Integrable functions within the theory of schemic motivic integration
aa r X i v : . [ m a t h . AG ] S e p INTEGRABLE FUNCTIONS WITHIN THE THEORY OF SCHEMIC MOTIVICINTEGRATION
ANDREW R. STOUTA
BSTRACT . We develop integrable functions within the theory of relative simplicial mo-tivic measures. C ONTENTS
1. Introduction 12. Permissible schemic functions 13. Permissible simplicial functions 44. Total functions 55. Convergence of sequences of total functions 66. Functions on limit sieves 87. Cach´e functions 11References 141. I
NTRODUCTION
In [5], the author developed the theory of measurable limit simplicial motivic sites (mea-surable motivic sites for short) in order to shed light on computational issues encounteredin [6]. The simpliical approach allowed for the prospect of defining motivic integrationover derived stacks, yet at another level it allowed the author to investigate classes of func-tions not yet appearing in the literature: permissible functions, total functions, and cach´efunctions along with an integrability condition. This work was heavily inspired by the de-finable approach to motivic integration as laid bare in the seminal work [1] of R. Cluckersand F. Loeser.Schemic motivic integration was created by H. Schoutens in [2] and [3]. One shouldconsult these works for terminology and background on this field. I gave a quick intro-duction to the subject in [5] within the framework of simplicial sieves. This current workheavily relies upon the results of these three papers.The motivation behind this work is the hope of producing a general change of variablesformula in the context of schemic motivic integration. This work was partially supportedby the chateaubriand fellowship, Prof. F. Loeser, and DSC research grant.2. P
ERMISSIBLE SCHEMIC FUNCTIONS
Let X P S h κ . By a function f : X Ñ N , we will mean a collection of set maps f p m q : X ˝ p m q Ñ N for each m P Fat κ such that for any embedding i : m ã Ñ m , then f p m qp i ˚ x q “ f p m qp x q . Note that thisasures us that the image of f p m q is naturally included into the image of f p m q . Given oneof these so-called functions, we can associate a contravariant functor Γ f : Fat κ Ñ Set defined by(2.1) Γ f p m q : “ Γ f p m q : “ tp x, n q P X ˝ p m q ˆ N | f p m qp x q “ n u . Note tha a morphism j : m Ñ m induces a map of sets j ˚ : X ˝ p m q Ñ X ˝ p m q by sendinga point x : m Ñ X to the point j ˚ x : m Ñ X defined by the composition x ˝ j . In thesame way, this induces a map of sets j ˚ : Γ f p m q Ñ Γ f p m q by sending p x, f p m qp x qq to p j ˚ x, f p m qp j ˚ x qq . For each m P Fat κ , this also gives rise to an assignment from ∆ ˝ to Set by sending a simplicial complex r n s to the set(2.2) Γ f p m qpr n sq : “ f p m q ´ p n q “ Γ f p m q X p X ˝ p m q ˆ t n uq . From this, we can see that Γ f p m qpr n sq can be identified with a subset of X ˝ p m q . Theinclusion of Γ f p m qpr n sq into X ˝ p m q will be natural in fat points. Thus, for each r n s P ∆ ˝ ,the presieve Γ f p´qpr n sq is actually a sieve with X as its ambient space.2.1. Definition.
Let X P S h κ and let f : X Ñ N be a function. We will say that f is a permissible function if the assignment from ∆ ˝ to Sieve κ defined by sending r n s to Γ f p´qpr n sq is in fact a functor. In this case, Γ f will denote the corresponding element in sSieve κ .2.2. Example.
For any element X of S h κ , we have the so-called positive constant func-tions which are precisely the functions such that for all m P Fat κ , f p m q is equal to somefixed n on X ˝ p m q . In this case, we have the following simple formula(2.3) τ n p Γ f q “ X and where τ m p Γ f q “ H when m ‰ n . Thus, it is immediate that every positive constant function is a permissible function. Wewill denote the collection of positive constant functions on X by K ` p X q .Given a scheme X P S h κ , we form the set of all permissible functions and denote itby S ` p X q . By definition, S ` p X q can be identified with a subset of sSieve κ via f ÞÑ Γ f .As such, the operations of ˆ , \ , Y and X restrict to operations on S ` p X q . Here, theseoperations are taking place in sSet . For example,(2.4) p Γ f Y Γ g qp m qpr n sq “ Γ f p m qpr n sq Y Γ g p m qpr n sq . We can introduce even more operations on the set S ` p X q . Given two elements f and g in S ` p X q , we form the function f ` g defined by(2.5) p f ` g qp m qp x q : “ p f p m q ` g p m qqp x q “ f p m qp x q ` g p m qp x q for each x P X ˝ p m q and for each m P Fat κ . Likewise, we form the function f ¨ g definedby(2.6) p f ¨ g qp m qp x q : “ p f p m q ¨ g p m qqp x q “ f p m qp x q ¨ g p m qp x q for each x P X ˝ p m q and for each m P Fat κ .2.3. Proposition.
