Differentiable rings, analytic rings, Nash rings and their applications to singularity theory (Singularity theory of differential maps and its applications)
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(2) 23. In. §4, we show the properties for \mathrm{C}^{\infty} ‐rings from the properties of \mathrm{C}^{\infty} ‐functions and its germs. First, have a localization of \mathrm{C}^{\infty}(M) at p as a set \mathrm{C}_{p}^{\infty}(M) of germs of \mathrm{C}^{\infty} ‐functions at p Second, we compare the difference of \mathbb{R} ‐derivations and \mathrm{C}^{\infty} ‐derivations of \mathrm{C}^{\infty}-\mathrm{r}\dot{\mathrm{m} gs.. we. .. In. §5,. germs. In. §6,. show the. we. we. for. properties. have that localizahons of. we. show the. from the. analytic‐rings. \mathrm{C}^{ $\omega$}(M). properties. of. properties. and its. analytic‐functions. p is not isomorphic to \mathrm{C}_{p}^{ $\omega$}(M) for Nash‐rings. From [4], we introduce the at. .. derivations, i.e. \mathbb{R} ‐derivations which satisfies Leibniz rule for Nash‐functions.. properties. of Nash‐. The kinds of functions. 2 2.1. Definition of functions. For. \mathrm{a}=($\alpha$_{1}, \ldots,$\alpha$_{n})($\alpha$_{i}\in\{0\}\mathrm{U}\mathrm{M}). Definition 1. ,. define. | $\alpha$|=$\Sigma$_{\mathrm{i}=1}^{n}$\alpha$_{i} and. I. A''\mathrm{C}^{\infty} ‐function(A smooth. for any positive integer r). \neq^{\mathrm{a}_X^{$\alph$}^{ \alph$} :=\displaytle\frac{\partil^{$\alph$_{1}+.\cdot\cdot.\cdot+$\alph$_{n}f \partilx_{1}^$\alph$_{1}\cdot8x_{n}^$\alph$_{n}. :. 0 and. $\alpha$!. :=$\Pi$_{i=1}^{n}$\alpha$_{i} !.. function) f : \mathbb{R}^{n}\rightarrow \mathbb{R} is afunction which satisfies:. $\alpha$\in(\{0\}\cup \mathbb{N})^{n}. \mathbb{R}^{n}\rightarrow \mathbb{R} which is continuous. with. on. | $\alpha$|\leq r. ,. there exists the r‐th. partial derivative. \mathbb{R}^{n}.. 2. A''\mathrm{C}^{ $\omega$} ‐fUnction(An. analyt.c function) f : \mathbb{R}^{n}\rightarrow \mathbb{R} is a \mathrm{C}^{\infty} ‐funciion which satisfies: for any p\in \mathbb{R}^{n} there exists an open neighborhood U such that $\Sigma$_{$\alpha$_{ $\alpha$}^{\urcorner} ^{1}.\neq_{x^{ $\alpha$} ^{\partial^{ $\alpha$} (p)(x-p)^{ $\alpha$} : U\rightarrow \mathbb{R} formally converges to f on U. ,. 3. For $\gamma$=0. ,. 1,2,. \cdots. ,. \infty,\mathrm{w},. satisfies:. there exists. There exists. real. a non‐zero. C^{ $\omega$} ‐Nashfunctions. a''\mathcal{N}^{r} ‐functionla. polynomial P(x,y) \in \mathbb{R}[x,y] (resp.. 1 Let $\eta$ be. $\eta$^{(r)}(x). =. $\eta$^{(r)}. $\eta$^{(r)}. function defined. The derivation. analytic function. derivation. a. of $\eta$. $\eta$(x). as. is. $\eta$'(x). =. =. is. \{ 0P_{r}(\displayst le\frac{1}x)e^{-\frac{1}x}. (x>0)_{for} (x\leq 0). f. :. \mathbb{R}^{n}\rightarrow \mathbb{R} is. P(x,f(x)). =0. a. a. real. \mathrm{C}^{r} ‐function which. on. \mathrm{C}^{W} ‐function) which is not the C^{ $\omega$} ‐function. a. function) from following examples.. Example 1. such that. Nash‐functions.. as. C^{\infty} ‐function. a. \mathrm{C}^{r} Nash function). \mathbb{R}^{n}(\mathrm{f}8lJ We call .. (resp.. the \mathcal{N}^{$\omega$_{-}. \left\{\begin{ar ay}{l} e^{-\frac{1}{X} (x>0)\ . $\eta$ is the \mathrm{C}^{\infty}- function but not he\ 0 (x\leq 0) \end{ar ay}\right. \left\{\begin{ar ay}{l } \frac{1}{x^{2} e^{-\frac{1}{ $\chi$} & (x>0) and continuous. Then, the r- th\ 0 & (x\leq 0) \end{ar ay}\right. \{ -x^{2}P_{r-1}'(x)+x^{2}P_{r-1}(x) $\eta$^{(r)}, $\eta$^{(r)}(0)=0.. polynomial P_{r}(x). =. x^{2}. (r=1) (r\geq 2). is continuous for any r\geq 0 Therefore, $\eta$ is a \mathrm{C}^{\infty} ‐function. For any r‐th derivation Therefore, \displaystyle \sum_{=0^{\urcorner_{n} }^{\infty 1}.$\eta$^{(r)}(0)x^{n} is not equal to $\eta$ on any neighborhood at 0. .. real‐polynomial p(x_{1},\ldots,x_{n}) : \mathbb{R}^{n}\rightarrow \mathbb{R} is a Nashfunction. p(x) is analytic and for a non‐zero real‐polynomial P(x,y) :=y-p(x) P(x,f(x))=0.. 2. A. ,. 3.. f (x):=\sqrt{1+x^{2}}:\mathbb{R}\rightarrow \mathbb{R}isnotareal-polynomid xAfunction f(x)isa nalytic ndforanon-zeroreal-polynomial (x,y):=y^{2}-(1+ utaNash\ovalbox{\t \small REJECT}),P(x,f(x) =0 b. 4.. 2.2. unction.. P. a. Afunction f(x) :=e^{X} : \mathbb{R}\rightarrow \mathbb{R} is analytic but not a Nashfunction. f(x) is analytic and there does not exists a non‐zero real polynomial P(x,y). such that. .. P(x,f(x))=0.. Nash manifolds. From Introduction in. [8],. a. subset of \mathbb{R}^{n} is called. form. semialgebraic subset if it is a finite union of sets of the. \{x\in \mathbb{R}^{n}|f_{i}(x)=0,g_{i}(x)>0\forall i=1, \cdots,k,j=1,\ldots,l\} where f_{1}. ,. \cdots. ,. f_{k},g_{1}. ,. \cdots. ,. g_{l}. are. real. polynomial functions on \mathbb{R}^{n}..
