Irreducible Affine Variety Research Articles

Representations Of Lie Algebras of Vector Fields on Pairing Between Gauge Modules and Rudakov Modules

For an irreducible affine variety X over an algebraically closed field of characteristic zero, we define two new classes of modules over the Lie algebra of vector fields on X: gauge modules and Rudakov modules. These modules admit a compatible action of the algebra of functions on X. We prove general simplicity theorems for these two types of modules, demonstrating their irreducibility under specific conditions. Additionally, we establish a pairing between gauge modules and Rudakov modules, highlighting the connections and interactions between these two classes of modules. We have established that gauge modules and Rudakov modules, corresponding to simple glN -modules, remain irreducible as modules over the Lie algebra of vector fields unless they appear in the de Rham complex. Additionally, we have studied the irreducibility of tensor products of Rudakov modules, providing a comprehensive understanding of these module structures and their applications.

Dimension Growth for Affine Varieties

Abstract We prove uniform upper bounds on the number of integral points of bounded height on affine varieties. If $X$ is an irreducible affine variety of degree $d\geq 4$ in ${\mathbb{A}}^{n}$, which is not the preimage of a curve under a linear map ${\mathbb{A}}^{n}\to{\mathbb{A}}^{n-\dim X+1}$, then we prove that $X$ has at most $O_{d,n,\varepsilon }(B^{\dim X - 1 + \varepsilon })$ integral points up to height $B$. This is a strong analogue of dimension growth for projective varieties, and improves upon a theorem due to Pila, and a theorem due to Browning–Heath-Brown–Salberger. Our techniques follow the $p$-adic determinant method, in the spirit of Heath-Brown, but with improvements due to Salberger, Walsh, and Castryck–Cluckers–Dittmann–Nguyen. The main difficulty is to count integral points on lines on an affine surface in ${\mathbb{A}}^{3}$, for which we develop point-counting results for curves in ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. We also formulate and prove analogous results over global fields, following work by Paredes–Sasyk.

Sums of even powers of [formula omitted]-regulous functions

We provide an example of a nonnegative k-regulous function on Rn for k≥1 and n≥2 which cannot be written as a sum of squares of k-regulous functions. We then obtain lower bounds for Pythagoras numbers p2d(Rk(Rn)) of k-regulous functions on Rn for k≥1 and n≥2. We also prove that the second Pythagoras number of the ring of 0-regulous functions R0(X) on an irreducible 0-regulous affine variety X is finite and bounded from above by 2dimX.

CHARACTERIZATION OF n-DIMENSIONAL NORMAL AFFINE SLn-VARIETIES

We show that any normal irreducible affine n-dimensional SLn-variety X is determined by its automorphism group seen as an ind-group in the category of normal irreducible affine varieties. In other words, if Y is an irreducible affine normal algebraic variety such that Aut(Y) ≃ Aut(X) as an ind-group, then Y ≃ X as a variety. If we drop the condition of normality on Y , then this statement fails. In case n ≥ 3, the result above holds true if we replace Aut(X) by \U0001d4b0(X), where \U0001d4b0(X) is the subgroup of Aut(X) generated by all one-dimensional unipotent subgroups. In dimension 2 we have some interesting exceptions.

The Wadge hierarchy on Zariski topologies

We study the relation of continuous reducibility, or Wadge reducibility, between subsets of an affine variety. We show that on any curve the relation of continuous reducibility is a bqo, though it may have large finite antichains. We determine the Wadge hierarchy on irreducible curves and on countable irreducible affine varieties of any dimension. Under a technical assumption of adequateness, we prove that on the class of finite differences of closed subsets of an irreducible affine variety the structure of the Wadge hierarchy depends only on the dimension of the affine variety and coincides with the difference hierarchy; we also show that this assumption of adequateness is satisfied by a large class of affine varieties. In contrast, we show that for large cardinalities the behaviour of the Wadge hierarchy outside the class of finite differences of closed sets can be much wilder, for example there may exist antichains of size the continuum.

Windows for cdgas

We study a Fourier-Mukai kernel associated to a GIT wall-crossing for arbitrarily singular (not necessarily reduced or irreducible) affine varieties over any field. This kernel is closely related to a derived fiber product diagram for the wall-crossing and simple to understand from the viewpoint of commutative differential graded algebras. However, from the perspective of algebraic varieties, the kernel can be quite complicated, corresponding to a complex with multiple homology sheaves. Under mild assumptions in the Calabi-Yau case, we prove that this kernel provides an equivalence between the category of perfect complexes on the two GIT quotients. More generally, we obtain semi-orthogonal decompositions which show that these categories differ by a certain number of copies of the derived category of the derived fixed locus. The derived equivalence for the Mukai flop is recovered as a very special case.