Let X P S h κ . The set S ` p X q is a semiring with respect to the opera-tions ` and ´ defined in formulas 2.4 and 2.5, respectively.Proof. The only thing that remains unclear is if ` and ´ are closed operations – meaning,if f, g P S ` p X q , then f ` g, f ¨ g P S ` p X q . Therefore, we show that f ` g is a permissiblefunction (as the proof for f ¨ g will follow along the same lines). To wit, we claim that the NTEGRABLE FUNCTIONS WITHIN THE THEORY OF SCHEMIC MOTIVIC INTEGRATION 3 assignment r n s ÞÑ Γ f ` g p´qpr n sq is a functor. Note that, for each m P Fat κ , we have thefollowing decomposition(2.7) Γ f ` g p m qpr n sq “ n ď t “ p Γ f p m qpr t sq Y Γ g p m qpr n ´ t sqq . Since f and g are permissable, we have expressed the assignment as a union of simplicialsieves which makes it indeed a functor. (cid:3) Because of Proposition 2.3, we may associate to each element X of S h κ its ringof permissible functions denoted by S p X q , which is formed by taking the associatedgrothendieck ring of S ` p X q with respect to the ` operation.2.4. Proposition.
Let X P S h κ . The assignment which sends any open set U of X to thering S p U q (resp., the semiring S ` p U q ) is a sheaf of rings (resp., a sheaf of semirings) onthe scheme X .Proof. For two open sets U Ă V of a scheme X , we have the restriction morphism S ` p V q Ñ S ` p U q given by sending f to the unique element of S ` p U q whose graph is ofthe form Γ f X p U q ‚ . Note that this is also enough to define the restriction homomorphimfrom S p V q to S p U q . The fact that S ` p´q and S p´q are presheaves on X is immediate.Let U i be an arbitrary cover of some open set U of X . We wish to show that the firstarrow in the diagram of semiring homomorphisms(2.8) S ` p U q Ñ ź i S ` p U i q ÑÑ ź i,j S ` p U i X U j q is an equalizer. This is straightforward. Letting f i P S ` p U i q be such that f i | U j “ f j | U i in S ` p U i X U j q , we can just form the function f on U defined by sending an x P U ˝ p m q to f i p m qp x q for any i such that x P p U i q ˝ p m q for all m P Fat κ . Uniqueness of this gluingholds because the corresponding graphs will be the same.We will let gr p U q : S ` p U q Ñ S p U q denote the grothendieck semiring homomorphismwith respect to the operation ` . Let f i P S p U i q be such that f i | U j “ f j | U i . As gr p U i q issurjective for each i , these functions lift to function g i P P ` p U i q such that(2.9) gr p U i X U j qp g i | U j q “ gr p U i qp g i q| U j “ gr p U j qp g j q| U i “ gr p U i X U j qp g j | U i q . Thus, there are a functions m ij , n ij P S ` p U i X U j q for each i and j such that g i | U j ` m ij “ g j | U i ` n ij . Thus, by gluing as before we arrive at unique functions g, m, n P S ` p U q such that g ` m “ g ` n . From this we find that gr p U qp m q “ gr p U qp n q whichmeans that the function f “ gr p U qp g q is such that f | U i “ f i . Uniqueness of this gluingfollows from a similar argument. (cid:3) Remark.
The assignment which sends an open set U of X to the grothendieck semiringhomomorphism gr p U q : S ` p U q Ñ S p U q is a morphism of semiring sheaves. Therefore,the stalk S x of P p´q at x P X is the grothendieck ring with respect to ` of the stalk S ` x of P ` p´q at x P X .One way to describe the functions in S p X q is to consider pairs p f, g q where f and g areelements of S ` p X q and where we mod out by the following equivalent relation: p f , g q „p f , g q if there exist some function h P S ` p X q such that f ` g ` h “ f ` g ` h .Here one should think of f as the positive part and g as the negative part. Further, this justmeans that elements of S p X q are functions F from X to N modulo „ . By this, we meanthat for each m P Fat κ we have a function from X ˝ p m q to N which is functorial with ANDREW R. STOUT respect to inclusions of fat points as before – with the added property that the assignmentfrom ∆ ˝ ˆ ∆ ˝ to Sieve κ given by(2.10) pr n s , r m sq ÞÑ Γ F ppr n s , r m sqq where Γ F ppr n s , r m sqq is the κ -sieve defined by sending each m P Fat κ to the set(2.11) Γ F p m q X p X ˝ p m q ˆ tp n, m quq is a functor – i.e., Γ F p´ , ´q is a bisimplicial κ -sieve. Since we therefore have use for suchobjects, we will denote by s n Sieve κ the category of all covariant functors ś nj “ ∆ ˝ Ñ Sieve κ , and we call such objects n - simplicial κ - sieves (or, bisimplicial κ -sieves andtrisimplicial κ -sieves for n “ and n “ , respectively.) The category s n Sieve κ isequivalent to the category of covariant functors ∆ ˝ Ñ s n ´ Sieve κ for any n ą .2.6. Example.
It is easy to check that K ` p X q is a sub-semiring of S ` p X q for X P S h κ .We define the ring of constant functions on X , denoted by K p X q , to be the image of K ` p X q in S p X q under the grothendieck semiring homomorphism gr p X q . We can identifyelements K p X q with a collection of functions f p m q : X ˝ p m q Ñ Z defined by f p m qp x q “ n for all x P X ˝ p m q , for all m P Fat κ , and for some fixed n P Z . This is because thegraphs of the positive and the negative part are instantly sieves.2.7. Remark.
In fact, the assignment which sends an open set U of X to K ` p U q (resp., K p U q ) is a sheaf of semirings (resp. a sheaf of rings) on X . Moreover, S ` p´q (resp., S p´q ) is a sheaf of K ` p´q -semialgebras (a sheaf of K p´q -algebras).3. P ERMISSIBLE SIMPLICIAL FUNCTIONS
Let X of s n Sieve κ . By a point of X p m q , we mean a tuple p x, ¯ m q such that x P X r ¯ m s p m q and ¯ m P N n . By a function f : X Ñ N , we mean a collection f p m q of set-theorectic functions from the set of points of X m to N with the property that f p m qp i ˚ x, ¯ m q “ f p m qp x, ¯ m q for all ¯ m P N n and all x P X r ¯ m s p m q whenver we have an embedding i : m Ñ m in Fat κ . Often we will suppress the brackets around ¯ m .3.1. Example.