(3) 24. Definition 2 \langle[8] ) Let M be a. topological space.. 1. A \mathrm{C}^{ $\omega$} ‐Nash manifold (A\mathcal{N}^{W} ‐manifold) is. a topological space M if there exists an open finite cover finite family \{V_{ $\alpha$}\}_{ $\alpha$} of open semialgebraic sets of \mathbb{R}^{n} and homeomorphisms $\phi$_{$\alpha$} : U_{ $\alpha$}\rightar ow V_{ $\alpha$} that $\psi$_{$\beta$^{\circ} q_{\mathrm{J}_{$\alpha$}^{-1} |_{$\psi$_{$\alpha$}(U_{$\alpha$}\capU_{$\beta$}) : $\varphi$_{ $\alpha$}(U_{ $\alpha$}\cap U_{ $\beta$}) \rightar ow$\psi$_{ $\beta$}(U_{ $\alpha$}\cap U_{ $\beta$}) is a \mathrm{C}^{$\gam a$} Nash diffeomorphism for any $\alpha,\ \beta$. \{U_{ $\alpha$}\}_{ $\alpha$} of M. such. a. ,. (U_{ $\alpha$}\cap U_{ $\beta$}\neq\emptyset). .. 2. A \mathrm{C}^{ $\omega$} Nash function is. for any. \mathrm{C}^{W} ‐function. f. :. M\rightarrow \mathbb{R} such that. f\circ$\phi$_{\overline{ $\alpha$} ^{1}. :. V_{ $\alpha$}\rightar ow \mathbb{R} is. a. \mathrm{C}^{ $\omega$} ‐Nashfunction. \mathcal{N}^{ $\omega$}(M). (resp. a \mathrm{C}^{W} ‐manifold, a\mathcal{N}^{ $\omega$} ‐manifold) \mathrm{M} define \mathrm{C}^{\infty}(M) (resp. \mathrm{C}^{ $\omega$}(M) of \mathrm{C}^{\infty} ‐functions (resp. \mathrm{C}^{$\omega$} ‐functions, \mathrm{C}^{$\omega$} ‐Nashfunctions) on M.. I. For a \mathrm{C}^{\infty} ‐manifold. Definihon 3. is. a. set. \mathrm{C}_{p}^{\infty}(M). (resp. \mathrm{C}_{p}^{ $\omega$}(M), tions) at p on M.. 2.. a. $\alpha$.. From the definition of. ,. \mathcal{N}_{p}^{ $\omega$}(\mathrm{M}). is. a. of germs of \mathrm{C}^{\infty} ‐functions (resp.. set. ,. C^{W} ‐functions, \mathrm{C}^{ $\omega$} ‐Nashfunc‐. functions, we have a following property.. \mathrm{C}^{\infty}(M)\supset \mathrm{C}^{W}(M)\supset \mathcal{N}^{ $\omega$}(\mathrm{M}) , \mathrm{C}_{p}^{\infty}(\mathrm{M})\supset \mathrm{C}_{p}^{ $\omega$}(M)\supset \mathcal{N}_{p}^{ $\omega$}(\mathrm{M}). .. We write \mathcal{K} for C^{\infty}, C^{ $\omega$}, \mathcal{N}^{ $\omega$} and \mathcal{K}^{ $\omega$} for \mathrm{C}^{ $\omega$}, \mathcal{N}^{ $\omega$}. ,. Definitions of. 3. rings. and their derivations. From Proposition 1.6.2. and 1.6.3. in [1], analytic functions on \mathbb{R}^{n} are closed under sums, products, and partial derivations. Moreover, analytic functions are closed under compositions from Proposition 1.6.7 in. [1].. From. Proposition. 3.1. in. [7], Nash functions. on. \mathbb{R}^{n}. are. closed under sums,. derivations.. For \mathrm{C}^{\infty} ‐functions (resp. \mathrm{C}^{ $\omega$} ‐functions, and Nash-\mathrm{f}\mathrm{u}\mathrm{n} $\alphaions), $ uct, and compositions of functions as a following proposition.. Proposition 1 \mathrm{C}^{\infty}(\mathbb{R}^{n}) (resp. \mathrm{C}^{ $\omega$}(\mathbb{R}^{n}), \mathcal{N}^{ $\omega$}(\mathbb{R}^{n}). f+g,f\cdot g. ho. ,. for any f,g,c_{1} algebra.. ,. \cdots. ,. \mathcal{K}(\mathbb{R}^{n}). c_{m} \in. and h \in. we can. is closed under sums,. (cl,. \cdots. ,. c_{m}. define. products,. operations. and. partial. of sum,. products, and compositions,. prod‐. i.e.. ) \in \mathcal{K}(\mathbb{R}^{n}). \mathcal{K}(\mathbb{R}^{m}) Therefore, C^{\infty}(\mathbb{R}^{n}) (resp. \mathrm{C}^{ $\omega$}(\mathbb{R}^{n}), \mathcal{N}^{w}(\mathbb{R}^{n}) .. 3.1. The definition of. From. Proposition 1, \mathrm{C}^{\infty} ‐functions (resp. analytic‐functions, Nash‐functions). is. an. \mathbb{R}-. rings on. 1\mathrm{R}^{n}. are. closed under. c_{m} ) \mapsto f\circ (cl, c_{m} ) by a \mathrm{C}^{\infty} ‐funchon (resp. an analytic‐function, a Nash‐ composition (cl, function) f on \mathbb{R}^{m} A \mathrm{C}^{\infty}-\mathrm{r}\dot{\mathrm{m} \mathrm{g} is defined in Definition 2.1. [5] by \mathrm{C}^{\infty} ‐functions. We define analytic‐rings. the. \cdots. \cdots. ,. ,. .. and. Nash‐rings. as same as. \mathrm{C}^{\infty}-\mathrm{r}\dot{\mathrm{m} gs with the. following definition.. \mathrm{C}^{\infty}-\mathrm{r}\dot{\mathrm{m} \mathrm{g} (differentiable ring) (resp. \mathrm{C}^{$\omega$} ‐ring(analytic‐ring), \mathcal{N}^{ $\omega$} ‐ring (Nash‐ring) satisfies that: for any l\in\{0\}\cup \mathbb{N} and any \mathrm{C}^{\infty} ‐map (resp. \mathrm{C}^{$\omega$} ‐map, \mathcal{N}^{ $\omega$} ‐map) f : \mathbb{R}^{l}\rightar ow \mathbb{R} (if l=0, f is a constant number of \mathbb{R}), there exists an operation $\Phi$_{f}^{\mathrm{C} : \mathfrak{C}^{l}\rightar ow \mathb {C} such fhat. Definition 4. ) is. a. 1. A. set\not\subset which. (a) for any k\in \{0\}\cup \mathrm{M} and any \mathrm{C}^{\infty} ‐maps (resp. \mathrm{C}^{ $\omega$} ‐maps, \mathcal{N}^{ $\omega$} ‐maps) g. (i=1, \cdots,k). :. \mathbb{R}^{k}\rightar ow \mathbb{R} and f_{i}. :. \mathbb{R}^{l}\rightar ow \mathbb{R}. ,. $\Phi$_{g}^{\mathb {C}($\Phi$_{f 1}^{\not\subset} (cl, (b) for all mojections. \cdots. ,. c_{l}. $\pi$_{i} (xl,. ..,. \cdots. ,. $\Phi$_{f k}^{\mathrm{C} (cl,. x_{l} ). \cdots. ,. c_{l}) ). =$\Phi$_{g\circ(f )}^{\mathrm{C} (c_{1}, \cdots, c_{l}) for all c_{1},. =x_{i}(i=1, \cdots,l). $\Phi$_{$\pi$_{i}^{\not\subset} (cl,. \cdots. ,. ,. c_{l} ) =c_{\mathrm{i}} for dl c1,. \cdots. ,. c_{l}\in \mathrm{C}.. \cdots,. c_{l}\in \mathrm{C},.
