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TITLE A Study on Smooth Varieties with Differentially Simple Coordinate Rings 
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KEYWORDS Derivations, Differentially simple rings, Algebraic sets, Singularities, Coordinate rings of smooth varieties 
ABSTRACT The notion of the algebraic sets was proved to be fundamental for the development of the Algebraic Geometry. In the paper at hands we study irreducible algebraic sets (varieties) with differentially simple coordinate rings. Starting from the fact that the coordinate ring of a singular variety does not admit simple derivations, we turn our attention to smooth varieties proving that the coordinate rings of the circle of the cylinder and of the real torus are differentially simple rings. However, this is not true for the coordinate rings of all smooth varieties; for example the coordinate ring of the real sphere does not admit simple derivations.

1. Introduction All he rings considered in this paper are commutative with identity and all the fields are of characteristic zero, while a local ring is understood to be a Noetherian ring with a unique maximal ideal. For general facts on commutative ring theory we refer to the books [13]. A prime ideal P of a ring R is said to be of height n, where n is an integer, n 1, if there exists a chain of distinct prime ideals decreasing from P of the form P P_{1} P_{2} … P_{n} and no longer one. We write then ht P = n Further, the Krull dimension of R, denoted by dim R, is defined to be the supremum of heights of all maximal ideals of R. Note that, if R is a finitely generated algebra over a field K, then all its maximal ideals have the same height. Therefore, if M is a maximal ideal of R, then dim R = ht M. A local ring is said to be regular if its dimension is equal to the minimal number of generators, say V(M), of its unique maximal ideal M. Generalizing, a ring R is said to be regular, if all its localizations R_{M} with respect to a maximal ideal M are regular local rings. In the paper at hands we study smooth varieties over a field K, whose coordinate rings admit simple Kderivations. The rest of the paper is organized as follows: Section II contains a brief account about the differential simplicity of a ring, which is needed for the purposes of the present work. In Section III some ideas from Algebraic Geometry are exposed, which are useful for the better understanding of the paper. Finally, Section IV contains the main paper’s results concerning the differential simplicity of the coordinate rings of a circle, of a cylinder and of the real torus.
2. Differentially Simple Rings Let R be a ring and let d be a derivation of R. Then an ideal I of R is called a dideal if d (I) I, and R is called a dsimple ring if it has no non zero, proper dideals. In the last case we say that d is a simple derivation of R. For reasons of brevity we shall write dI instead of d (I). A dsimple ring R contains the field F = {x R: dx = 0} and therefore it is either of characteristic zero, or of a prime number p. In earlier works [4, 5] we have studied the dsimplicity of a commutative ring. If R is a dsimple ring of characteristic p, then things are quite simple; namely R is a 0dimensional quasilocal ring and therefore, if R is a domain, then R is a field [5; Theorem 1.5]. On the contrary, if R is of characteristic zero and d is a derivation of R, then no general criterion is known to decide whether or not R is a dsimple ring, unless if R is a onedimensional (as a ring) finitely generated algebra of the form K[y_{1}, y_{2},…, y _{n}] over a field K . In fact, in this case R is a dsimple ring, if, and only if, R =(dy_{1}, dy_{2},…, dy _{n}) [5; Theorem 2.4]. Several examples of dsimple rings of dimension greater than one and even of infinite dimension are exposed in [4, 5], such as the polynomial rings in finitely and infinitely many variables and the Laurent polynomial rings over a field K, the regular local rings of finitely generated type [6] , etc. Another important result that we are going to use later in this paper states that, if R is a dsimple Gring  a wide class of rings containing all finitely generated algebras over fields and all complete local rings, and being closed under localization [3; pp. 249257]  of characteristic zero, then R is a regular ring [7; Theorem 1].
