M ⊆ Z q ⩾0 be an affine monoid, i.e., a finitely generated submonoid of Z q .We fix {γ 1 , . . ., γ u } a minimal set of generators of M. Let ZM denote the group generated by M and C(M) = R ⩾0 M the cone generated by M.

BACKGROUND
In this section we include some preliminary information that is needed in the rest of the paper.
Affine monoids and affine semigroup rings.For proofs of the claims in this subsection and further information about affine monoids we refer the reader to Bruns and Gubeladze's book [6].
x q ] be the affine semigroup associated to M. As a k-vector space, R is generated by the monomials {x α | α ∈ M}.We note that the monomial ideals of R are precisely those generated by {x α | α ∈ U} for some ideal U ⊆ M.Under this correspondence, prime monomial ideals of R correspond to prime ideals of M. For every M-module U ⊆ Q q we have a corresponding R-module RU := {x α+η | α ∈ M, η ∈ U} in the algebraic closure of k(x 1 , . . ., x q ).Moreover, we have dim(R) = rank(M).
Graded algebras and modules.A non-negatively graded algebra A is a ring that admits a direct sum decomposition A = j⩾0 A j of Abelian groups such that A i • A j ⊆ A i+ j .It follows from this that A 0 is a ring, and each A i is an A 0 -module.If we let A + = j>0 A j , then A + is an ideal of A, called the irrelevant ideal.
Throughout this manuscript we will make the assumption that A is Noetherian or, equivalently, that there exist finitely many elements a 1 , . . ., a n ∈ A + such that A = A 0 [a 1 , . . ., a n ], which can be assumed to be homogeneous of degrees d 1 , . . ., d n .In this case, note that A is a quotient of a polynomial ring A 0 [x 1 , . . ., x n ] by a homogeneous ideal.
A Z-graded A-module is an A-module N that admits a direct sum decomposition N = j∈Z N j of Abelian groups, and such that A i • N j ⊆ N i+ j .As a consequence, each N i is an A 0 -module.Moreover, if N is Noetherian there exists i 0 ∈ Z such that N i = 0 for all i < i 0 ; on the other hand, if N is Artinian there exists j 0 ∈ Z such that N j = 0 for all j > j 0 .
Given a Z-graded A-module N, and an integer j ∈ Z, we define the shift N( j) as the Z-graded A-module whose i-th graded component is N( j) i = N i+ j .In particular, A(− j) is a free graded A-module of rank one with generator in degree j.
Graded local cohomology and Castelnuovo-Mumford regularity.In this subsection we recall general properties of local cohomology.We refer the interested reader to Brodmann and Sharp's book on this subject [3].Let k be a field and S = k[x 1 , . . ., x n ], with deg(x i ) = d i > 0. Let N be a finitely generated Z-graded S-module.If we let m = (x 1 , . . ., x n ), then the graded local cohomology modules H i m (N) are Artinian and Z-graded.