(4) 25. 2. A. morphism between \mathcal{K} ‐rings \mathrm{C},\mathfrak{D}. $\Phi$_{f}^{\mathfrak{D} ( $\phi$ (c1), We. \cdots. ,. $\psi$(c_{n}) ). is. a. $\psi$:\not\subset\rightar ow \mathfrak{D}. map. = $\psi$ 0$\Phi$_{f}^{\not\subset} (cl,. \cdots. such that. c_{n} ) for all. ,. f\in \mathcal{K}(\mathbb{R}^{n}),c_{1}. ,. \cdots. ,. c_{n}\in\not\subset.. give examples of rings and homomorphisms.. Example 2. 1. Let \mathbb{R} be a set. f\in \mathcal{K}(\mathbb{R}^{l}) 2. Let M be. a. of real numbers.. \mathbb{R} has. a. structure. of \mathcal{K} ‐ring by the operation. $\Phi$_{[}^{\mathb {R}. \mathbb{R}^{l}\rightar ow \mathbb{R} for. :. as. $\Phi$_{f}^{\mathrm{R} (r_{1},\ldots,r_{l}) :=f (rl, \mathcal{K} ‐manifold.. f\in \mathcal{K}(\mathbb{R}^{l}) as. \mathcal{K}(M). $\Phi$_{f}^{\mathcal{K}(M)} (cl,. has. \cdots. structure. a. ,. c_{l} ). \cdots. r_{l} ) for all r_{1}. ,. ,. \cdots. r_{l}\in \mathbb{R}.. ,. of \mathcal{K} ‐ring by the operation. :=f\circ (cl,. \cdots. ,. c_{l} ) for all c_{1} ,. \cdots. ,. $\Phi$_{f}^{\mathcal{K}(\mathrm{M}). :. \mathcal{K}(M)^{l}\rightar ow \mathcal{K}(\mathrm{M}) for. c_{l}\in \mathcal{K}(M). .. f:M\rightarrow N be a \mathcal{K} ‐mapp ing of \mathcal{K} ‐manifolds. Its pullback f^{*}:\mathcal{K}(N)\rightarrow \mathcal{K}(M) defined as f^{*}(c) :=c\circ f(c\in \mathcal{K}(N)) is a morphism of \mathcal{K} ‐rings.. 3. Let. Definition 5 Let ￠ be. \mathcal{K} ‐ring. An \mathbb{R} ‐point of \not\subset is. a. \mathcal{K}-rings. Example 3. Let M be a \mathcal{K} ‐manifou and p. a. defined. point of M. \mathcal{K}(M). as a. has. an. surjective homomorphism \mathbb{R} ‐point e_{p}. e_{p}(f)=f(p) for any f\in \mathcal{K}(M) 3.2. The \mathbb{R} ‐algebra structure of \mathcal{K} ‐nngs. From. Proposition 1,. :. p. :. \mathrm{C}. \mathbb{R}. \rightarrow. of. \mathcal{K}(M)\rightarrow 1\mathrm{R} by the operation. .. for any \mathcal{K} ‐manifold \mathrm{M} the \mathcal{K}-\mathrm{r}\dot{\mathrm{m} \mathrm{g}\mathcal{K}(M) is the \mathbb{R}‐algebra. As same as \mathcal{K}(\mathrm{M}) any the natural \mathbb{R} ‐algebra structure. From the operations $\Phi$_{f} of \mathcal{K}‐rings in Definition 4, define of the \mathbb{R} ‐algebra as ,. ,. \mathcal{K}-\mathrm{r}\dot{\mathrm{m} \mathrm{g}\not\subset has. operations. We. \bullet. the addition. \bullet. the. .. the scalar. see. on. \not\subset. by c+c'. multiplication on. :=$\Phi$_{(x,y)\mapsto x+y}(c,c'). \mathb {C} by c\cdot c'. multiplication by. that elements 0 and 1 in ￠. \bullet. 0_{\mathrm{C} :=$\Phi$_{\emptyset\mapsto 0}(\emptyset) and. \bullet. 1_{\mathrm{C} :=$\Phi$_{\emptyset\mapsto 1}(\emptyset). :=$\Phi$_{(x,y)\mapsto xy}(c,c'). $\lambda$\in \mathbb{R} by $\lambda$ c are. ,. ,. and. :=$\Phi$_{x\mapsto $\lambda$ x}(c). .. given by. .. An ideal of the. \mathrm{C}^{\infty}-\mathrm{r}\dot{\mathrm{m} \mathrm{g} is defined as an ideal of the commutative \mathbb{R}‐algebra ([5]). Then, an ideal of the analytic‐ring (resp. Nash‐ring) is defined as an ideal oỈ the commutative \mathbb{R}‐algebra as same as \mathrm{C}^{\infty}-\mathrm{r}\dot{\mathrm{m} gs. We have Hadamards Lemma for \mathrm{C}^{\infty} ‐functions, \mathrm{C}^{ $\omega$} ‐functions, and Nash‐functions as a following corollary. Corollary 1. For any \mathcal{K} ‐functions. f\in \mathcal{K}(\mathbb{R}^{n}). ,. there exists n-\mathcal{K} ‐functions g_{1}. f(x+y)-f(x)=\displaystyle \sum_{i=1}^{n}y_{i}g_{i}(x,y) for Therefore, we define \not\subset be. g_{1}. ,. a. \cdots. a. all. as an. \cdots. ,. g_{n}\in \mathcal{K}(\mathbb{R}^{2n}). such that. x=(x_{1},\ldots,x_{n}),y=(y_{1},\ldots,y_{n})\in \mathbb{R}^{n}.. quotent \mathcal{K}-\mathrm{r}\dot{\mathrm{m} \mathrm{g} by an ideal of the \mathcal{K} ‐rmg. \mathcal{K} ‐ring and I\subset \mathrm{e} be an ideal g_{n}\in \mathcal{K}(\mathbb{R}^{2n}) such that. ,. \mathbb{R} ‐module. From. Definition 2.7. in. as same as. Corollary 1, for any f\in \mathcal{K}(\mathbb{R}^{n}). ,. ,. $\Phi$_{f}^{ $\epsilon$}(c_{1}+i_{1}, \ldots,c_{n}+i_{n})-$\Phi$_{f}^{\mathb {C} for any c_{1} ,. \cdots. ,. c_{n}\in \mathrm{C} and i_{1}. ,. \cdots. ,. i_{n}\in 1 We .. $\Phi$_{f}^{\mathrm{C}/\mathrm{J} (c_{1}+1, \ldots,c_{n}+I) :=$\Phi$_{f}^{\not\subset} (cl,. (cl,. can. \cdots. ,. \cdots. ,. c_{n}. define. c_{n}. ). a. ). =\displaystle\sum_{j=1}^{ni_{j}$\Phi$_{g i}^{\not\subet}. (cl,. \cdots. ,. c_{n} ,. quotient \mathcal{K}-\mathrm{r}\dot{\mathrm{m} \mathrm{g}e/I. i\mathrm{l}. ,. \cdots. ,. i_{n} ). as. +I for any f\in \mathcal{K}(\mathbb{R}^{n}),c_{1}+I. ,. \cdots. ,. [5].. Let. there exists. c_{n}+I\in\not\subset/I..