Consider the constant functor p´q ‚ : Sieve κ Ñ s n Sieve κ which sendsa sieve X to the n -simplicial sieve p X q ‚ which is defined as the functor which sends σ P ś nj “ ∆ ˝ to p X q σ : “ X . Now let X P Sieve κ and let f : X Ñ N be a func-tion. The constant functor also extends to functions by defining g : “ p f q ‚ : X Ñ N by g p m qp x, ¯ m q : “ f p m qp x q for all ¯ m P N n and all m P Fat κ . More succintly, we have(3.1) S ` p X q Ă S ` pp X q ‚ q for any X P Sieve κ .Given one of these so-called functions, we can associate a functor Γ f p´qp ¯ m q : Fat κ Ñ Set defined by m ÞÑ Γ f p m qp ¯ m q : “ Γ f p m q p ¯ m q“ tp x, ¯ m, n q | x P X r ¯ m s p m q , f p m qp x, ¯ m q “ n, for some n P N u (3.2)for each ¯ m P N n . Thus, we have an assignment ∆ ˝ Ñ s n Sieve κ given by(3.3) r n s ÞÑ Γ f p m qp ¯ m qpr n sq : “ Γ f p m qp ¯ m q X p X ¯ m p m q ˆ t n uq . If this assignment is actually a functor (i.e., it is a element of s n ` Sieve κ ), then we saythat f : X Ñ N is a permissible n - simplicial function . Given a simpilicial sieve X , NTEGRABLE FUNCTIONS WITHIN THE THEORY OF SCHEMIC MOTIVIC INTEGRATION 5 we again denote the set of permissible simplicial functions on X by S ` p X q . As before,the operations ˆ , \ , Y , and X , induce operations on S ` p X q via the identification withgraphs as before. Furthermore, we have binary operations ` and ¨ via pointwise additionand pointwise multiplication which is defined analogously to Definition 2.5 and 2.6. Forexample, given f, g P S ` p X q , we define f ` g to be the function which sends a point p x, ¯ m q of X p m q to f p m qp x, ¯ m q ` g p m qp x, ¯ m q for all m P Fat κ . Proposition.
Let X P s n Sieve κ . Then, the assignment which sends an admissibleopen set U to S ` p U q is a sheaf of semirings. Thus, we may form the grothendieck ring S p U q of S ` p U q with respect to ` for each open set S ` p U q . Therefore, the assignmentwhich sends an open set U to S p U q is a sheaf of rings, and this gives rise to a morphism ofsheaves of semirings gr p´q : S ` p´q Ñ S p´q which induces the grothendieck morphismof semirings S `p x, ¯ m q Ñ S p x, ¯ m q for any point of p x, ¯ m q of X p m q and for any m P Fat κ .Proof. The proofs of these statements are essentially the same as the proofs of the corre-sponding statements in Proposition 2.3 and 2.4. (cid:3)
Example.
Likewise, given an element X P s n Sieve κ , we can define the semiringof positive constant functions K ` p X q (resp., the ring of constant functions K p X q ) in acompletely analogous way to the schemic definition. Further, the assignment which sendsan admissible open set U to K ` p U q (resp., K p U q ) is a sheaf of semirings (resp., a sheaf ofrings) on X . Furthermore, S ` p´q (resp., S p´q ) is a sheaf of K ` p´q -semialgebras (resp.,a sheaf of K p´q -algebras) on X .4. T OTAL FUNCTIONS
Consider the ring(4.1) A : “ Z r L , L ´ , p ´ L ´ i q i ą s . Given an element X P s n Sieve κ , we construct the ring of total functions on X , denotedby T p X q , by considering the smallest ring generated by elements of A , elements of S p X q ,and objects of the form L α where α is an element of S p X q .For each real number q ą , we have an evaluation ring homomorphism ν q : A Ñ R defined by sending L to q . Given elements a and b in A , we say that a ě b if ν q p a q ě ν q p b q for all real q ą . This induces a partial order on A . We define the following semiring(4.2) A ` : “ t a P A | ν q p a q ě , @ q ą u . This valuation map extends to T p X q in the natural way. For X P s n Sieve κ , we define the semiring of total functions , denoted by T ` p X q , by(4.3) T ` p X q : “ t f P T p X q | ν q p f q ě , @ q ą u Finally, we define a partial order on T p X q by setting f ě g if f ´ g P T ` p X q . In theabove definitions, we followed § . of [1].4.1. Example.
Consider any fat point m over κ . Then, S p m q “ K p m q . Thus, it is imme-diate that(4.4) T p m q – A and T ` p m q – A ` . Let p´q ‚ : Sieve κ Ñ s n Sieve κ be the trivial functor for some n ą . Then, T pp m q ‚ q isequal to the smallest ring generated by A , functions from N n to Z , and objects L α where α is a function from N n to Z modulo the n -simplicial identities on the associated graphs. ANDREW R. STOUT
Example.