3. Ideas from Algebraic Geometry This section contains the background from Algebraic Geometry which is necessary for the understanding of the rest of the paper. For general facts on Algebraic Geometry we refer to the books [8, 9]. Definition 1: Let K be a field. Then the Cartesian product K^{n} = k x k x…x k (ntimes) is called an affine space over K, and its elements a = (a_{1}, a_{2},,…,, a_{n}), are called points of K^{n}, with a_{1}, a_{2},,…,, a_{n }in K being the coordinates of the point a.^{ }. Definition 2: Let K[x_{1}, x_{2},….,x_{n}] be a polynomial ring over a field K . Then a subset Y of K^{n }is called analgebraic set over K^{n}, if there exists a nonempty subset S of K[x _{1}, x_{2},….,x_{n}] such that Y = {a K^{n}: f(a) = 0, f S}. We write then Y = U(S), while, if S = {f} is a singleton set, we write for simplicity Y = U (f). Examples: U(0) = K^{n}, U(1) = , U(x_{1}^{2}+x_{2}^{2}1) = unit circle, U(x_{2}x_{1}^{2}) = parabola, U(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}1) = unit sphere, U(x_{1}^{2}x _{2}^{2}x_{3}^{2}) = cone, etc. Definition 3: Let Y be an algebraic set of an affine space over a field K. Then, it is easy to check that the set I = J(Y) = {f K[x_{1}, x_{2},….,x_{n}]: f(a)=0, a Y} is an ideal of K[x_{1}, x_{2},….,x_{n}], called the ideal of Y. As a special case J( ) = K[x_{1}, x_{2},….,x_{n}], while if K is an infinite field , then J(K^{n}) = 0. Obviously, if Y= U (S), then J(Y) S. Further we have: Proposition 4: If Y is an algebraic set over K^{n}, then Y = U(J(Y)) Proof: If a Y, then, by Definition 3, f(a) = 0, for all f in J(Y). Hence, by definition 2, a is in U(I(Y)), therefore Y U(J(Y)). Conversely, let Y = U(S) and let a U(J(Y)). Then, by Definition 2, f(a) = 0 for all f in J(Y). But J(Y) S, therefore f(a) = 0 for all f in S, which means that a Y. Thus U(J(Y)) Y. Definition 5: Let Y be an algebraic set of an affine space over a field K and let I= J(Y). Then the factor ring R = K[x_{1}, x_{2},….,x_{n}] / I is called the coordinate ring of Y. Further, the dimension of Y is defined to be equal to the dimension of R. Definition 6: Let Y be an algebraic set of an affine space over a field K, let R be the coordinate ring of Y and let a be a point of Y. It is easy then to check that the set J(a) = {f K[x_{1}, x_{2},….,x_{n}]: f(a) = 0} is a maximal ideal of the polynomial ring K[x_{1}, x _{2},….,x_{n}]. Therefore, the image M_{a} = J(a)/I of J(a) in R is a maximal ideal of R. Then, the local ring R_{a} = R_{Ma} is called the local ring of a in Y. Definition 7: An algebraic set of an affine space over a field K is said to be irreducible if it cannot be written as the union of two smaller algebraic sets. In this paper an irreducible algebraic set is called a variety. Definition 8: Let Y be a variety over K^{n} and let a be a point of Y. Then an element (b_{1}, b_{2}, …, b_{n}) of K^{n} is said to be a tangent to Y at a, if for all f in J(Y) we have that = 0. It is easy to check that the set Ta of all tangents of Y at a is a vector space over K called the tangent space of Y at a.. It is well known that the dimension of the vector space T_{a } is equal to the minimal number, say V(M_{a}), of generators of the maximal ideal M_{a} of the local ring of a in Y (Definition 6). But, by the generalized principal ideal theorem [2; Theorem 152) ht M_{a} V(M_{a}). But R is a finitely generated algebra, therefore dim R = ht M_{a}, therefore dim R dim Ta. Definition 9: Let Y be a variety over K^{n}. Then a point a of Y is said to be a simple point, if the dimension of Y is equal to the dimension of the tangent space T_{a}. Otherwise a is called a singular point of Y. A variety Y has always simple points, while it may have or not singular points. Definition 10: A variety Y over K^{n} not having singular points is called asmooth variety, otherwise it is called a singular variety. It is now easy to prove the following result: Proposition 11: If the coordinate ring of a variety Y over K^{n} is a regular ring, then Y is a smooth variety. Proof: Let a be any point of Y and let R_{a} be the local ring of a in Y. Since R is a regular ring, dim R_{a} = V(M_{a}). But R is a finitely generated Kalgebra, therefore dim R = ht M_{a} .= dim R_{a} = V(M_{a}) = dim T_{a }. This shows that a is a simple point of Y, therefore Y is a smooth variety.