(5) 26. Localizations and local. 3.3. Definition 6 1. We call. \mathcal{K} ‐ring. a. with following. \mathb {C}_{p}. (a) There exists. rings. for \mathrm{C}^{\infty} ‐rings) Let\not\subset be. (Joyce [5]. a. \mathcal{K} ‐ring and p. an. localization. properties a. \mathbb{R} ‐point. of\not\subset at. of ￠.. p.. unique morphism $\pi$_{ $\rho$} : \mathrm{C}\rightar ow \mathrm{C}_{p} such that. a. $\pi$_{\mathrm{p} (s). is invertible in. \mathrm{t}_{p} for all s\in p^{-1}(\mathbb{R}\backslash \{0\}). (1). .. (b) If there exists a morphism $\phi$ :\not\subset\rightarrow \mathfrak{D} which satisfies (1), there exists a unique morphism ỷpp such that. $\psi$_{p}\circ$\pi$_{p}= $\phi$.. 2. A \mathcal{K} ‐ring \mathrm{C} is called. Any localization \mathrm{e}_{p}. \mathcal{K} ‐local. a. ring if ￠ has a unique maximal ideal m_{\mathbb{C}. \mathcal{K}-\mathrm{r}\dot{\mathrm{m} \mathrm{g}\not\in: at any \mathbb{R}-pontp is a. of. \mathcal{K}‐local. which. :. \mathrm{C}_{p}\rightar ow \mathfrak{D}. satisfies \mathrm{C}/m_{\mathbb{C} \cong \mathbb{R}.. ring with a maximal ideal m_{p}\subset \mathrm{e};_{p}.. as f(p) there exists a e_{p}(f) \mathcal{K}(M)_{\mathrm{p} :=\mathcal{K}(M)_{e_{p}} of \mathcal{K}(M) at e_{p} with a homomorphism $\pi$_{p} : \mathcal{K}(M)\rightar ow \mathcal{K}(\mathrm{M})_{ $\rho$} defined as $\pi$_{p}(f) =f/1 The set \mathcal{K}_{p}(M) of germs of \mathcal{K} ‐functions on M at p has a homomorphism $\phi$_{\mathrm{P} : \mathcal{K}(M) \rightarrow \mathcal{K}_{p}(\mathrm{M}) defined as $\varphi$_{p}(f) :=\lceil f,M]_{p}. For any f \in \mathcal{K}(M) with f(p) \neq 0, $\psi$_{\mathrm{P} (f) [f, M]_{\mathrm{p} has an invertible element [_{7}^{1},f^{-1}(\mathbb{R}\backslash \{0\})]_{\mathrm{p} . Then, there exists a unique homomorphism $\iota$_{p} : \mathcal{K}(M)_{p}\rightarrow \mathcal{K}_{p}(M) such that $\iota$_{ $\rho$}\circ$\pi$_{\mathrm{p} =$\varphi$_{p}.. For. a. \mathcal{K}‐manifold M and its. localization. point. p with. \mathbb{R}‐point e_{p} defined. an. =. ,. .. :=. 3.4 For. \mathbb{R} ‐derivations and \mathcal{K}‐denvations. [3], \mathbb{R}‐derivations. \mathrm{C}^{\infty} ‐nng. on a. derivahons). as same as. Definition 7. Ưoyce [5]. \mathrm{C}^{\infty}-\mathrm{r}\dot{\mathrm{m} \mathrm{g} are. defined. \mathbb{R} ‐derivations of the \mathbb{R} ‐algebra. \mathrm{C}^{\infty} ‐derivations. as. We define \mathrm{C}^{ $\omega$} ‐derivations. [5].. defined in. are. (analytic‐derivations) \mathrm{C}^{\infty} ‐derivations with the following definition.. for \mathrm{C}^{\infty} ‐rings). 1. An \mathbb{R} ‐derivation is. an. Suppose ￠ is a. \mathcal{K} ‐ring, and \mathfrak{M}. a. and \mathcal{N}^{ $\omega$} ‐derivations. on a. (Nash‐. C‐module.. \mathbb{R} ‐linear map d :\not\subset\rightarrow \mathfrak{M} with. d(c_{1}c_{2})=c_{2}d(c_{1})+c_{1}d(c_{2}) for any c_{1},c_{2}\in\not\subset. 2. A \mathcal{K}‐derivation is. d(. 3. Let d. :. for any. $\Phi$_{f} (cl,. \cdots. an. ,. \mathrm{C}\rightarrow \mathfrak{M} be. \mathbb{R} ‐linear map d. c_{n}) ) a. ỷJ \circ d=d'. .. Then. we. :. x_{2}\partial[=x_{1}, \neq_{x_{1} ^{\mathrm{a} =x_{2} For. a. Example4. r_{x_{1}^{$\delta$}. ,. \cdots. ,. $\sigma$\displaytle\frac{\partil}{x_n}. a. Let U be ,. ) \cdot d(c_{i}) for any n\in \mathbb{N},f\in \mathcal{K}(\mathbb{R}^{n}),c_{1}. ,. \cdots. ,. ,. c_{n}\in \mathrm{C}.. an. \mathbb{R} ‐derivation since there exists. a. function f(x_{1},x_{2})=x_{1}x_{2} with. product of \mathcal{K} ‐rings. ,. open set. of \mathbb{R}^{n}. \mathrm{C}^{\infty}(U) (resp. \mathrm{C}^{ $\omega$}(U), \mathcal{N}^{ $\omega$}(U). c_{n}\in \mathcal{K}(U) $\Sigma$^{n}c_{i}^{\partial_{-} ,. :. any i=1 ,. \cdots. ,. \mathcal{K} (U) \rightar ow \mathcal{K} (Ư) is. examples of \mathrm{C}^{\infty} ‐derivations for \mathrm{C}^{\infty} ‐mainfolds.. n,f\in \mathcal{K}(U) a. the Nash‐function. is closed under. i.e.. TherefOre,for c_{1}. \cdots. a. f on an open set \mathbb{R}^{n} a partial derivation of f is also following example of derivations as partial derivations.. an. ,. .. \displaystyle\frac{\partialf}{\partialx_{i}\in\mathcal{K}(U) for We show. c_{n}. (K=\mathbb{R},\mathcal{K}) We call. Nash‐function. Therefore, we have. ,. a pair (\mathfrak{M},d) a K‐cotangent module for \mathrm{C} if unique morphism $\varphi$ : \mathfrak{M}\rightarrow \mathfrak{M}' of \mathrm{C}‐modules such that ($\Omega$_{\mathrm{C},K},d_{\mathrm{C}_{v}K}) for the K‐cotangent module for \mathbb{C}. We have that any \mathcal{K} ‐derivation is a. \cdots. \mathrm{C}\rightar ow \mathfrak{M}' there exists. write. which is. e\rightarrow M with. =\displayst le\sum_{i=1}^{n}$\Phi$_{\parti l}\#_{x i} (cl,. K‐derivation. K ‐derivation d'. :. .. \mathcal{K}‐derivation.. partial. ([7]).. denvations.