Fix a σ P ś n ´ mj “ ∆ ˝ where n, m P N with n ě m . Let τ σ : s n Sieve κ Ñ s m Sieve κ be the functor which sends an n -simplicial sieve X to the m -simplicial sievedefined by(4.5) σ ÞÑ X p σ ˆ σ q , @ σ P m ź j “ ∆ ˝ . This functor also induces a semiring homomorphism τ σ : T ` p X q Ñ T ` p τ σ p X qq (andthereby a ring homomorphism τ σ : T p X q Ñ T p τ σ p X qq ) defined by sending a positivetotal function f : X Ñ N to the unique positive total function g : “ τ σ p f q : τ σ p X q Ñ N defined by sending a point p x, σ q to f p m qp x, σ ˆ σ q for all m P Fat κ .4.3. Definition.
Let X P s n Sieve κ and consider an element f P T p X q . Let Y : “ τ σ p X q for some σ P ś n ´ mj “ ∆ ˝ where n, m P N with n ě m . We say that f is σ - summable over Y if the sequence p ν q p τ i p f qqq i P N n ´ m is summable in R for all real q ą . Definition.
Let X P s n Sieve κ and consider an element f P T p X q . Let Y : “ τ σ p X q for some σ P ś n ´ mj “ ∆ ˝ where n, m P N with n ě m . We say that f is σ - integrable over Y if there exists a function g P T p Y q such that(4.6) ν q p g q “ ÿ i P N n ´ m ν q p τ i p f qq , @ q ą . If such a g exists, we will denote it by µ τ σ p X q p f q or by µ σ Y p f q . We will denote the subringof T p X q formed by all σ -integrable functions over Y by I σ Y T p X q Remark.
We can also define the notion for positive total functions. Let X P s n Sieve κ and consider an element f P T ` p X q . Let Y : “ τ σ p X q for some σ P ś n ´ mj “ ∆ ˝ where n, m P N with n ě m . We say that f is σ - integrable over Y if there exists a function g P T ` p Y q such that(4.7) ν q p g q “ ÿ i P N n ´ m ν q p τ i p f qq , @ q ą . If such a g exists, we will denote it by µ τ σ p X q p f q or by µ σ Y p f q . We will denote the subringof T ` p X q formed by all σ -integrable functions over Y by I σ Y T ` p X q Theorem.
Let X P s n Sieve κ and consider an element f P T p X q . Let Y : “ τ σ p X q for some σ P ś n ´ mj “ ∆ ˝ where n, m P N with n ě m . If f is σ -integral over Y , then it is σ -summable over Y . Moreover, the set map (4.8) µ σ Y : I σ Y T p X q Ñ T p Y q , f ÞÑ µ σ Y p f q is a ring homomorphism.Proof. This is immediate. (cid:3)
Remark.
To obtain the converse in 4.6, we either have to either investigate a subsetof the ring of total functions which are summable or we have to search for an appropriateover-ring of A .5. C ONVERGENCE OF SEQUENCES OF TOTAL FUNCTIONS
We say that a sequence of elements p a i q in A is q - convergent to a ‹ if p ν p a i qq convergesto p ν q p a ‹ qq in R for a fixed a real number q ą . Moreover, we say that p a i q is convergentto a ‹ if p ν p a i qq converges to p ν q p a ‹ qq in R for all real q ą . NTEGRABLE FUNCTIONS WITHIN THE THEORY OF SCHEMIC MOTIVIC INTEGRATION 7
Lemma.
A sequence p a i q in A is convergent to a ‹ P A if and only if it is q -convergentto a ‹ P A for some transcendental q .Proof. This follows from the fact that the evaluation ring homomorphism ν q : A Ñ R isinjective whenever q is transcendental. (cid:3) Example.
Let p a i q be a sequence such that a i “ L n i for almost all i . Assume alsothat it is q -convergent to a ‹ P A for some real q ą . Then,(5.1) | ν q p a i q ´ ν q p a j q| “ q min t n i ,n j u | q | n i ´ n j | ´ | , @ i, j . Since | ν q p a i q ´ ν q p a j q| converges to zero, either n i diverges to ´8 or p n i q is a cauchysequence. However, a cauchy sequence of integers is eventually constant. Thus, either n i diverges to ´8 or n i “ N for all sufficiently large i where N is some fixed interger N .Thus, a ‹ is either equal to or L N for some integer N. Let X P s n Sieve κ and let p f i q be a sequence of elements in T p X q . We say that p f i q is q - convergent to f ‹ P T p X q if for all m P Fat κ and all points p x, ¯ m q of X p m q , thesequence p f i p m qp x, ¯ m qq is q -convergent to f ‹ p m qp x, ¯ m q for some fixed real number q ą .Furthermore, we say that p f i q is convergent to f ‹ P T p X q if it is q -convergent to f ‹ for all q ą .5.3. Lemma.
Let X P s n Sieve κ and let p f i q be a sequence of elements in T p X q . Then, p f i q is convergent to f ‹ P T p X q if and only if p f i q is q -convergent to f ‹ P T p X q for sometranscedental q . Example.
Let p f i q be a sequence of total functions such that f i “ L β i where β i P S p X q for all i . Assume that p f i q is q -convergent to f ‹ for some fixed q ą . For each m P Fat κ and for each point p x, ¯ m q of X p m q , we have that f ‹ p m qp x, ¯ m q is either equal to or L N for some integer N P Z . We define a n -simplicial subsieve C of X by(5.2) C p m q “ Supp p f ‹ p m qq Let C : X Ñ N be the characteristic function of C . Then, f ‹ is equal to the function C ¨ L β for some β P S p X q . Again, it is easy to check that p f i q is convergent to f ‹ in thiscase.Let ¯ A be the ring Z r L srr L ´ ss Ą A and ¯ A ` to be the ring largest subring contained inboth ¯ A and A ` . For X P s n Sieve κ , we define ¯ T p X q “ ¯ A b A T p X q ¯ T ` p X q “ ¯ A ` b A ` T ` p X q . (5.3)5.5. Theorem.
