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Beyond perturbation 1: de Rham spaces 7 1 0 2 Dennis Borisov Kobi Kremnizer b University of Goettingen, Germany University of Oxford, UK e [email protected] [email protected] F 7 February 28, 2017 2 ] Abstract G D It is shownthatif one usesthe notionof -nilpotentelements due ∞ to Moerdijk and Reyes, instead of the usual definition of nilpotents to . h define reduced C∞-schemes, the resulting de Rham spaces are given t a as quotients by actions of germs of diagonals, instead of the formal m neighbourhoods of the diagonals. [ 2 Contents v 8 7 1 Radicals and reductions in C∞-algebra 4 2 1.1 Six radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6 1.2 Three reductions . . . . . . . . . . . . . . . . . . . . . . . . . 8 0 . 1 2 Two kinds of de Rham space 11 0 7 2.1 Regularity conditions . . . . . . . . . . . . . . . . . . . . . . . 11 1 2.2 De Rham spaces as colimits of pre-sheaves . . . . . . . . . . . 13 : v 2.3 Reformulation in terms of topological rings . . . . . . . . . . 16 Xi 2.4 De Rham groupoids . . . . . . . . . . . . . . . . . . . . . . . 20 r a 3 Some differential operators and further questions 24 A Proof or Lemma 3 26 B Proof of Proposition 8 28 Introduction As it is known ([4]), in algebraic geometry one can define connections, differential operators etc. without ever mentioning derivatives or any kind of 1 limiting procedure. One only needs tohave the notion of infinitesimal neighbourhoods given by nilpotent elements. Looking at what happens when one contracts such neighbourhoods, one arrives at objects that have some prescribed behaviour along these contractions. The results of contractions are usually called de Rham spaces, and the objects that live on them are called crystals. Choosing to work with linear objects (i.e. sheaves of modules) one arrives at D-modules and linear differential operators (see e.g. [5]). This technique can be applied also to differential geometry, once we use the theory of C -rings to take an algebraic-geometric approach (e.g. [9]). ∞ However, C -rings are much more than just commutative R-algebras, and ∞ thereismorethanoneway todefinecrystalsindifferential geometry because there is more than one notion of nilpotence. It was observed in [8] that apart from the usual nilpotent elements, C - ∞ rings can have -nilpotent ones, which are defined as follows: a A is ∞ ∈ -nilpotent, if the C -ring A a 1 obtained by inverting a is 0. Here it ∞ − ∞ { } is important that this inverting happens in the category of C -rings. For ∞ 1 example x C∞(R)/(e−x2)is -nilpotent, but notnilpotent. Clearly every ∈ ∞ nilpotent element is also -nilpotent. ∞ UsingtheusualnotionofnilpotenceonegetsdeRhamspacesthatcanbe described as quotients by actions of formal neighbourhoods of the diagonals. This is true in both algebraic and differential geometry. In the differential- geometric literature, however, one usually talks about jet bundles, instead of the formal neighbourhood of the diagonal and sheaves of modules over such neighbourhoods. The difference is in name only. Something completely different happens when we apply de Rham space formalism to contracting -nilpotent neighbourhoods. Instead of quotients ∞ by actions of formalneighbourhoods ofdiagonals we getquotients by actions of germs of diagonals. As in the formal case, these C -rings of germs carry ∞ a linear topology given by order of vanishing at the diagonal. Since in differential geometry there are many more orders of vanishing than just