(6) 27. Example. 5 Let \mathrm{M} be. 1. For any. a. $\Gamma$(T^{*}M). \mathrm{C}^{\infty} ‐manifold and. f\in \mathrm{C}^{\infty}(M) define a ,. \mathrm{C}^{\infty} ‐section. be the set. df. of \mathrm{C}^{\infty} ‐sections. M\rightarrow T^{*}M. :. to the. cotangent bundle T^{*}\mathrm{M} on. M.. by. df (v) :=v(f) for any x\in M and v\in T_{x}\mathrm{M}. Define an \mathbb{R}‐derivation d : \mathrm{C}^{\infty}(M)\rightarrow $\Gamma$(T^{*}M) 2. Let V:M\rightarrow TM bea \mathrm{C}^{\infty} ‐vectorfield. of M. as. d(f) :=df.. V_{x}\in T_{x}M for x\in M Define V(f)\in \mathrm{C}^{\infty}(\mathrm{M}) by. as. .. (V(f))(x):=V_{X}(f) regard V : \mathrm{C}^{\infty}(M)\rightar ow \mathrm{C}^{\infty}(\mathrm{M}). We. Differentiable. 4 4.1. as an. .. \mathbb{R} ‐derivation.. rings. Germs of C^{\infty} ‐frnctions. Lemma 1 Let M be. a. \mathrm{C}^{\infty} ‐manifold, p. a. point of M and. Ư. an. open. neighborhood at. p. of M.. $\eta$\in \mathrm{C}^{\infty}(M) and an open neighborhood V at p of Ư such that. 1. There exists. $\eta$(x)=\left\{ begin{ar ay}{l 1(x\inV)\ 0(x\inM\backslashU). \end{ar ay}\right. 2. For any f\in \mathrm{C}^{\infty}(U) there exists ,. 4.2. such that. \cdot. \mathrm{C}^{\infty} ‐mam fold.. a. define. a. morphism. Suppose that p\in M is. a. point and. $\pi$_{p}^{U} \mathrm{C}^{\infty}(U)\rightar ow \mathrm{C}_{p}^{\infty}(M) of \mathrm{C}^{\infty}-\mathrm{r}\dot{\mathrm{m} gs :. 1, we have. From Lemma. a. an. open neighborhood of M at p We .. $\pi$_{p}^{U}(f) :=[f, U]_{p}.. following property about germs of \mathrm{C}^{\infty} ‐functions on a. C^{\infty} ‐manifold.. :. morphism i_{p} : \mathrm{C}^{\infty}(M)_{p}\rightar ow \mathrm{C}_{p}^{\infty}(M) From the following corollary, $\iota$_{p} (\mathrm{C}_{p}^{\infty}(M), $\pi$_{\mathrm{p} ^{\mathrm{M} ) is regarded as a localization of \mathrm{C}^{\infty}(M) at p.. We have defined the. phism and. .. For a \mathrm{C}^{\infty} ‐ring C^{\infty}(M) isomorphism \mathrm{C}^{\infty}(M)_{e_{ $\rho$}}\cong \mathrm{C}_{p}^{\infty}(\mathrm{M}). Corollary 3 (Joyce [5]). and its \mathbb{R} ‐point e_{\mathrm{P}. :. is isomor‐. \mathrm{C}^{\infty}(M)\rightar ow \mathbb{R} by a point p\in M. ,. we. have. .. 4.3. Derivations of \mathrm{k}‐iet determined C^{\infty} ‐nngs. \mathrm{C}^{\infty}(\mathrm{M}) 4.1.. U is. as. $\pi$_{p}^{U} \mathrm{C}^{\infty}(U)\rightar ow \mathrm{C}_{ $\rho$}^{\infty}(M) is surjective.. Corollary 2. the. f|_{V}\equiv g|v.. The localizations for C^{\infty} ‐nngs. Let M be can. g\in \mathrm{C}^{\infty}(M). [6].. is embedded to. $\Pi$_{p\in \mathrm{M} \mathbb{R} by f. We define k‐jet determined. as. Definition 8 (Yamashita [9]) Let \mathrm{C} be. homomorphism j_{p}^{k}. :. m_{p^{\infty} :=\displaystyle \bigcap_{k\in \mathbb{N} m_{p^{k} J ￠ is. a. \mathrm{C}\rightar ow \mathrm{C}_{p}/m_{\mathrm{p}^{k+1}} .. Define i^{k}. k‐iet determined. \mapsto. the. a. \{f(p)\}_{ $\rho$\in M} Then \mathrm{C}^{\infty}(\mathrm{M}) is point generalization of point determined. .. \{0\}\mathrm{U}\mathrm{M}\mathrm{U}\{\infty\} For an \mathbb{R}‐point p of ￠,define a :=$\pi$_{p}(c)+m_{p}^{k+1} for c\in C (If k=\infty we mean m_{p^{k+1}} by. \mathrm{C}^{\infty} ‐ring and k \in. by i_{\mathrm{P} ^{k}(c) if i^{}. is. .. ,. :\not\subset\rightar ow$\Pi$_{\mathrm{p}:\mathrm{C}\rightar ow \mathb {R} \mathrm{C}_{p}/m_{p}^{k+1}. \mathrm{C}^{\infty} ‐ring. determined in Def nition. as. injective.. Theorem 1 (Yamashita [9]) Let \mathrm{C},\mathfrak{D} be \mathrm{C}^{\infty} ‐rings, \mathbb{N}\cup\{\infty\} Suppose that CD is k‐iet determined.. $\psi$. :. i^{k} :=(j_{p}^{k})_{p:\mathrm{C}\rightar ow \mathrm{R} .. \mathrm{C}\rightar ow \mathfrak{D}. a. homomorphism of \mathrm{C}^{\infty} ‐rings. and k\in. \{0\}\cup. .. Then any \mathbb{R} ‐derivation V:\mathbb{C}\rightarrow \mathfrak{D} Let M be. Example 6. \{f(p)\}_{p\in M}. a. over. \cdot. $\varphi$ is a \mathrm{C}^{\infty} ‐den vation.. \mathrm{C}^{\infty} ‐manifold. There exists. an. injection i^{0}. Then, \mathrm{C}^{\infty}(M) be a 0 ‐jet determined \mathrm{C}^{\infty} ‐ring.. .. :. \mathrm{C}^{\infty}(M) \displaystyle \rightar ow\prod_{\mathrm{p}\in M}\mathb {R} defined as 1^{ $\theta$}(f). Th\ell 7 $\phi$ ore from Theorem 1, for a smooth mapping f : \mathrm{M}\rightarrow N of \mathrm{C}^{\infty} ‐manifolds, any \mathbb{R}‐derivation V:\mathrm{C}^{\infty}(N)\rightarrow \mathrm{C}^{\infty}(M) along f^{*}:\mathrm{C}^{\infty}(N)\rightarrow \mathrm{C}^{\infty}(\mathrm{M}) is \mathrm{C}^{\infty} ‐derivation. ,. :=.