In the standard graded case, reg(N) has a well-known interpretation in terms of graded Betti numbers of N. In our setup this is still the case, but the degrees of the algebra generators of S must be taken into account.For i ∈ Z ⩾0 , let β S i (N) = a 0 (Tor S i (N, k)) ∈ Z ∪ {−∞}.As another way to see this, for a non-zero graded S-module T let β (T ) be the maximum degree of an element in a minimal homogeneous generating set of T .If is a minimal graded free resolution of N where c := pd(N) is the projective dimension of N, then In the standard graded case, that is, when d 1 = . . .= d n = 1, then reg(N) = Reg(N).In our more general scenario, we still have the following relation between the two notions of regularity.Proof.We may assume that n > 0, otherwise the claim is trivial.We prove the statement by induction on c = pd S (N).If c = 0 then N is free, and it is clear that Reg(N) = β (N).On the other hand, a i (N) = 0 for all i ̸ = n, while a n (N) = a n (S) + β , and the base case follows.Now assume that c ⩾ 1.We have a graded short exact sequence 0 → Ω → F 0 → N → 0, where F 0 is the first free module in a minimal free resolution of N, and pd S (Ω) = c − 1.By induction we have that Reg(Ω) = reg(Ω) , and therefore the above equality gives that We now show that max{reg(N), reg(Ω) − 1} = reg(N).The short exact sequence 0 If we had reg(Ω) − 1 > reg(N), then necessarily reg(Ω) = a n (Ω) + n, and looking at top degrees in the above exact sequence we also conclude that a n (Ω) = a n (F 0 ).On the other hand, Proposition 3.1], and so, reg(Ω) ⩽ reg(N), a contradiction.Thus we always have that reg(Ω) − 1 ⩽ reg(N), and by (2.1) the inequality Reg(N) ⩽ reg(N) + ∑ n i=1 (d i − 1) is proved.For the reverse inequality, first observe that the above isomorphisms give that a i (N) = a i+1 (Ω) for all i < n − 1, while the exact sequence yields that a n−1 (N) ⩽ a n (Ω).Since n > 0 and Ω is a submodule of a free module, it has positive depth, and thus a 0 (Ω) = 0.It follows that max{a since the inequality Reg(Ω) − 1 ⩽ Reg(N) always holds.Now, the above exact sequence on local cohomology also gives that a , and the proof is complete.□

PURITY OF M-MODULES AND (SEMI)NORMAL
2) be the monoid generated by {(2, 1), (1, 2)}.We have that the inclusion In Figure 1 we represent the elements of M with circles and the ones from ( 3 2 , 3 2 ) + M with multiplication signs.The shaded region is included to illustrate that ( 3 2 , 3 2 ) + M is obtained as a translation of M.
In the following proposition, we provide equivalent statements for Definition 3.1 in a particular case.
Proposition 3.3.Let V ⊆ Q q be an M-module and α ∈ V .The following statements are equivalent: First, assume (i) and let γ ∈ M and β ∈ V with γ + β ∈ α + M. Thus, β ∈ α + M and then (ii) follows.Now, assume (ii) and let β ∈ V be such that Finally, assume (iii).Let β ∈ V \ (α + M) and γ ∈ M. Assume by means of contradiction that □ We now discuss seminormality and normality, which are the main subjects of study in this manuscript.We refer to the work of Bruns, Li and Römer [8] to reader in seminormal rings.
The following alternative characterization of seminormality and normality is useful for the proof of our main results.While it might be already known to experts, we record it here with a proof for convenience of the reader.For a related result in prime characteristic we refer to the work of Bruns, Li, and Römer [8, Section 6].Proposition 3.5.Let M be an affine monoid.
(1) M is seminormal if and only if there exists m ∈ Z >1 such that 1 m M ∩ ZM = M.
(2) M is normal if and only if 1 m M ∩ ZM = M for every m ∈ Z >1 .
Proof.We note that the containment 1 m M ∩ ZM ⊇ M holds trivially for any m ∈ Z >1 .For (1), assume that 1 m M ∩ ZM ⊆ M for some m ∈ Z >1 .Let α ∈ ZM be such that 2α ∈ M and 3α ∈ M. We can find nonnegative integers a and b such that m = 2a + 3b, and thus mα = a(2α) + b(3α) ∈ M. It follows by our assumption that α ∈ M, and thus M is seminormal.Conversely, let F be the set of all faces of C(M) (of any dimension); we note that F is a finite set.For any F ∈ F we consider the finitely generated Abelian group G F = (ZM ∩ QF)/(Z(M ∩ F)).Let p ≫ 0 be a prime number such that the ideal (p) is not associated to G F as a Z-module for any Motivated by the previous result, we consider the following definition.
Definition 3.6.We set Remark 3.7.As a consequence of Proposition 3.5, we deduce that We now see that A (M) is a multiplicative set.

Proof. First assume that mn
□ Now, we consider a set that records the pure translations of M in 1 m M. In Section 5 we see that this set corresponds to the free summands of k 1 m M as a k[M]-module.