Let X P s n Sieve κ and f P T p X q . Assume that f is σ -summable over Y for some σ P ś n ´ mj “ ∆ ˝ with n ě m . Then, there is a g P ¯ T p Y q such that (5.4) ν q p g q “ ÿ i P N n ´ m ν q p τ i p f qq , @ q ą . Moreover, g is in T p Y q if and only if f is subject to a presburger condition in the last n ´ m coordinates.Proof. If f is a constant function into ¯ A or if f P S p X q , then the statement followsimmediately. By properties of limits, this leaves us to prove the statement when f is of theform L β for some β P S p X q . Consider the partial sums(5.5) s i ,...,i k “ τ i p f q ` ¨ ¨ ¨ ` τ i k p f q ANDREW R. STOUT
Then for each m P Fat κ and each point p x, ¯ m q of X m , we have(5.6) s i ,...,i k p m qp x, ¯ m q “ L β p m qp x, ¯ m ˆ i q ` ¨ ¨ ¨ ` L β p m qp x, ¯ m ˆ i k q By definition of summability, this implies that ν q p s i ,...,i k p m qp x, ¯ m qq converges to somereal number r . Therefore, ν q p β p m qp x, ¯ m ˆ i j qq diverges to ´8 as the sum of the coor-dinates of i j P N n ´ m tends toward . This is enough to show that there is an element a ‹ p m qp x, ¯ m q P ¯ A such that ν q p a ‹ p m qp x, ¯ m qq is equal to r “ lim ν q p s i ,...,i k p m qp x, ¯ m qq .Let g : X Ñ ¯ A be the function defined by g p m qp x, ¯ m q : “ a ‹ p m qp x, ¯ m q . It is clearthat g P ¯ T p Y q . The proof of the last assertion is completely analogous to the proof ofTheorem-Definition 4.5.1 of [1]. (cid:3) Definition.
Let X P s n Sieve κ and consider an element f P T p X q . Let Y : “ τ σ p X q for some σ P ś n ´ mj “ ∆ ˝ where n, m P N with n ě m . We say that a function f P ¯ T p X q is weakly σ - integrable over Y if there is a g P ¯ T p Y q such that(5.7) ν q p g q “ ÿ i P N n ´ m ν q p τ i p f qq for all real q ą . Furthermore if f P ¯ T ` p X q , then we say that it is weakly σ - integrableover Y if there is a g P ¯ T ` p Y q such that(5.8) ν q p g q “ ÿ i P N n ´ m ν q p τ i p f qq for all real q ą . In either case, we will often write µ σ Y p f q or µ τ σ p X q p f q for g . Finally,we will denote the sub-ring (resp., sub-semiring) of ¯ T p X q (resp. ¯ T ` p X q ) formed by allweakly σ -integrable functions over Y by I σ Y ¯ T p X q (resp., I σ Y ¯ T ` p X q ).5.7. Remark.
Note that as in the case of Theorem 4.6, µ σ Y defines a ring homomorphism(resp., a semiring homomorphism) µ σ Y : I σ Y ¯ T p X q Ñ ¯ T p Y q (resp., µ σ Y : I σ Y ¯ T ` p X q Ñ ¯ T ` p Y q ).5.8. Theorem.
Let X P s n Sieve κ and consider an element f P T p X q . Let Y : “ τ σ p X q for some σ P ś n ´ mj “ ∆ ˝ where n, m P N with n ě m . Then, we have the followinginjective morphisms I σ Y T ` p X q b A ` ¯ A ` ã Ñ I σ Y ¯ T ` p X q I σ Y T p X q b A ¯ A ã Ñ I σ Y ¯ T p X q . (5.9) Proof.
Naturally, we investigate the multiplication homomorphism from I σ Y T ` p X q b A ` ¯ A ` to I σ Y ¯ T ` p X q . For injectivity, consider the finite sum ř f i b a i with f i P I σ Y T ` p X q and a i P ¯ A . Then, ř f i ¨ a i “ if and only if f i ¨ a i “ for all i . It is immediate thenthat either a i “ or f i “ . Thus, the multiplication semiring homomorphism is injective.The rest follows. (cid:3)
6. F
UNCTIONS ON LIMIT SIEVES
Definition.
Let X P s n Sieve κ . Let z P y Fat κ and choose a point system X such that z “ inj lim X . For each m P X , let X m P sSieve κ be such that there is natural inclusion(6.1) X m ã Ñ ∇ q m X . We define X ‹ : “ proj lim X m and call it a limit n - simplicial sieve at the point z withrespect to the point system X . We call X the base of X ‹ . NTEGRABLE FUNCTIONS WITHIN THE THEORY OF SCHEMIC MOTIVIC INTEGRATION 9
Definition.
Let F p´q be any of the sheaves of semirings S ` p´q , T ` p´q , ¯ T ` p´q orany of the sheaves of rings S p´q , T p´q , ¯ T p´q . Then, for a limit n -simplicial sieve X ‹ wedefine(6.2) F p X ‹ q : “ inj lim F p X m q . Remark.