(7) 28. The condition of k‐jet determined is need to Theorem 1. From D.. example. that all \mathbb{R} ‐derivation. are. not \mathrm{C}^{\infty} ‐derivations.. Joyce [5]. Remark 5.5.,. we. have. Let ￠ be a \mathrm{C}^{\infty} ‐ring \mathrm{C}^{\infty}(\mathbb{R}) Por the \mathbb{R} ‐cotangent module ($\Omega$_{\mathbb{C},\mathrm{R} ,d_{\mathrm{C},\Re}) d_{\mathrm{C},\mathrm{R}. Example 7 Ưoyce [5]). an. .. :\not\subset\rightar ow$\Omega$_{\mathrm{C},\mathrm{R} is an \mathbb{R}‐derivation but not a \mathrm{C}^{\infty} ‐derivation. generated by d(x) $\Omega$_{\mathrm{C},\mathrm{R} is not afinitely generated generated ,. $\Omega$_{\mathrm{C},\mathrm{C} \infty. afinitely generated. is. module.. In fact, for the. \not\subset‐module. .. C‐. exponential e^{x}\in \mathrm{C}^{\infty}(\mathbb{R}) e^{x}d_{\mathrm{C},\mathrm{R} (x)-d_{\not\subset j\mathrm{R} (e^{x})\neq 0 in $\Omega$\not\subset,\mathrm{R}. ,. Analytic‐rings. 5. Germs of. 5.1. analytic functions. 1, for a \mathrm{C}^{\infty} ‐manifold M and a point p there exists neighborhoods V\subset U of M at p such that $\eta$|_{V}\equiv 1 and From Lemma. For. ,. connected \mathrm{C}^{ $\omega$} ‐manifold. a. tendable to M have Lemma 2 Let \mathrm{M} be. exists. a. a. unique function g\in \mathrm{C}^{\infty}(\mathrm{M}). connected \mathrm{C}^{ $\omega$} ‐manifold and f be. \mathrm{C}^{\infty} ‐function $\eta$ \in. $\eta$|_{M\backslash \overline{U} \equiv 0.. f be a. \neq_{x^{ $\alpha$} ^{\partial^{ $\alpha$} (p)=0 for any Germs of. Let \mathrm{M} be. a. real-C^{ $\omega$} ‐function. on. \mathbb{R}^{n}.. a. f=0 on. analytic rings. and Nash. inclusion of open connected subsets in M We .. ,. real-\mathrm{C}^{ $\omega$} ‐function \mathbb{R}^{n}. on. M.. M. ex‐. if and only if there such that. rings Suppose. can. define. that. x. M is. \in. morphisms. From Lemma 2 and Lemma 3,. we. have. $\pi$_{p}^{U}. are. injective.. 1. $\rho$ uv and. maximal ideal. a. a. point. of \mathcal{K}^{ $\omega$}-\mathrm{r}\dot{\mathrm{m} gs. and i. V. :. \rightarrow. U is. an. as. ,. $\pi$_{p}^{U}:\mathcal{K}^{ $\omega$}(U)\ni f\mapsto[f, U]_{p}\in \mathcal{K}_{p}^{ $\omega$}(M). a. f=0 on. if and only if there exists a point p\in \mathbb{R}^{n}. $\rho$_{\mathrm{U}V}:\mathcal{K}^{ $\omega$}(U)\ni f\mapsto f|v\in \mathcal{K}^{ $\omega$}(V). 2. For. and open. $\alpha$.. \mathrm{C}^{ $\omega$} ‐manifold(oesp. \mathcal{N}^{ $\omega$} ‐manifold).. Corollary 4. \mathrm{C}^{\infty}(M). M a germ [f, U]_{p} of \mathrm{C}^{ $\omega$} ‐imctions which is whose germ is [f, U]_{p} for following lemmas.. non‐empty open subset U\subset M such that f|_{U}\equiv 0.. Lemma 3 Let. 5.2. a. (resp. \mathcal{N}^{ $\omega$} ‐manifold). a. .. following corollary.. m_{p}=\{f\in \mathcal{K}^{ $\omega$}(M)|f(p)=0\} of \mathcal{K}^{ $\omega$}(\mathrm{M}) m_{p}^{\infty}=\mathrm{n}_{i=1}^{\infty}m_{p}^{i}=0. ,. We have defined the morphism i_{p} : \mathrm{C}^{ $\omega$}(M)_{p}\rightar ow \mathrm{C}_{p}^{ $\omega$}(M) $\iota$_{\mathrm{p} is not isomorphism. For example, take an analytic function f(x) := \displaystyle \frac{1}{1-x} on (-1,1) and a point 0\in \mathbb{R}. f(x) =$\Sigma$_{i=0}^{\infty}x^{n} on (-1,1) We cant take g\in \mathrm{C}^{W}(\mathbb{R}) such that [g,\mathbb{R}]_{0}=[f, (-1.1)]_{0} Therefore, $\iota$_{0} : \mathrm{C}^{ $\omega$}(\mathb {R})_{0}\rightar ow \mathrm{C}_{0}^{ $\omega$}(\mathb {R}) is not surjective, moreover not isomorphism. .. .. .. Nash‐rings. 6 6.1. The condition of Nash functions. Suppose that. U\subset \mathbb{R}^{n} be. For Nash‐functions. by Kähler‐differential. on. ,. Theorem 2 (lshikawa‐Yamashita [4]\rangle A real analyticfunction there exists a Nashfunction g\in \mathcal{N}^{ $\omega$}(U)(g\neq 0) such that. f\in \mathrm{C}^{ $\omega$}(U). g(df-\displaystyle \sum_{i=1}^{n}\frac{\partial f}{ $\delta$ x_{i} dx_{i})=0 in. spaces. open connected semialgebraic subset. \mathbb{R}^{n} we have a following theorem for Nash‐derivations of Nash‐rings.. an. $\Omega$_{\mathrm{C}^{ $\omega$}(U),\mathb {R} and $\Omega$_{\mathrm{C}^{\infty}(U),\mathb {R}. is. a. Nash‐function if and only if.