Remark 3.11.We note that V m (M) ̸ = / 0 if and only if m ∈ A (M).As a consequence we have In the following remarks we observe that V m (M) is compatible with projections onto faces of C(M) and with isomorphisms of monoids.Remark 3.13.For every face Remark 3.14.Since every isomorphism of monoids ϕ : M → M ′ extends to an isomorphism of groups ϕ : We now describe basic properties V m (M).In particular, we see that this set is finite.
m M be such that w − β ∈ ZM.Thus, from The following lemma provides useful facts about the sets V m (M).We recall that the Minkowski sum of two subsets A, B ⊆ R q is defined as Proof.We begin with the containment ⊇ in (1).
Proof.By assumption we have that α m M, it follows by Proposition 3.15 that γ i ∈ V m (M).However, this contradicts Remark 3.12, and therefore α / ∈ V m (M).For the second claim, recall that α∈V m (M) (α + M) is a disjoint union of M-modules (see proof of Lemma 3.17 We now define a new numerical invariant for seminormal monoid.This number plays an important role in our main results.This invariant is inspired by the F-pure threshold of a ring [24].This is because the F-pure threshold of a standard graded algebra can be described as the supremum among the degrees of a minimal generator of a free summand of R 1/p e [11].However, the F-pure threshold of R = k[M] can be different than the pure threshold of M (see Example 3.23 and Remark 3.24).In Proposition 4.4 we prove that for normal monoids this invariant is rational.We now discuss how the pure threshold of a monoid M can be obtained from any increasing sequence in A (M). Proposition 3.21.Let {m t } t∈Z ⩾1 be the elements of A (M) ordered increasingly.Then, In particular, lim for any m ∈ A (M).
Proof.If mpt(M) = 0, the result follows.We assume mpt(M) > 0. Let b = max{|γ 1 |, . . ., |γ u |} and for any n For each 1 ⩽ i ⩽ u let 0 ⩽ r i < m ′ be such that c i ≡ r i (mod m ′ ).By Proposition 3.15 and Lemma 3.17 (1) we have Since ε was chosen arbitrarily, the result follows.□ We now compute some examples of pure thresholds.We note that this invariant depends on the grading given by the embedding M ⊆ Z q .Example 3.22.Let M be generated by d 1 e 1 , . . ., d q e q ∈ Z q , where d i ∈ Z >0 and {e 1 , . . ., e q } is the canonical basis in Z q .Then, is the Veronese subring of order t of a polynomial ring k[x 1 , . . ., x q ] with the grading deg(x i ) = 1.We have that and therefore mpt(M) = q.We point out that, if k has prime characteristic, then fpt(k[M]) = q t [13, Example 6.1].
Remark 3.24.It follows from Example 3.23 that mpt(M) may differ from fpt(k[M]) even when M is normal.This is not surprising since fpt(k[M]) is independent of the presentation of k[M] as a quotient of a polynomial ring, while we have already observed that mpt(M) heavily depends on the degrees of the generators and on the embedding of M.
The following construction allows us to provide bounds for depths of affine semigroup rings (see Section 5).In Proposition 3.26 we justify the terminology used in the definition.Definition 3.25.We define the pure prime of M by mV m (M).Proposition 3.26.Let M be an affine monoid.Then P(M) is a prime ideal of M.
Proof.Since P(M) = m∈Z m>1 (M \ mV m (M)), it follows from Corollary 3.16 that P(M) is an ideal of M. Now, let a, b ∈ M \ P(M) and m, n ∈ A (M) be such that α := a m ∈ V m (M) and β := b m ∈ V n (M).We claim that a+b mn ∈ V mn (M) which implies a + b ̸ ∈ P(M), finishing the proof.Indeed, suppose a+b mn ̸ ∈ V mn (M), then from Proposition 3.15 it follows that which contradicts Lemma 3.17 (2).□ We obtain the following theorem that relates P(M) with the normality of M.