the finite ones, the infinitesimal theory we get here is much richer. An immediate benefit of this richer theory is having many more differential operators, than justthe polynomial ones. Another consequence becomes apparent when we consider the opposite procedure – summation. Instead of the usual Taylor series, that characterize behavior of functions in relation to algebraic monomials, we need to work with functions having arbitrary vanishing properties, i.e. we need to go totrans-series (e.g. [11])and beyond. Consider the deformation theory one gets from this: in addition to spec- 2 ifying behavior of functors with respect to nilpotent extensions, we should consider also -nilpotent extensions. Since all -nilpotent extensions add ∞ ∞ up to germs, the deformation theory necessarily goes beyond perturbations. In turn this leads to a completely new definition of derived geometry. Differ- ential gradedmanifolds orsimplicial C -ringswillnotsufficeanymore, since ∞ decomposition according to degree/simplicial dimension reflects decomposition according to finite orders of vanishing. The present paper is the first in a series examining this rich infinitesimal theory and possibly some of its applications. Here is the contents of the paper: InSection1.1weconsider6differentradicalsofidealsinC -rings. Three ∞ ofthemarewellknownincommutative algebra(nilradical, Jacobsonradical, and intersection of all maximal ideals having R as the residue field), while another two come from considering different Grothendieck topologies on the category of C -spaces. The main actor of this paper – the -radical – is ∞ ∞ specific to the C -algebra. ∞ In Section 1.2 we prove that just as nilradical satisfies the strong functoriality property, so does the -radical. We give proofs for some useful ∞ facts relating to algebra of these radicals, preparing the ground for de Rham spaces. BeforewecandefinedeRhamspacesforbothnilpotentand -reductions ∞ we introduce regularity conditions in terms of injectivity with respect to the reduction functors. This is done in Section 2.1. Then in Section 2.2 we define the two kinds of de Rham spaces. Here we also prove the central result that these de Rham spaces can be built using certain neighbourhoods of the diagonals. The main ingredient in the proof is the strong functoriality from Section 1.2. Section 2.3 is dedicated to presenting these neighbourhoods of the diagonals asspectra of C -rings equipped with linear topologies. Inthe nilpotent ∞ case these spectra are just the usual formal neighbourhoods, while in the - ∞ case they are the germs of diagonals. De Rham groupoids appear in Section 2.4. We show in particular that both in the nilpotent and the - cases the de Rham spaces we construct ∞ are weakly equivalent (as simplicial sheaves) tothe nerves of the corresponding de Rham groupoids. We show that often (e.g. for all manifolds) these nerves consist of the formal neighbourhoods and respectively of the germs of diagonals in all cartesian powers. Finally in Section 3 we look at the differential operators one obtains from - de Rham groupoids. We postpone a detailed analysis to another ∞ 3 paper, butwedoshowthattheseoperatorsgobeyondperturbation, i.e.they provideinfinitesimaldescriptionofdeformationsoftheidentity morphismon a manifold whose infinite jets at the identity vanish. Acknowledgements PartofthisworkwasdonewhenthefirstauthorwasvisitingMax-Planck InstitutfürMathematikinBonn,severalshortvisitstoOxfordwerealsovery helpful. Hospitality and financial support from both institutions are greatly appreciated. This work was presented in a series of talks in Göttingen, and the first author is grateful for the comments from C.Zhu and V.Pidstrygach. 