(8) 29. From Theorem 2 about Nash‐fumctions and Nash‐derivations, Theorem 3 For any. Suppose. that \mathfrak{A} is. f\in \mathcal{N}^{W}(\mathbb{R}^{n}). a. and c_{1}. Nash‐ring \cdots. ,. V( is. nilpotent element in. a. an. \mathfrak{A} ‐module with. an. have. a. following property.. \mathbb{R} ‐derivation V. \mathfrak{A}\rightarrow \mathfrak{M}.. :. c_{n}\in \mathfrak{A},. $\Phi$_{f} (cl,. \cdots. ,. c_{n}) ). -\displayst le\sum_{i=1}^{n}$\Phi$_{\parti l}\#_{x i} (cl,. \cdots. ,. c_{n}. ) V(c_{\mathrm{i} ). M.. Ideals of Nash. 6.2. ,. and \mathfrak{M} be. we. rings. For \mathrm{C}^{\infty} ‐rings, all. finitely generated \mathrm{C}^{\infty} ‐nngs are not finitely presented \mathrm{C}^{\infty}-\mathrm{r}\dot{\mathrm{m} gs, i.e. \mathrm{C}^{\infty}-\mathrm{r}\dot{\mathrm{m} gs \mathrm{C}^{\infty}(\mathbb{R}^{n})/\langle f_{1} f_{k}\in \mathrm{C}^{\infty}(\mathbb{R}^{n}) f_{k}\}_{\mathrm{C}^{\infty}(\mathrm{R}^{n})} by f_{1} From Corollary 1.5.5.\dot{\mathrm{m}} p41 [8] and Theorem 8.7.15 in [2], we have a following proposition.. form. ,. \cdots. ,. ,. Proposition 2 (Shiota I8]) Suppose. bedding g_{1}. ,. \cdots. ,. \mathcal{N}^{ $\omega$}(\mathrm{M}) g_{k}\in I such thfft l=\{g_{1} i:M\rightarrow \mathbb{R}^{m}.. is. ,. a. \cdots. ,. \cdots. ,. that M be Noether. of the. .. an. ring,. g_{k}\rangle_{\mathcal{N}^{ $\omega$}(M)}.. afine \mathcal{N}^{ $\omega$} ‐manifold, such that there exists an \mathrm{C}^{$\omega$} Nash em‐ i.e. for any ideal I\subset \mathcal{N}^{ $\omega$}(\mathrm{M}) there exisfs finitefunctions ,. Therefore, any finite generated \mathcal{N}^{ $\omega$} ‐ring is finitely presented.. The localizations of. 6.3 For. sheaf O_{\mathrm{X} on if a sequence. a. only. Nash‐rings. topological space X, a sequence \cdots \mathcal{F}^{i-1}\rightar ow\overline{\prime\prime}i\rightar ow\overline{\prime r^{ $\iota$+1}}\rightar ow\cdots of \mathcal{O}_{X} \cdots \mathcal{F}_{p}^{i-1}\rightar ow \mathcal{F}_{p}^{i}\rightar ow\overline{f}_{p}^{i+1}\rightar ow\cdots of \mathrm{O}_{X,p} is exact for any p\in \mathrm{X} ([3]).. a. Proposition 3 (Shiota [8]). fully flat on \mathcal{N}^{ $\omega$}(\mathb {R}^{n})_{e_{P}. ,. For. Nash‐ring \mathcal{N}^{ $\omega$}(\mathbb{R}^{n}) and its of \mathcal{N}^{ $\omega$}(\mathb {R}^{n})_{e_{p}. a. \mathbb{R} ‐point. e_{t^{J}. :. \mathcal{N}^{ $\omega$}(\mathbb{R}^{n}). \rightar ow \mathbb{R},. i.e. any sequence ... .. is exact if and. \mathcal{N}_{r^{ $\omega$} (\mathb {R}^{n}). is faith‐. \rightarrow\Re_{ $\eta$-1}\rightarrow\Re_{ $\eta$}\cdot\rightarrow\Re_{i+1}\rightarrow\cdots,. if and only if the sequence of \mathcal{N}_{p}^{ $\omega$}(\mathb {R}^{n}). is exact. \cdots\rightar ow \mathfrak{R}\otimes \mathcal{N}_{p}^{ $\omega$}(\mathbb{R}^{n})\rightar ow \mathfrak{R}_{ $\eta$}\cdot\otimes,\mathcal{N}^{ $\omega$}(\mathbb{R}^{n})\rightar ow \mathfrak{R}\otimes \mathcal{N}_{\mathrm{p} ^{ $\omega$}(\mathbb{R}^{n})\rightar ow\cdots is exact. For. a. \mathrm{C}^{\infty}-\mathrm{r}\dot{\mathrm{m} \mathrm{g}\mathrm{C}^{\infty}(M). ,. above. Proposition satisfies since. \mathrm{C}_{p}^{\infty}(M)\equiv C^{\infty}(\mathrm{M})_{e_{ $\rho$}}. for any. point p\in M. Analytic‐ringed spaces and Nash‐ringed spaces. A. A \mathrm{C}^{\infty}-\mathrm{r}\dot{\mathrm{m} ged space is defined in. functor. [5]. We define \mathrm{C}^{ $\omega$}-\mathrm{r}\dot{\mathrm{m} ged spaces (resp. Nash‐ringed spaces) Spec as same as \mathrm{C}^{\infty}-\mathrm{r}\dot{\mathrm{m} gs with the following definitions.. The definition of. A.l. Definition 9 \mathrm{x}. 2. A a. analytic‐ringed spaces. I. A\mathcal{K}^{ $\omega$} ‐ringed space. \underline{\mathrm{X} = (\mathrm{X},\mathcal{O}_{X}). is. a. and. Nash‐ringed. topological. space X with. morphism f=(f,f^{*}):(\mathrm{X}, \mathrm{O}_{\mathrm{X} )\rightarrow(\mathrm{Y}, \mathcal{O}_{Y}) of \mathcal{K}^{ $\omega$} ‐ringed spaces morphism f^{\overline{\#} : O_{\mathrm{Y} \rightar ow f_{*}(O_{\mathrm{X} ) of sheaves of \mathcal{K}^{ $\omega$} ‐rings on Y.. 3. A local \mathcal{K}^{ $\omega$} ‐ringed space X. x\in \mathrm{X}.. =. (\mathrm{X}, O_{X}). is. a. is. a. \mathcal{K}^{ $\omega$} ‐ringed space for which. and. a. spaces a. sheaf O_{X} of \mathcal{K}^{ $\omega$} ‐nngs. continuous map. O_{X,x}. are. f. :. \mathcal{K}^{ $\omega$} ‐local. on. \mathrm{X}\rightarrow $\gamma$ and. rings for. all.