Theorem 3.27.Let M be an affine monoid.Then M is normal if and only P(M) = / 0.
Proof.We begin with the forward direction.Since M is normal, by Remark 3.14 we can assume that M is a submonoid of Z n ⩾0 for some n ∈ Z >0 and such that M = ZM ∩ Z n ⩾0 [6, Theorem 2.29].Fix a ∈ M ⊆ Z n ⩾0 and chose m ∈ Z >1 bigger than every entry in a.By Remark 3.7 we have m ∈ A (M) and m ∈ A (Z n ⩾0 ).By the choice of m, it is clear that Thus, the left hand side expression in By assumption there exists m ∈ A (M) such that nb ∈ mV m (M).Therefore, On the other hand, f ∈ ZM, then f ∈ 1 m M − nb m ∩ ZM = M, which finishes the proof.□ Corollary 3.28.There exists a face Proof.The first part follows from Proposition 3.26 and the correspondence between prime ideals of monoids and faces of their cones [6,Proposition 2.36].By Theorem 3.27 to show that . Moreover, we may assume M is seminormal.We note that where the last inclusion follows from Remark 3.13.Since we always have the other inclusion The previous proposition allows us to define the following invariant of affine monoids, the pure dimension.As we see in Corollary 3.30, this new notion measures how far a monoid is from being normal.In Theorem 5.7 we use this invariant to provide lower bounds for the depth of affine semigoup rings.We finish this section with the following example.

ASYMPTOTIC GROWTH OF NUMBER OF PURE TRANSLATIONS
In the short section, we study the asymptotic behavior of the number of elements in the sets V m (M).Throughout we adopt the same notation from Section 3. Definition 4.1.Let M be a seminormal affine monoid, and let F M be its pure prime face.For every m ∈ A (M) we define Moreover, we set When M is normal, there is a simple description of B(M) as the region in Notation 4.2.We prove that these regions coincide in Lemma 4.3, which also includes important properties of B(M).( Proof.We begin with (1).Let {g 1 , . . ., g l } be a minimal set of generators of M ∩ F M and consider the region , then Proposition 3.15 implies g i ∈ V m (M) which contradicts Remark 3.12.We conclude V m (M), and then B m (M), is contained in F M \ Γ which is bounded.Now, let ∂ and • denote boundary and interior on RF M , respectively.Let µ denote the dim(RF M )-dimensional Lebesgue measure on RF M .We note that for any Therefore, for any r > 0 and any , where B(r, x) denotes the ball in RF M with radius r and center x.Therefore, there exists a real c < 1 such that for any such r and x we have µ(∂ B(M)∩B(r,x)) µ(B(r,x)) < c.By Lebesgue's density theorem [19, Corollary 2.14], we conclude µ(∂ B(M)) = 0.
We continue with (2).Let {m t } t∈Z ⩾1 be the elements of A (M) ordered increasingly.For each t ∈ Z ⩾1 set p t = m 1 • • • m t and notice p t ∈ A (M) by Lemma 3.8.The conclusion now follows from Lemma 3.12 (1).Now we prove (3).Let m t ∈ A (M) and α ∈ 1 m t M ∩ B(M), it suffices to show α ∈ V m t (M).By (2), we have α ∈ B p i (M) for some i.We may assume i ⩾ t and then m t divides p i .Therefore, there by Proposition 3.15 and Lemma 3.17, which finishes the proof.
We finish with (4).If M is normal we have Lemma 3.11].Thus, the equality B(M) = ∆ follows as the set ∪ m∈Z ⩾1 Z m M is dense in R q .□ From Lemma 4.3 (4) we obtain that the pure threshold of normal monoids is rational.