1 Radicals and reductions in C -algebra ∞ Let C R be the category of C -rings, we denote by C R C R the ∞ ∞ ∞ fg ∞ ⊂ full subcategory of finitely generated C -rings. By definition (e.g. [9] §I.1) ∞ C R is the category of product preserving functors R Set, where R ∞ is the algebraic theory of C -rings, i.e. it is the categor∞y →having Rn ∞ ∞ { }n∈Z≥0 as objects and C -maps as morphisms. Objects of C R are quotients of ∞ ∞ fg C (Rn) by ideals ([9], §I.5). ∞ n 0 { } ≥ The category C R is both complete and co-complete (e.g. [1] Cor. 1.22, ∞ Thm. 4.5). We denote the coproduct in this category by . For a C -ring ∞ ⊗ A we denote by Spec(A) the corresponding object in the opposite category C Rop. And given C Rop we write C ( ) for the corresponding ob- ∞ ∞ ∞ X ∈ X jectofC R. Foranarbitrary C Rop wedenoteby therepresentable ∞ ∞ X ∈ X pre-sheaf homC∞Rop( , ): C∞Rop Set. − X → 1.1 Six radicals LetA C RbeaC -ring,asR containsthetheoryR ofcommutative, ∞ ∞ alg ∈ ∞ associative,unitalR-algebras,everyR -congruenceonAisgivenbyanideal ∞ of the underlying commutative R-algebra. The converse is also true (e.g. [9], prop. I.1.2) and we can identify R -congruences with ideals. There are ∞ several notions of radical ideals in this context, we consider 6 of them. Definition 1. Let A A be an ideal ≤ 1. the nilradical of A is n√ilA := f A k Z s.t. fk A , >0 { ∈ |∃ ∈ ∈ } 2. the -radical of A is ∞√A:= f A (A/A) f 1 = 0 , − ∼ ∞ { ∈ | { } { }} 4 3. the Jacobson radical of A is √JA := m, where M(A) is the set of T M(A) maximal ideals containing A, 4. the R-Jacobson radical of A is R√JA := m, where M (A) M(A) R T ⊆ MR(A) consists of ideals with R as the residue field, 5. theR-jetradicalofAis√jA := f A p: A R g A s.t. J (f)= { ∈ |∀ → ∃ ∈ ∞p J (g) , where J (g) is the infinite jet of g at p, ∞p } ∞p 6. the R-germ radical of A is √g A := f A p: A R g A s.t. f = p { ∈ |∀ → ∃ ∈ g , where g is the germ of g at p. p p } The nilradical, Jacobson and R-Jacobson radicals are borrowings from commutative algebra. The R-jetand R-germ radicals appear indefining var- ious Grothendieck topologies on C R op (e.g. [9] §III) and natural topolo- ∞ fg gies on C -rings ([3]). We will be mostly concerned with the -radical, ∞ ∞ that was introduced in [8] §2. First we need to compare these radicals. Proposition 1. For any A, A as above there is a sequence of inclusions n√ilA ∞√A √JA R√JA, ⊆ ⊆ ⊆ and each one of these inclusions can be strict. Proof. The only inclusion that is not obvious is ∞√A √J A. Let α A, and ⊆ ∈ suppose there is m M(A), s.t. α / m. Then [α] A/m is invertible, and ∈ ∈ ∈ hence (A/A) α 1 cannot be trivial, i.e. α / ∞√A. − { } ∈ In the following examples we put A := C (R). ∞ 1. Let m C (R) consist of functions having 0-jet at 0 R. Then ∞0 ⊂ ∞ ∈ C (R)/m = R[[x]] (Borel lemma), and hence nil m = m . On the ∞ ∞0 ∼ p ∞0 ∞0 other hand the filter V R f m , V = p f(p) = 0 coincides { ⊆ |∃ ∈ ∞0 { | }} with the filter of closed subsets of R containing 0. Therefore from Lemma 2.2 in [8] we conclude that ∞ m = m , the ideal of functions p ∞0 0 vanishing at 0. So nil m ( ∞ m . p ∞0 p ∞0 2. Let mg C (R) consist of functions having 0-germ at 0 R. Accord- ingto0[8⊂], pa∞ge 329 the ring C (R)/mg is local, and hence∈J mg = m . ∞ 0 p 0 0 On the other hand, using Lemma 2.2 from [8] we see that ∞ mg = mg. p 0 0 So ∞ mg ( J mg. p 0 p 0 5 3. LetA