On the preimage of a sphere by a polynomial mapping

Let X be an irreducible complex affine variety of dimension greater than one and let f:X rightarrow mathbb {C}^m be a polynomial mapping. Let {|}*{|} be a semialgebraic norm on mathbb {C}^m. Then for R large enough the sets f^{-1}(B_R), f^{-1}(S_R), X{setminus } f^{-1}(B_R) are all connected, where B_R={ zin mathbb {C}^m : |z|le R} and S_R={ zin mathbb {C}^m : |z| = R}. As an application we show that if F is a counterexample to the Jacobian Conjecture, then the non-properness set of F has a non-trivial link at infinity.

Representations of Lie algebras of vector fields on affine varieties

For an irreducible affine variety X over an algebraically closed field of characteristic zero we define two new classes of modules over the Lie algebra of vector fields on X—gauge modules and Rudakov modules, which admit a compatible action of the algebra of functions. Gauge modules are generalizations of modules of tensor densities whose construction was inspired by non-abelian gauge theory. Rudakov modules are generalizations of a family of induced modules over the Lie algebra of derivations of a polynomial ring studied by Rudakov [23]. We prove general simplicity theorems for these two types of modules and establish a pairing between them.

Lie algebras of vector fields on smooth affine varieties

ABSTRACTWe reprove the results of Jordan [18] and Siebert [30] and show that the Lie algebra of polynomial vector fields on an irreducible affine variety X is simple if and only if X is a smooth variety. Given proof is self-contained and does not depend on papers mentioned above. Besides, the structure of the module of polynomial functions on an irreducible smooth affine variety over the Lie algebra of vector fields is studied. Examples of Lie algebras of polynomial vector fields on an N-dimensional sphere, non-singular hyperelliptic curves and linear algebraic groups are considered.

Automorphism groups of affine varieties and a characterization of affine $n$-space

We show that the automorphism group of affine -space determines up to isomorphism: If is a connected affine variety such that as ind-groups, then as varieties. We also show that every torus appears as for a suitable irreducible affine variety , but that cannot be isomorphic to a semisimple group. In fact, if is finite-dimensional and if , then the connected component is a torus. Concerning the structure of we prove that any homomorphism of ind-groups either factors through where is the Jacobian determinant, or it is a closed immersion. For we show that every nontrivial homomorphism is a closed immersion. Finally, we prove that every nontrivial homomorphism is an automorphism, and that is given by conjugation with an element from .

On nilpotent and solvable Lie algebras of derivations

Let K be a field and A be a commutative associative K-algebra which is an integral domain. The Lie algebra DerKA of all K-derivations of A is an A-module in a natural way, and if R is the quotient field of A then RDerKA is a vector space over R. It is proved that if L is a nilpotent subalgebra of RDerKA of rank k over R (i.e. such that dimRRL=k), then the derived length of L is at most k and L is finite dimensional over its field of constants. In case of solvable Lie algebras over a field of characteristic zero their derived length does not exceed 2k. Nilpotent and solvable Lie algebras of rank 1 and 2 (over R) from the Lie algebra RDerKA are characterized. As a consequence we obtain the same estimations for nilpotent and solvable Lie algebras of vector fields with polynomial, rational, or formal coefficients. Analogously, if X is an irreducible affine variety of dimension n over an algebraically closed field K of characteristic zero and AX is its coordinate ring, then all nilpotent (solvable) subalgebras of DerKAX have derived length at most n (2n respectively).

Lehmer points and visible points on affine varieties over finite fields

AbstractLet V be an absolutely irreducible affine variety over $\mathbb{F}_p$. A Lehmer point on V is a point whose coordinates satisfy some prescribed congruence conditions, and a visible point is one whose coordinates are relatively prime. Asymptotic results for the number of Lehmer points and visible points on V are obtained, and the distribution of visible points into different congruence classes is investigated.

Wronskian of derivations

An associative multilinear polynomial depending on 16 variables and being skew-symmetric with respect to 12 of them is presented. This polynomial provides us with a mapping recovering the algebra of regular functions of an irreducible affine variety from any smooth involutive distribution of dimension 2.

The Beilinson equivalence for differential operators

Let D be the ring of differential operators on a smooth irreducible affine variety X over C , or, more generally, the enveloping algebra of any locally free Lie algebroid on X . The category of finitely generated graded modules of the Rees algebra D ˜ has a natural quotient category P D which imitates the category of modules on P r o j of a graded commutative ring. We show that the derived category D b ( P D ) is equivalent to the derived category of finitely generated modules of a sheaf of algebras E on X which is coherent over X . This generalizes the usual Beilinson equivalence for projective space, and also the Beilinson equivalence for differential operators on a smooth curve used by Ben-Zvi and Nevins in [6] to describe the moduli space of left ideals in D .