4. Main results In this section we present examples of smooth varieties, whose coordinate rings admit simple derivations, i.e. they are differentially simple rings. We emphasize that this is not always true; e.g. it is well known that the coordinate ring S= of the real sphere, although it is regular, admits no derivation d , such that S is dsimple. [7; Section 3, Example (iii)]. We start with the following result showing that only the coordinate rings of smooth varieties can admit simple derivations: Theorem 12: Let k be a field, let n be a non negative integer, and let Y be a singular variety over k . Then the coordinate ring R of Y admits no simple derivations. Proof: Assume that there exists a derivation d of R, such that R is a dsimple ring. But R, being a finitely generated Kalgebra is a Gring, therefore R is a regular ring [7; Theorem 1]. Consequently, by Proposition 11, Y is a smooth variety, which contradicts our hypothesis.. Next we are going to present some characteristic examples of smooth varieties having dsimple coordinate rings, for suitably chosen kderivations d of them. First, we study the case of the circle: Theorem 13: The coordinate ring R= of the unit circle defined over a field k admits kderivations d such that R is a dsimple ring. Proof: Since P = ( is a prime ideal of k[ , we have that dim R = 2 – ht P and therefore, by the principal ideal theorem [2; Theorem 142), dim R = 1. Consider the kderivation d of k[ defined by d =a , d =a , for some non zero element a of k. Since dP P, d induces a kderivation of R, denoted also by d. Given f in k[ , set = f + P, then = a. Thus the result follows by Theorem 2.4 of [5], stating that, if d is a kderivation of an onedimensional finitely generated kalgebra, say R=k[ ], then R is dsimple, if, and only if, R = ( ). Next, we consider the case of the cylinder: Theorem 14: The coordinate ring C= of the cylinder defined over a field k is dsimple for suitable kderivation d of C. Proof: Reconsider the coordinate ring R of the unit circle. We can write R=k [ ] and C = k [ ] = R [x ]. The derivation d = of k [ induces a derivation of R, denoted also by d. The above derivation d is the special case of the derivation of Theorem 13 for a = 1, therefore R is a dsimple ring. We shall show that d can be extended to a derivation of C, such that C is also a dsimple ring. For this, observe first that any F in R can be written in the form F= , where n is a non negative integer and g is in k [ ], for each i. But , therefore F = + , with f, g in k [ ]. This expression of f is unique, because + = gives that f . Thus, on comparing the degrees of both sides with respect to , it turns out that h = 0 and therefore f = g = 0. We shall show further that dF , for all F in R. In fact, dF = d Thus, if dF= , by the uniqueness of this expression, it follows that = and (1 (1) . But f = for some non negative integer t, with in k, for each i. Thus, equating the leading coefficients in (1), we find that  (t+1) =0. Therefore =0, which contradicts (1). Consider now the derivation + of k [ , which induces a derivation of C, whose restriction in R is d. Denote the above derivation by d as well. We shall show then that C is a dsimple ring. In fact, assume that I is a proper non zero dideal of C = R[x ] and let A be the set of the leading coefficients of all the polynomials of least degree, say n, in I.. Obviously then A is a proper non zero ideal of R Let a be in A, then there is a polynomial F in I, such that F = ax + terms of lower degree. But dF = (da) x + terms of lower degree is also in I and therefore da is in A.Hence A is a dideal of R, so A=R. This means that there exists a monic polynomial, say G=x + terms of lower degree, in I. Then dG =nx + terms of lower degree is also in I and has degree less than n, therefore d G = 0. In particular n + d( ) = , or d ( )= , a contradiction, since  = F is in R. Remark: The varieties whose ideals are principal, like the circle and the cylinder, are called hypersurfaces. Next we study the case of the real torus obtained by rotating a cycle around an axis in its plane which does not intersect it (Figure 1). Figure 1: The surface of a torus For this, we need the following Lemma: Lemma 15: Let A= R [ ] be a polynomial ring over the field R of the real numbers. Then the R derivation d of A defined by d , d(x , with in R for each i = 1, 2,…n, induces a derivation of the ring T= , denoted also by d, such that T is dsimple if, and only if, are linearly independent over the ring Z of integers. Proof: Set I= . Since d (x ) = 0 for each i, is d I , and therefore d induces a derivation of T= . Consider now the polynomial ring B=C [ ], where C denotes the field of complex numbers and set I = BI. Then d extends to a Cderivation of B by d (f + ig) = df + idg for all f, g in A. It is easy to check that I is a dideal of B, therefore d lifts to a C derivation of . It is straightforward to show that is a dsimple ring, if, and only f, is a dsimple ring. Set y and y , for j = 1,2,….,n. Then we have that , , and y , for each j and therefore we can write = C [y , y ]. But d (y + I . Therefore the result follows by Theorem 3.5 of [4], stating that C [y ,y ] is dsimple, if, and only if, the b 's are linearly independent over Z. For n =2 the ring T= of Lemma 15 is the coordinate ring of the real torus, considered as a 2dimensional surface in 4 dimensions. Thus one obtains the following result: Theorem 16: The coordinate ring T of the real torus is dsimple for suitable Rderivations d of T. Proof: Set d = , where a is an irrational number and apply Lemma.15. Remark: Combining the above results with Corollary 3.6 of [10] one obtains examples of simple skew polynomial rings over the coordinate rings of the circle, the cylinder and the real torus.
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Cite this paper Michael Gr. Voskoglou. (2017) A Study on Smooth Varieties with Differentially Simple Coordinate Rings. International Journal of Mathematical and Computational Methods, 2, 5359 