Let X ‹ be a limit n -simplicial sieve. Let F p´q be any of the sheaves of semir-ings S ` p´q , T ` p´q , ¯ T ` p´q or any of the sheaves of rings S p´q , T p´q , ¯ T p´q . Thenthe assignment which sends an admissible open U of X ‹ to F p U q is a sheaf of semir-ings (or rings) on X ‹ . Furthermore, if F ` p´q the sheaf of semirings S ` p´q (resp., T ` p´q , ¯ T ` p´q ) and let F p´q the sheaf of rings S p´q (resp., T p´q , ¯ T p´q ). Then,the direct limit of the grothendieck semiring homomorphisms on each admissible openinduces the grothendieck morphism of sheaves of semirings S ` p´q Ñ S p´q (resp., T ` p´q Ñ T p´q , ¯ T ` p´q Ñ ¯ T p´q ).6.4. Remark.
All the material from § § n -simplicial sieves.Let X P s n Sieve κ . The n -simplicial arc operator ∇ q m operates on a permissible func-tion f : X Ñ N (i.e., f P S ` p X q ) by acting on the associated graph. In other words, wemay define a set map(6.3) ∇ q m : S ` p X q Ñ S ` p ∇ q m X q which sends a function f to the unique function ∇ q m f whose graph is the n -simplicialsieve ∇ q m Γ f .6.5. Proposition.
Let X P s n Sieve κ . Then, the set map ∇ q m : S ` p X q Ñ S ` p ∇ q m X q isa semiring homomorphism.Proof. Let f, g P S ` p X q . First, it is clear that ∇ q m p f ¨ g q “ ∇ q m f ¨ ∇ q m g . Consider thedecomposition(6.4) Γ f ` g p n qpr n sq “ n ď t “ p Γ f p n qpr t sq Y Γ g p n qpr n ´ t sqq . This will almost always be a disjoint union, and in that case, we can just apply Theorem4.8 of [3] to obtain the result. In order for this decomposition to not be a disjoint union,there must exists a positive integer t such that t “ n and f p n qp x, ¯ m q “ g p n qp x, ¯ m q forsome point p x, ¯ m q of X n for some n P Fat κ . Thus, suppose this is the case and define asubsieve C of X by(6.5) C r ¯ m s p n q “ t x P X r ¯ m s p n q | f p n qp x, ¯ m q “ g p n qp x, ¯ m qu . Let h “ C p f ´ g q which is equal to zero by definition where C is the characteristicfunction of C . Then, f ` g “ f ` g ` h “ p f ` C f q ` p g ´ C g q , which means thatwe may reduce to the case of proving the result for f ` C f . This again reduces furtherto proving that ∇ q m ¨ f “ ¨ ∇ q m f . Since the arc operator preserves multiplication and ∇ q m k “ k for any constant function k on X , the result follows. (cid:3) We may define a set map(6.6) ∇ q m : S p X q Ñ S p ∇ q m X q which sends a function f to the unique function ∇ q m f whose graph is the n -simplicialsieve ∇ q m Γ f modulo the equivalence relation discussed at the end of § Proposition.
Let X P s n Sieve κ . ∇ q m : S p X q Ñ S p ∇ q m X q is the ring homomor-phism induced by ∇ q m : S ` p X q Ñ S ` p ∇ q m X q via the grothendieck semiring homomor-phism.Proof. First, we need to show that Definition 6.6 is well-defined. Thus, let F and G be el-ements of S p X q represented by pairs p f , f q and p g , g q , respectively, which are equiva-lent. Then, we have defined ∇ q m F and ∇ q m G to be the function represted by pairs p f , f q and p g , g q , respectively, which have the property that f i is the unique function whosegraph is ∇ q m Γ f i and g i is the unique function whose graph is ∇ q m Γ g i for i “ , . Fromthis it follows that p f , f q and p g , g q are also equivalent. From this the proposition alsofollows. (cid:3) Remark.
Since ∇ q m acts on all of the generators of T p X q (where we set ∇ q m a “ a for a P ¯ A ), we also have ring homomorphisms T p X q Ñ T p ∇ q m X q and ¯ T p X q Ñ ¯ T p ∇ q m X q . Likewise, we have semiring homomorphisms T ` p X q Ñ T ` p ∇ q m X q and ¯ T ` p X q Ñ ¯ T ` p ∇ q m X q .6.8. Remark.
Let F p´q be any of the sheaves of semirings S ` p´q , T ` p´q , ¯ T ` p´q or anyof the sheaves of rings S p´q , T p´q , ¯ T p´q . Given fat points m , n such that n ě m , thenthe relative simplicial arc operator ∇ q m { n induces a (semi)ring homomorphism(6.7) ∇ q m { n : F p ∇ q n X q Ñ F p ∇ q m X q via sending a function f to the unique function whose graph is of the form ∇ q m { n Γ f .Moreover, this is a directed system.6.9. Proposition.
Let F p´q be any of the sheaves of semirings S ` p´q , T ` p´q , ¯ T ` p´q or any of the sheaves of rings S p´q , T p´q , ¯ T p´q . Let X be a point system. Let ρ m { n : ∇ q m X Ñ ∇ q n X be the induced map given by n ď m . We write ρ m for ρ m { Spec p κ q . The directedsystem ρ ˚ m { n : F p ∇ q n X q Ñ F p ∇ q m X q is such that (6.8) ∇ q m “ ρ ˚ m . Thus, for any limit n -simplicial sieve X ‹ and any f P F p X ‹ q , we have (6.9) f “ ∇ q z g where g P F p X m q for some m P X .Proof. immediate. (cid:3) Proposition.