Proof.The statement follows readily from Lemma 4.3 (4) and the equality mpt We now turn our focus to asymptotic growth of the number of elements in the sets V m (M).We define the following limit, which we prove exists in Theorem 4.6 Definition 4.5.Let M be a seminormal affine monoid.Set s = mpdim(M) and let {m t } t∈Z ⩾1 be the elements of A (M) ordered increasingly.We define the pure ratio of M as We define the pure signature of M as In the following theorem we show that mpr(M) exists as a limit, and that it equals the relative volume of B(M).Here, by relative volume with respect to a lattice L ⊆ H of rank r in an rdimensional hyperplane H ⊆ R q , denoted by vol L , we mean the r-dimensional volume in H normalized such that any fundamental domain of L has volume one. .Therefore, by taking the limit t → ∞ we obtain that the limit exists and is equal to vol Z(M∩F M ) (B(M)).We note that vol Z(M∩F M ) (B(M)) is positive since V m (M) has interior points of F M (see Corollary 3.28).The last statements follow from Corollary 3.30 and Lemma 4.3 (4).□ Theorem 4.6 is related to previous computations done for the F-signature of normal semigroup rings [21,26].We end this section with a question motivated by Proposition 4.4.This question is open, to the best of our knowledge, for seminormal monoids that are not normal.
Question 4.8.Let M be a seminormal affine monoid.Is mpr(M) a rational number?

APPLICATIONS TO AFFINE SEMIGROUP RINGS
Throughout this section we adopt the following notation.
and zero otherwise.For an ideal I ⊆ M, we denote by x I the corresponding M-homogeneous R-ideal, For an M-homogeneous element f = x α ∈ R, we denote by log( f ) = α ∈ M the corresponding element in M. Remark 5.2.We note that R ∼ = R 1/m via the k-algebra map given by x α → x α/m .Proposition 5.3.Let M be an affine monoid, and let α ∈ 1 m M.Then, φ m α is a map of R-modules if and only if α ∈ V m (M).
Proof.We note that φ m α is a map of R-modules if and only if for every γ ∈ M and β ∈ 1 m M we have φ m α (x γ x β ) = x γ φ m α (x β ).By the definition of φ m α these are equivalent to φ m α (x γ x β ) ̸ = 0 implies x γ φ m α (x β ) ̸ = 0, or equivalently to, The conclusion now follows from Proposition 3.3.□ In the next result, we use the semigroup splitting threshold to provide a bound for the Castelnuovo-Mumford regularity of affine semigoup rings.We refer the reader to Section 2 for information about a-invariants and regularity.Moreover, if we present R as S/I, where S = k[x 1 , . . ., x u ] and each x i has degree d i := deg(x i ) = |γ i | the degree of γ i for i = 1, . . ., u, and I ⊆ S is a homogeneous ideal, then Proof.We can assume that M is seminormal.Let m ∈ A (M) be such that m > 1, which exists by Proposition 3.5 (1).Fix t ∈ Z ⩾0 and α ∈ V m t (M).From Proposition 5.3 it follows that φ m t α gives a splitting of the homogeneous injective map R(−|α|) → R 1/m t defined as multiplication by x α .Thus, for each i, the induced map . By taking the maximum value of |α| over all α ∈ V m t (M) and letting t → ∞, by Proposition 3.21 we obtain that a i (R) ⩽ −mpt(M) as desired.The inequality for regularity follows by definition, and the last equality by the relation between the rank of semigroups and dimension of semigroup rings (see e.g.[7, p.257]).
Finally, the inequalities involving β (I) follow at once from the fact that β

and the previous inequalities. □
We now compute the pure threshold for a normal monoid that is Gorenstein.This follows previous work done for the F-pure threshold [11, Theorem B], which was motivated by a conjecture posted by Hirose, Watanabe and Yoshida [13].