(9) 30. A.2. The. Example 8. examples Let \mathrm{M} be. a. of. analytic‐ringed spaces. and. Nash‐ringed. spaces. \mathcal{K}^{ $\omega$} ‐manifold.. an open set U\subset M define O_{M}(U) as a set offunctions \mathrm{s} : U\rightar ow\coprod_{x\in U}\mathcal{K}^{ $\omega$}(M)_{e_{\mathrm{p} } such that for any point x \in Ư, there exists a open neighborhood V\subset U at p and elements c,d\in \mathcal{K}^{W}(M)(d(p)\neq 0) which satisfll \mathrm{s}(q)=$\pi$_{q}(c)$\pi$_{q}(d)^{-1} for any point q\in U.. 1. For. 2.. ,. Define \mathrm{O}_{M}' as \mathcal{O}_{M}'(U) :=\mathcal{K}^{ $\omega$}(U) for any open set. Then, (M, O_{ $\lambda$ 4}) and A.3. (M, O_{M}') are \mathcal{K}^{$\omega$} ‐ringed spaces.. The definition of. Definition 10. (a). Ooyce [5]). Spec 1. For a. \mathrm{C}^{\infty}-nng\mathrm{C} define a ,. Define a topological space X￠ . \bullet \bullet. as. Define a set X_{\mathrm{C} :=\{x : \mathrm{e}\rightar ow \mathbb{R}|x is a \mathbb{R}‐point of\not\subset define c_{*}:X_{\mathrm{C}}\ni x\mapsto x(c)\in \mathbb{R}. Set a topology of X￠ as a smallest topology T_{\mathrm{C} such. For each c\in\not\subset. ,. For each. 2.. Therefore define. O_{x_{\mathrm{c} }(U). as a. set. that c_{*}is continuous for all c\in \mathbb{C}.. offunctions. \mathrm{s} :. U\rightarrow\coprod_{x\in U}\mathbb{C}_{x}. with following. x\in U, s(x)\in \mathfrak{c}_{x} is satisfied.. U is covered. for. spaceX￠ as followings.. ,. properties \bullet. \mathrm{C}^{\infty} ‐ringed. followings by C^{\infty} ‐ring C.. (b) For an open subset U\subset \mathrm{X}_{\mathrm{C} define. \bullet. U\subset M.. some. by open set V with c,d\in \mathrm{C}(\forall x\in V, $\pi$_{x}(d)\neq 0) $\pi$_{X}(c)$\pi$_{x}(d)^{-1}=s(x)(\forall x\in V) ,. is. satisfied.. the following \mathrm{C}^{\mathrm{o}\mathrm{o} ‐ringed space. Spec \mathrm{C} :=(\mathrm{X}_{\mathrm{C} ,\mathrm{O}_{X_{\mathrm{c} }). .. References [1] Ralph. P Boas and Harold P Boas. A. 1996.. primer of realfunctions,. Vol. 13.. Cambridge University Press,. [2] Jacek Bochnak, Michel Coste, and Marie‐Frangoise Roy. Géométrie algébrique réelle (Real algebraic geometry), Vol. 12. Springer Science & Business Media, 1987. [3] Robin Hartshorne. Algebraic Geometry, volume 52 of Graduate New York, 1977.. Texts in Mathematics.. [4] Go‐o Ishikawa and Tatsuya Yamashita. Leibniz complexity of Nash functions arXiv preprint arXiv:1509.08261, 2015.. on. Springer‐Verlag, differentiations.. [5] Dominic Joyce. Algebraic geometry over \mathrm{C}^{\infty}-\mathrm{r}\dot{\mathrm{m} gs. arXiv preprint arXiv:l00l.0023v6, 2015. [6] Ieke Moerdijk and Gonzalo E Reyes. Models for smooth infinitesimal analysis. Springer Science & Business Media, 2013.. [7] Rodolphe Ramanakoraisina. Complexité des fonctions de Nash. Communications in Algebra, Vol. 17, No. 6, pp. 1395‐1406, jan 1989.. [8] Masahiro Shiota. Nash manifolds, Lectures Notes in Math., 1269. Springer‐Verlagỷ 1987.. [9] Tatsuya Yamashita. Derivations 4822, jun 2016.. on a. \mathrm{C}^{\infty} ‐ring. Communications in. Algebra,. Vol. 44, No. 11, pp. 4811‐.
(10) 31. Department of Mathematics University. Hokkaido. Hokkaido 065‐0006. JAPAN \mathrm{E} ‐mail address:. tatsuya‐y@ynath.sci.hokudai.ac.jp \mathrm{j}$\zeta$^{\backslash }\ovalbox{\t \smal REJECT}_{\grave{\mathrm{J} \ovalbox{\t \smal REJECT} $\lambda$\not\cong $\lambda$\not\cong\ovalbox{\t \smal REJECT} \mathfrak{B}\not\cong\ovalbox{\t \smal REJECT} \mathscr{X}\neq\leftrightar ow\ovalbox{\t \smal REJECT} $\iota$ p \lfo rỈ\rflo r T. \grave{\mathrm{J}\ovalbox{\t\smal REJ CT}$\Psi$.
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