Let X be an n -simplicial sieve and let m P Fat κ and let Y : “ τ σ p X q for some σ P ś n ´ mj “ ∆ ˝ where n, m P N with n ě m . If f is (weakly) σ - integrable over Y , then ∇ q m f is (weakly) σ -integrable over Y . Moreover, (6.10) µ σ Y p ∇ q m f q “ ∇ q m µ σ Y p f q . Proof. immediate. (cid:3)
Let X be a limit n -simplicial sieve, X its points system with z “ inj lim X , and let Y : “ τ σ p X q for some σ P ś n ´ mj “ ∆ ˝ where n, m P N with n ě m . If f is in I σ Y ¯ T p X q (or, I σ Y T p X q ), then(6.11) µ σ Y p f q “ ∇ z µ σ Y p g q for some g P ¯ T p X m q (resp., T p X m q ) where m P X . Therefore, we have the followingproposition. NTEGRABLE FUNCTIONS WITHIN THE THEORY OF SCHEMIC MOTIVIC INTEGRATION 11
Proposition.
Let X be a limit n -simplicial sieve. Let Y : “ τ σ p X q for some σ P ś n ´ mj “ ∆ ˝ where n, m P N with n ě m . Then, there are isomorphisms I σ Y T ` p X q – inj lim I σ Y T ` p X m q I σ Y T p X q – inj lim I σ Y T p X m q I σ Y ¯ T ` p X q – inj lim I σ Y ¯ T ` p X m q I σ Y ¯ T p X q – inj lim I σ Y ¯ T p X m q . (6.12) 7. C ACH ´ E FUNCTIONS
Let !ם be a partial n -simplicial motivic site (which we will simply call a motivic site forbrevity) over a limit n -simplicial sieve Z . In [5], I defined the notion of the grothendieckrings Gr p !ם q and Gr p Mes !ם q where Mes !ם is the motivic site of all measurable limit n -simplicial sieves in !ם .An n -simplicial subsieve Y of Z that is an element of !ם may not be measurable. Weform the ring T p Z q by taking as generators the functions of the form Y where Y is ameasurable n -simplicial subsieve of Z and the element L ´ . We also define ¯ T p X q : “ T p X q b A ¯ A . Note that there is an injective ring homomorphism T p Z q ã Ñ Gr p Mes !ם q defined by sending Y to r Y s . More specifically, the ring structure on T p Z q is therestriction of the ring structure on Gr p Mes !ם q .We define the ring of cach´e functions on Z over A with respect to !ם by(7.1) C p Z , !ם q : “ Gr p Mes !ם q b T p Z q T p Z q and the ring of cach´e functions on Z over ¯ A with respect to !ם by(7.2) ¯ C p Z , !ם q : “ C p Z , !ם q b A ¯ A . For ease of notation, we will sometimes write C p Z q for C p Z , !ם q and ¯ C p Z q for ¯ C p Z , !ם q as the underlying motivic site will usually be explicitly stated.We also define the ring of σ - integrable cach´e functions on Z over Y “ τ σ p X q withrespect to !ם and coefficients in A by(7.3) I σ Y C p Z , !ם q : “ Gr p τ σ p Mes !ם qq b T p Y q I σ Y T p Z q and the ring of σ - integrable cach´e functions on Z over Y “ τ σ p Z q with respect to !ם andcoefficients in ¯ A by(7.4) I σ Y ¯ C p Z , !ם q : “ Gr p τ σ p Mes !ם qq b ¯ T p Y q I σ Y ¯ T p Z q . Again, we will sometimes write I σ Y C p Z q for I σ Y C p Z , !ם q and I σ Y ¯ C p Z q for I σ Y ¯ C p Z , !ם q when the underlying motivic site is fixed. Note that we have an inclusion of rings I σ Y C p Z q ã Ñ I σ Y ¯ C p Z q .7.1. Lemma.
Let !ם be a motivic site relative to Z P s n Sieve κ . Then, there is a surjectivemorphism of motivic sites (7.5) τ σ p Mes !ם q Ñ Mes τ σ p !ם q for any m -simplicial complex σ with n ě m . Thus, we have a surjective ring homomor-phism (7.6) Gr p τ σ p Mes !ם qq Ñ Gr p Mes τ σ p !ם qq . Proof. immediate. (cid:3)
In general, the assignment which sends an admissible open U of Z to F p U q is not asheaf on Z . Thus, we will denote the sheafification of F p´q by a F p´q .7.2. Theorem.