C (R)consistoffunctionshavingcompactsupport. Clearly cs ∞ ⊂ this is a proper ideal of C (R), and hence √JA = C (R). It is also ∞ cs ∞ 6 clear that M (A ) = , i.e. R√JA = C (R). So √JA ( R√J A . R cs cs ∞ cs cs ∅ The other two radicals do not fit in this chain of inclusions. It is clear that we always have √g A √jA R√JA, and these inclusions can be strict (e.g. applied to mg C (⊆R)). Th⊆e following examples show that this is the 0 ⊂ ∞ most we can say in the general situation. Example 1. Consider(x2) C (R). Clearly nil (x2)= m ,while j (x2)= ∞ 0 ⊂ p p (x2). Thus j (x2) ( nil (x2). p p Consider again the ideal A C (R) of functions with compact sup- cs ∞ ⊂ port. It is easy to see that √g A = C (R), while as before √J A = C (R). cs ∞ cs ∞ 6 Thus √JA ( √g A . cs cs In summary we have the following relationship between the six radicals where each inclusion can be strict: n√ilA ∞√A √JA R√JA √jA √g A. (1) ⊆ ⊆ ⊆ ⊇ ⊇ Wefinishthissectionwithausefulcharacterization of -radicals. Itisinan ∞ obvious sense a generalization of the nilpotence condition. For any C -ring ∞ A, we can identify A with homC∞Rop(Spec(A),R), and then, going to the category of pre-sheaves on C Rop, the natural map R hom(R,R) R, ∞ × → evaluated at Spec(A), gives a morphism of C -rings ∞ A (A C (R)) // A, ∞ × ⊗ which we denote by (g,f) f(g) for f A C (R), g A. For example, ∞ 7→ ∈ ⊗ ∈ denoting a generator of C (R) by x, we can choose f := xn A C (R) ∞ ∞ ∈ ⊗ forsome n N, and thenf(g) = gn sincethe operation (g,f) f(g) canbe ∈ 7→ written as the composition Spec(A) Γg Spec(A) R f R, where Γ is the g −→ × −→ graph of g. Clearly f = xn factors through the projection Spec(A) R R, × → since xn is constant along Spec(A). Altogether we have that f(g) = gn. Definition 2. Let A C R, an element g A is -nilpotent, if there is ∞ ∈ ∈ ∞ f A C (R), s.t. f(g) =0 and f becomes invertible in A C (R 0 ). ∞ ∞ ∈ ⊗ ⊗ \{ } This notion has an obvious extension to non-trivial ideals: let A A ≤ be any ideal, g A is -nilpotent relative to A, if there is f A C (R), ∞ ∈ ∞ ∈ ⊗ 6 s.t. f(g) A and f becomes invertible in (A/A) C (R 0 ). However, ∞ ∈ ⊗ \{ } it brings nothing new: let A := A/A, denote the projection φ: A A, we → have a commutative diagram in C Rop: ∞ Spec(A) Γφ(g) // Spec(A) Rf◦(Φ×IdR) //R × Φ Φ IdR = (cid:15)(cid:15) × (cid:15)(cid:15) Spec(A) // Spec(A) R //R, Γg × f whereΦcorrespondstoφ. Theng is -nilpotent relativetoA, iff Γ Φ = g ∞ ◦ ◦ 0. Since φ is surjective and C (R) is free, it is clear that this notion is ∞ equivalent to -nilpotence of φ(g) in A. ∞ One can say that for such g the “degree of nilpotence” is at most f, meaning the rate of vanishing of f at Spec(A). Notice that this “degree of nilpotence” is not only possibly infinite, but can also vary on Spec(A). Proposition 2. Let A C R, and let A A be any ideal. Then ∞ ∈ ≤ g ∞√A the class of g in A/A is -nilpotent. ∈ ⇐⇒ ∞ Proof. First we bring everything to A/A using the following simple lemma. Lemma 1. Let φ: A A be a surjective morphism in C R, and let 1 2 ∞ → A A be an ideal. Then φ 1( ∞√A) = ∞ φ 1(A). ≤ 2 − p − Proof. For any a A consider the commutative triangle of C -morphisms 1 ∞ ∈ A ww♦♦♦♦♦♦♦♦♦♦♦♦ 1 ❖❖❖❖❖❖❖❖❖❖❖❖'' (A /φ 1(A)) a 1 //(A /A) (φ(a) 1 . 