) for all p ≫ 0 (see for instance by [9,Lemma 4.3] adapted to the positively graded case).If L is any field extension of k, and m is the homogeneous maximal ideal of R, then we have graded isomorphisms ).Thus, we may assume that k is a perfect field of characteristic p > 0. We can write R = S/I, where S = k[T 1 , . . ., T u ], each T i maps to a generator x γ i of R and deg(T Let n = (T 1 , . . ., T u ).As f e / ∈ n [p e ] by Fedder's criterion [12], there is a monomial T n 1 1 • • • T n u u in its support with 0 ⩽ n i ⩽ p e − 1 for all i.This implies that the map S/I → (S/I) Proof.We can assume that M is seminormal.Let S = k[y 1 , . . ., y u ] endowed with the M-grading given by deg(y i ) = γ i .We set a surjection of k-algebras ρ : S → R by y i → x γ i .Let m = (x γ 1 , . . ., x γ u ) ⊆ R and η = (y 1 , . . ., y u ) ⊆ S. We note that ρ(η) = m.Set J = Ker(ρ), so that R = S/J.
Set t = depth(R) = min{i | Ext u−i S (R, S) ̸ = 0}.We first show that Ann R Ext u−t S (R, S) ⊆ x P(M) ; we proceed by contradiction.Suppose that there exists an M-homogeneous element f ∈ R such that f ∈ Ann R Ext u−t S (R, S) \ x P(M) .Let m ∈ Z ⩾0 be such that log( f ) ∈ mV m (M).Since the multiplication map Ext u−t S (R, S) f → Ext u−t S (R, S) is the zero map, we have that H t m (R) f → H t m (R) is the zero map by Matlis duality [18].Thus, H t m (R 1/m ) ) is the zero map as well.Since the composition of → R is the identity, we have the same for the composition Since the middle map is zero, we have that H t m (R) = 0, which is not possible because t = depth(R).Since Ann R Ext u−t S (R, S) ⊆ x P(M) , we have that (5.1) dim Ext u−t S (R, S) ⩾ dim R/x P(M) = mpdim(M).
Since S is a Gorenstein ring, its injective resolutions as S-module is given by where E j = ht(p)= j E(S/p) is the direct sum of the injective hulls of all the prime ideals in S of height j [2].Therefore, (5.2) dim Ext u−t S (R, S) ⩽ t.
We note that dim k (R/I e ) = |V p e (M)|, and that x P(M) is the splitting prime of R [1].It follows that mpdim(M) = sdim(R), and the result follows for the ratios.We now discuss the claim about F-signature.We have that char(k) ∈ A (M) for normal monoids by Proposition 3.5.The result follows because the F-signature coincides with the F-splitting ratio for strongly F-regular rings, and R is strongly F-regular if and only if M is normal.□ We end this section with a monoid version of Kunz's characterization of regularity [16].
Theorem 5.9.Let M be an affine monoid.Then, |V m (M)| = m rank(M) for some m ∈ Z >0 if and only if M ∼ = Z t >0 for some t.
Proof.Since |V m (M)| = m rank(M) , we have that |V m t (M)| = m t rank(M) by Lemma 3.17

4 .
of M-modules and (semi)normal affine monoids 5 Asymptotic growth of number of pure translations 13 5. Applications to affine semigroup rings 15 1. INTRODUCTION

Lemma 3 . 8 .
Let m, n ∈ Z >1 .Then mn ∈ A (M) if and only if both m ∈ A (M) and n ∈ A (M).
(3)), and thus α∈V m (M) (α + M) ⊆ 1 m M is pure.As a consequence, the ZM-module Z1  m M contains α∈V m (M) Z(α + M) as a free direct summand, and thus |V m (M)| ⩽ m rank(M) , where the latter is the rank of Z 1 m M as a ZM-module.□
which shows a ∈ mV m (M).We continue with the backward direction.Let f ∈ ZM be such that n f ∈ M for some n ∈ Z >1 .By Proposition 3.5 it suffices to show f ∈ M. Write f = a − b with a, b ∈ M, then na ∈ nb + M. Thus, n(a, b) = b + (n − 1)(a, b), where (a, b) denotes the ideal of M generated by the set {a, b}.It follows that (n + r)(a, b) = rb + n(a, b) for every r ∈ Z >0 .Hence, ra + nb ∈ (n + r)(a, b) = rb + n(a, b) ⊆ rb + M for every r ∈ Z >0 .