Let Z be a limit simplicial sieve and let F p´q be the any of the functors ofthe form C p´ , F p´qq , ¯ C p´ , F p´qq , I σ ´ C p´ , F p´qq , I σ ´ ¯ C p´ , F p´qq where F is any func-tor from limit n -simplicial sieves to the category of motivic sites which are closed underinfinite union. Then Gr p Mes !ם F p Z q | ´ q is a flasque sheaf on Z . Also, the assignmentwhich sends an admissible open U of Z to F p U q is a presheaf on Z .Proof. We just need to prove that S p´q : “ Gr p Mes !ם F p Z q | ´ q is in fact a sheaf as therest of the statements are trivial. Therefore, let U be an admissible open of Z and let U i be an arbitrary cover of U by admissible opens. For each, i , let s i P S p U i q be such that s i | U j “ s j | U i . If s i “ r S i s for some n -simplicial sieve S i P F p U i q Ă F p U q , then thistranslates to(7.7) r S i X U j s “ r S j X U i s in S p U q . We define the gluing of the S i to be the element r S s where S “ Y S i . Such asieve is clearly measurable. In this vain, proving unicity amounts to showing that if(7.8) r S X U i s “ r T X U i s , then r S s “ r T s which is easily shown to be the case by using the scissor relation. (cid:3) Analogues of the results in § From now on let F p´q be the any of the presheavesof the form C p´ , F p´qq , ¯ C p´ , F p´qq , I σ ´ C p´ , F p´qq , or I σ ´ ¯ C p´ , F p´qq where F is afunctor which sends an n -simplicial sieve to the category of motivic sites which are closedunder arbitrary union and such that(7.9) ∇ q m S P Mes F p ∇ q m Z q , for all S P Mes F p Z q . We have(7.10) F p Z q – inj lim F p Z m q , and the simplicial arc operator ∇ q m defines a morphism of ringed sieves from p Z m , F p´qq to p Z , F p´qq where Z is the base of Z by tensoring the action of ∇ q m on the grothendieckring of the motivic site Mes F p Z q and the action of ∇ q m on the other factor as defined in § f of F p Z q is of the form(7.11) f “ ∇ q z g for some g P F p Z m q and for some m P X where z “ inj lim X is the point system defining Z . Thus, if f is an element of I σ Y C p Z , F p Z qq or I σ Y ¯ C p Z , F p Z qq , we have(7.12) µ σ Y p f q “ ∇ q z µ σ Y p g q for some g in C p Z m , F p Z m qq (or, respectively, ¯ C p Z m , F p Z m qq ) for some m P X .7.4. Remark.
An analogous theorem to Theorem 10.1.1 of [1] holds for both I σ ´ C p´ , F p´qq and I σ ´ ¯ C p´ , F p´qq . We will take this subject up in [4].
NTEGRABLE FUNCTIONS WITHIN THE THEORY OF SCHEMIC MOTIVIC INTEGRATION 13
Spealization to measurability.
Here we apply the notion of the category of familiesof sieves indexed by N n as per the comments in § . of [5]. We denote this category by i n Sieves κ ,Note that there are a large amount of measures coming from the grothendieck ring ofa motivic site. By choosing a family of limit sieves Z , we are fixing the point system X and limit point z for each element of a motivic site !ם relative to Z . We fix an ultra-filter „ on X and a non-negative real number Q . We denote by µ Q the cooresponding measuredefined by(7.13) µ „ z , X ,Q p X ‹ q : “ ulim r X m s L ´ r Q ¨ dim ∇ q m X s . Restricting to the sub-motivic site !ם Q of elements of !ם which are measurable with respectto „ and Q , then µ Q restricted to Gr p !ם Q q L ‚ factors through the composition(7.14) Gr p !ם Q q L ‚ Ñ Gr p sSieve S q L ‚ Ñ ź „ Gr p sSieve S q L ‚ where the map on the far right is the diagonal homomorphism. We can realize this as acach´e function I X defined inductively by β m : “ r Q ¨ dim ∇ q m X s on X p m qz X p m q and β m “ β m on X p m q where m ď m and(7.15) I X p m q “ X m ¨ L ´ β m . for all m P X and otherwise. Note that β m P S p X q . Clearly then, since X P !ם Q , we havethat I X is σ -integrable for all σ . In this way, we can see that the notion of integrability inthis paper specializes to the notion of measurability [5].Now, we define(7.16) ż σ X dµ Q : “ ż σ X dµ Q : “ µ X σ p I X q . For each σ , we can extend this linearly through elements of the pullback of the the ringof σ -integrable total functions on Z along the structure morphism. We may also extendlinearly through elements of the grothendieck ring. Thus, we have defined a pairing of N n and X σ I Z σ C p Z , !ם Q q which sends p σ, c q to the element(7.17) ż σ c P T p X σ q determined by Equation 7.16. We call this element the geometric integral of c along the σ -simplex. This is a natural extension of the idea of measure in [5]. It is important to notethat we may apply the pushforward of the structure morphism j to T p X σ q and tensoringby gives us an element of C p Z σ q . It could be interesting to investigate the properties ofthis integral relative to the action of the measure µ Z σ more closely. The important thingto note is that if µ Z σ factors in this way, then we are actually computing something like aschemic geometric motivic integral in the vein of [3], [6], or [5].7.6. Topological realization.
In [5], we defined the notion of the topological motivic site.Likewise, geometric realization should turn a total function into functions from a CW-complex to A . This is possibly something to investigate further.We also defined the notions of homotopy grothendieck rings and topological homotopygrothendieck rings. Thus, we may tensor the ring of total functions on X by the homotopygrothendieck ring to obtain the notion of the ring of homotopy cach´e functions C h p X q which will come equipped with a surjective ring homomorphism(7.18) C h p X q Ñ C p X q . R EFERENCES[1] R. Cluckers & F. Loeser
Constructible motivic functions and motivic integration , Invent. math. 173, 23-121(2008) 1, 4, 5, 7.4[2] H. Schoutens
Schemic Grothendieck Rings I , websupport1.citytech.cuny.edu/faculty/hschoutens/PDF/SchemicGrothendieckRingPartI.pdf (2011). 1[3] H. Schoutens Schemic Grothendieck Rings II , websupport1.citytech.cuny.edu/faculty/hschoutens/PDF/SchemicGrothendieckRingPartII.pdf (2011). 1, 6, 7.5[4] A. Stout A primitive change of variables formula for schemic motivic integration , to appear 7.4[5] A. Stout
Measurable motivic sites , http://arxiv.org/abs/1306.4056
1, 7, 7.5, 7.5, 7.5, 7.6[6] A. Stout
Stability theory for schemes of finite type and schemic motivic integration , arxiv.org/abs/1212.1375
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