1 − − 2 − { } { } Both arrows out of A are initial among those C -morphisms that kill 1 ∞ φ 1(A) and invert a. Therefore the horizontal arrow must be an isomor- − phism. Hence a ∞ φ 1(A) φ(a) ∞√A a φ 1( ∞√A). ∈ p − ⇔ ∈ ⇔ ∈ − Now we can assume A= 0. Let x be a generator of C (R), g A we have ∞ ∀ ∈ A{g−1} ∼= ((A⊗C∞(R))/(g−x)){x−1}. Therefore g ∞√0 x ∞ (g x). The latter is equivalent to existence ∈ ⇔ ∈ p − of some f (g x), that becomes invertible in (A C (R)) x 1 ([8] page ∞ − ∈ − ⊗ { } 329). Composing Spec(A) Γg Spec(A) R f R we get f(g) = 0. Con- −→ × −→ versely, from f(g) = 0 we get f (g x), as f(g) = 0 means f has to be in ∈ − the ideal of the graph of g, which is (g x). − 7 1.2 Three reductions To define reductions we need to investigate functoriality properties of the radicals. We start with the following elementary lemma. Lemma 2. Let φ: A A be a morphism in C R, and let A A , 1 2 ∞ 1 1 → ≤ A A be any ideals, s.t. φ(A ) A . Then 2 2 1 2 ≤ ⊆ nil A φ 1( nil A ), ∞ A φ 1( ∞ A ), RJ A φ 1(RJ A ). p 1 ⊆ − p 2 p 1 ⊆ − p 2 p 1 ⊆ − p 2 Proof. The nilradical of A is the intersection of prime ideals containing A, and pre-images of prime-ideals are again prime ideals. Maximal ideals with residue field R are kernels of surjective maps onto R, and any C -morphism to R is surjective (elements of R are the structure ∞ constantsofthetheoryofC -rings). Thereforepre-imagesofmaximalideals ∞ with residue field R are again such ideals. According to [8] page 329, α ∞√A , iff β A , s.t. β becomes invert- 1 1 ∈ ∃ ∈ ible in A α 1 . Then φ(β) A , and we conclude that φ(α) ∞√A . 1 − 2 2 { } ∈ ∈ In fact, for the nilradical and - radical a much stronger result is true. ∞ Proposition 3. Let φ: A A be a morphism in C R. Then 1 2 ∞ → φ 1( n√il0) = nil Ker(φ), φ 1(∞√0) = ∞ Ker(φ). − − p p Proof. For the nilradical the statement is clear: φ(a)k = 0 ak Ker(φ). ⇔ ∈ The case of -radical. Let a be a set of generators of A as a i i I 1 ∞ { }∈ C -ring, and let b be a set of generators of A obtained by enlarging ∞ j j J 2 φ(a ) . Letψ{: C} ∈(RI)֒ C (RJ)betheC -morphism corresponding i i I ∞ ∞ ∞ t{o Ψ:}R∈J RI given by I→ J.1 Clearly φ, ψ make up a commutative → ⊆ diagram with the projections π : C (RI) A , π : C (RJ) A . 1 ∞ 1 2 ∞ 2 → → Weclaimitisenoughtoprovethepropositionforψ. Indeed,fromLemma 2 we know that ∞ Ker(φ) φ 1( ∞√0). This inclusion is an equality, iff − p ≤ π 1(∞ Ker(φ)) = π 1(φ 1( ∞√0)), but π 1(φ 1( ∞√0)) = ψ 1(π 1( ∞√0)), 1− p 1− − 1− − − 2− and from Lemma 1 we know that π 1( ∞ Ker(φ)) = ∞ π 1(Ker(φ)) = 1− p p − ∞ ψ 1(π 1(0)) and ψ 1(π 1( ∞√0)) = ψ 1( ∞ π 1(0)). q − 2− − 2− − q 2− So it is enough to prove that A C (RJ) ∞ ψ 1(A) = ψ 1(∞√A). ∀ ≤ ∞ p − − From Lemma 2 we already have one direction, it remains to show that ∞ ψ 1(A) ψ 1( ∞√A). (2) p − ⊇ − 1Wesay f ∈C∞(Rn), if f factors into a projection RI →Rk and a smooth Rk →R. 8 For an a C (RI) being in ψ 1(∞√A) is equivalent to existence of f A, ∞ − ∈ ∈ s.t. p ψ(a)(p) = 0 = p f(p)= 0 RJ ([8]Lemma 2.2). We canchoose finit{e s|ubsets I I}, J{ |J, s.t. I } ⊆J and a C (RI′), f C (RJ′). ′ ′ ′ ′ ∞ ∞ ⊆ ⊆ ⊆ ∈ ∈ We denote m := I , m+n:= J , A := C (Rm+n) A. ′ ′ ′ ∞ | | | | ∩ The statement a ∞ ψ 1(A) is equivalent to existence of h ψ 1(A), ∈ p − ∈ − s.t. p a(p) = 0 = p h(p) = 0 . The following lemma implies then (2). { | } { | } Lemma 3. Let Ψ: Rm+n Rm be the projection. Let V Rm be a closed → ⊆ subset. Let f C (Rm+n) be such that p f(p) =0 = Ψ 1(V). There are ∞ − ∈ { | } g C (Rm+n), h C (Rm), s.t.