Lemma 4 . 3 .
Let M be a seminormal affine monoid.Then (1) B(M) is a bounded set and it has volume, i.e., its boundary has measure zero in the dim(RF M )-dimensional Lebesgue measure on RF M .(2) There exists an increasing sequence {p t } t∈Z ⩾1 ⊆ A (M) such that B p t ⊆ B p t+1 for every t ∈ Z ⩾1 and B(M) = t∈Z ⩾1 B p t (M).

Theorem 4 . 6 .
Let M be a seminormal affine monoid.We have thatmpr(M) = vol Z(M∩F M ) (B(M)) > 0.In particular, M is normal if and only if mps(M) > 0. Furthermore, in this case mps(M) ∈ Q >0 .Proof.By Lemma 4.3 (1), the characteristic function χ B(M) is Riemann integrable.Now, by Lemma 4.3 (3) and Corollary 3.28 we haveV m t (M) = Z m t (M ∩ F M ) ∩ B(M).Thus, |V m t (M)| m s tis a Riemann sum for χ B(M) with normalized volumes of the cells and mesh the diameter of a fundamental domain for Z(M∩F M ) m t

Notation 5 . 1 .
Given an affine monoid as in Notation 1.1, we
a minimal free resolution of R over S, then c = ht(I) and F c = S(−D − a d (R)), where D = ∑ u i=1 d i .The minimal free resolution of F e • : 0 → F e c → . . .→ F e 0 = S → S/I [p e ] → 0 of S/I [p e ] is such that F e c = S(p e (−D − a d (R))).The comparison map F e • → F • induced by the natural surjection S/I [p e ] → R in homological degree c is S(p e (−D − a d (R))) → S(−D − a d (R)).Furthermore, it is given, up to an invertible element, by multiplication by f e [27, Lemma 1].Since such a map is homogeneous of degree zero, we conclude that deg( f e ) = (p e − 1)(D + a d (R)).
e) splits, and so, β (e) ∈ V p e (M).Note that |β (e)| = ∑ u i=1 |γ i |(p e − 1 − n i ) p e = (p e − 1)D − ∑ u i=1 n i d i p e = (p e − 1)D − deg( f e ) p e = −a d (R) p e − 1 p e .We have that mpt(M) ⩾ lim e→∞ |β (e)| = −a d (R) by Proposition 3.21.As the other inequality always holds by Theorem 5.4, we have equality.□From the previous result, one may wonder if the converse is true.In particular, as fpt(k[M]) = a d (k[M]) implies that k[M] is Gorenstein if k[M]has a structure of standard graded k-algebra[22].This motivates the following question.Question 5.6.Assume that M is normal of rank d.If mpt(M) = −a d (R), is R is Gorenstein?We now provide a bound for the depth of R, which recovers Hochster's result that normal semigroup rings are Cohen-Macaulay [14, Theorem 1].Theorem 5.7.Let M and R be as in Notation 5.1.Then, mpdim(M) ⩽ depth(R).

Proposition 5 . 8 .
Let M and R be as in Notation 5.1.If char(k) ∈ A (M), then mpr(M) is the F-splitting ratio of R. As a consequence, if char(k) is a prime number and M is normal, then mps(M) equal to the F-signature of R. Proof.Let p = char(k) and m be the maximal homogeneous ideal in R. Let I e = { f ∈ R | φ ( f 1/p e (3).Then,mps(R) = lim t→∞ |V m (M)| m rank(M) = 1.Hence, mpdim(M) = rank(M), and so, M is a normal monoid by Corollary 3.30.We have that mps(R) = 1.Then, F p [M] is a regular graded F p -algebra, because s(R) = 1 by Proposition 5.8 and the characterization of regular rings via F-signature [15,Corollary 16].Moreover, M has a set of rank(M) minimal generators.Hence, M ∼ = Z t >0 .□