ψ(h) = fg, and x g(x) = 0 =Ψ 1(V). ∞ ∞ − ∈ ∈ { | } One proves this lemma by finding an h C (Rm), s.t. p h(p) = 0 = ∞ ∈ { | } V andh ΨvanishesonΨ 1(V)fasterthanf. Thisiseasygiventheclassical − ◦ theory of infinite orders ([6]). Details are in Appendix A. A similar statement forthe R-Jacobson radical is wrong in general. Con- sider C (R)/mg C (R 0 )/(mg), where mg C (R) consists of func- ∞ 0 → ∞ \{ } 0 0 ≤ ∞ tions having 0 germ at 0 R. This morphism is injective because a smooth ∈ function in a punctured neighbourhood can be extended to the puncture in at mostoneway, yet R√J 0inthe domain ism /mg, the idealof functions that 0 0 vanish at 0 R, while in the codomain it is the entire ring. ∈ The following lemma answers the natural question of whether by composing different radicals we get anything new. Lemma 4. Let A C R, and let A A be any ideal. Then ∞ ∈ ≤ nqil n√ilA= n√ilA, ∞q ∞√A= ∞√A, RqJ R√JA= R√JA, ∞q n√ilA= ∞√A, RqJ ∞√A = R√JA, RqJ n√ilA = R√JA, nqil ∞√A= ∞√A, nqil R√JA = R√JA, ∞q R√JA = R√JA. Proof. In the first row the only non-trivial statement is ∞ ∞√A = ∞√A. p According to Proposition 2, if a ∞ ∞√A, there is f A C (R), s.t. ∈ p ∈ ⊗ ∞ f becomes invertible in A C (R 0 ) and f(a) ∞√A. In turn this ∞ ⊗ \ { } ∈ implies existence of g A C (R) with the same invertibility properties, ∞ ∈ ⊗ and s.t. g(f(a)) A. Consider f and g as elements of A C (R2), via ∞ ∈ ⊗ the two different inclusions A C (R) ⇒ A C (R2). Then we see that ∞ ∞ ⊗ ⊗ g(f(a)) = (g(f))(a). Indeed, let x,y be generators of C (R2), then g(f(a)) ∞ is the class of g modulo (a x)+(f y), which is the same as (g(f))(a)) − − (reversing the order of division). Inverting x implies inverting f, and then 9 modulo (f y) it means inverting y, which implies inverting g, thus g(f) − becomes invertible in A C (R 0 ) and we conclude that a ∞√A. ∞ ⊗ \{ } ∈ Each radical preserves the inclusion relation between ideals (Lemma 2), hence the other rows follow from the first because of Proposition 1. Lemma 2 implies that A A/ n√il0, A A/ ∞√0, A A/ R√J 0 canoni- 7→ 7→ 7→ cally extend to functors R ,R ,R : C R C R. nil pt ∞ ∞ ∞ −→ Definition 3. We will call these functors the nilpotent, - and point reduc- ∞ tions. If nilpotent, - or point radical of 0 is again 0, the C -ring will be ∞ ∞ called respectively reduced, - reduced and point reduced. ∞ The corresponding full subcategories of C R consisting of reduced C - ∞ ∞ rings will be denoted by C R , C R , C R . It is clear (Proposition 1) ∞ nil ∞ ∞ pt ∞ that we have a sequence of adjunctions C∞R oo Rnil // C∞Rnil oo R∞ //C∞R oo Rpt //C∞Rpt ∞ with the right adjoints being inclusions of full subcategories. Lemma 4 tells us of course that R R = R , R R = R . We finish this section nil pt pt ∞ ◦ ∞ ◦ ∞ with the following simple lemma. Lemma 5. Let A C R, and let A ,A A be any ideals, then ∞ 1 2 ∈ ≤ nil A A = nil A nil A , ∞ A A = ∞ A ∞ A . p 1∩ 2 p 1∩ p 2 p 1∩ 2 p 1∩ p 2 Let A ,A C R, and let A A , A A be any ideals. Then 1 2 ∞ 1 1 2 2 ∈ ≤ ≤ nil (A )+(A ) (nil A )+(nil A ), ∞ (A )+(A ) (∞ A )+( ∞ A ), p 1 2 ≥ p 1 p 2 p 1 2 ≥ p 1 p 2 where (A ) A A is generated by A A ֒ A A and so on. 1 1 2 1 1 1 2 ≤ ⊗ ≤ → ⊗ Proof. Since A A A , A A A , we have ∞√A A ∞√A , 1 2 1 1 2 2 1 2 1 ∩ ≤ ∩ ≤ ∩ ≤ ∞√A A ∞√A (Lemma 2). We choose a surjective α: C (RS) A. 1 2 2 ∞ ∩ ≤ → Since α 1( ∞√ ) = ∞ α 1( ) (Lemma 1), we can assume A = C (RS). − − p − − ∞ According to [8], Lemma 2.2 f ∞√A ∞√A , iff g A , g A , 1 2 1 1 2 2 ∈ ∩ ∃ ∈ ∃ ∈ s.t. p g (p)= 0 = p f(p)= 0 = p g (p) = 0 . Then g g A A 1 2 1 2 1 2 { | } { | } { | } ∈ ∩ and p (g g )(p) = 0 = p f(p) = 0 , therefore f ∞√A ∞√A implies 1 2 1 2 { | } { | } ∈ ∩ f ∞√A A . The nilpotent case is well known. 1 2 ∈ ∩ The second statement is obvious in the nilpotent case. In case of ∞√ , − choosing generators we can assume A , A are free (Lemma 1), and then the 1 2 claim becomes obvious due to characterization of f ∞√A above. ∈ 10