Finite Generation of the Group of Eigenvalues for Sets of Derivations or Automorphisms of Division Algebras

Let D be a division algebra such that D⊗ D o is a Noetherian algebra, and Δ be a set of derivations or automorphisms of the division algebra D, then the group Ev(Δ) of common eigenvalues (i.e., weights) is a finitely generated abelian group. Typical examples of D are the quotient division algebra Frac (𝒟 (X)) of the ring of differential operators 𝒟 (X) on a smooth irreducible affine variety X over a field K of characteristic zero, and the quotient division algebra Frac (U (𝔤)) of the universal enveloping algebra U(𝔤) of a finite dimensional Lie algebra 𝔤.

Rational and replacement invariants of a group action

Group actions are ubiquitous in mathematics. They arise in diverse areas of applications, from classical mechanics to computer vision. A classical but central problem is to compute a generating set of invariants. We consider a rational group action on the affine space and propose a construction of a finite set of rational invariants and a simple algorithm to rewrite any rational invariant in terms of those generators. The construction comes into two variants both consisting in computing a reduced Gröbner basis of a polynomial ideal. That polynomial ideal is of dimension zero in the second variant that relies on the choice of a cross-section, a variety that intersects generic orbits in a finite number of points. A generic linear space of complementary dimension to the orbits can be chosen for cross-section. When the intersection of a generic orbit with the cross-section consists of a single point, the rewriting of any rational invariant in terms of the computed generating set trivializes into a replacement. For general cross-sections we introduce a finite set of <i>replacement invariants</i> that are algebraic functions of the rational invariants. Any rational invariant can be rewritten in terms of those by simple substitution. We have therefore obtained an algebraic formulation of the moving frame construction of Fels and Olver [2], providing a bridge between the algebraic theory of polynomial and rational invariants [7, 1], and the differential-geometric theory of local smooth invariants, [6]. In this abstract we formalize our main results in the case where K is an algebraically closed field of characteristic zero. Several examples, both classical and original, are treated in the poster. Take an algebraic group <i>G</i> given by an unmixed dimensional ideal <i>G</i> in a polynomial ring K[λ<inf>1</inf>,...,λ<inf>ℓ</inf>]. A rational action of the group <i>G</i> on <i>K</i><sup><i>n</i></sup> is given as the rational map: [EQUATION] where <i>h,g</i>1,...,<i>g</i><inf><i>n</i></inf> are polynomial functions in K[λ<inf>1</inf>,...,λ<inf>ℓ</inf>,<i>z</i><inf>1</inf>,...,<i>z</i><inf><i>n</i></inf>]. An element <i>p</i>/<i>q</i> ∈ <i>K</i>(<i>z</i>) is a <i>rational invariant</i> if <i>p</i>(λ·<i>z</i>)<i>q</i>(<i>z</i>) = <i>p</i>(<i>z</i>)<i>q</i>(λ<i>z</i>) mod <i>G</i>. The set of rational invariants forms a (finitely generated) field K(<i>z</i>)<sup><i>G</i></sup>. A <i>cross-section of degree d</i> > 0 is an irreducible affine variety <i>P</i> that intersects generic orbits in exactly <i>d</i> simple points. When generic orbits are of dimension <i>r</i> > 0, a generic linear affine space of codimension <i>r</i> is a cross-section. Consider a new set of variables <i>Z</i> = (<i>Z</i><inf>1</inf>,...,<i>Z</i><inf><i>n</i></inf>). The ideal (<i>Z</i> - λ · <i>z</i>) is the saturation by <i>h</i> of the ideal generated by the polynomials <i>h</i>(λ,<i>z</i>)<i>Z</i><inf><i>i</i></inf> - <i>g</i><inf><i>i</i></inf>(λ,<i>z</i>), 1 ≤ <i>i</i> ≤ <i>n</i>. We define the following two elimination ideals: [EQUATION] where <i>P</i> ⊂ K[<i>Z</i>] is the ideal of the chosen cross-section <i>P</i>. The variety of <i>O</i> in K<sup><i>n</i></sup> x K<sup><i>n</i></sup> is the closure of the graph of the action. If we consider a projection on the second component K<sup><i>n</i></sup>, the fiber above <i>P</i> is the variety of <i>I</i>. The extensions <i>O</i><sup><i>e</i></sup> and <i>I</i><sup><i>e</i></sup> of the ideals <i>O</i> and <i>I</i> to K(<i>z</i>)[<i>Z</i>] are both equidimensional of respective dimension <i>r</i> and 0. They are the heart of our construction. The following results are valid for any term order chosen on <i>Z.</i>

A Question of Rentschler and the Dixmier Problem

The following theorem is the main result of the paper. THEOREM. Let R and T be somewhat commutative algebras with the same holonomic number and let φ: R → T be an algebra homomorphism. Then every holonomic T-module M is (via φ) a holonomic R-module and has finite length as an R-module. When applied to the Weyl algebra this result gives a positive answer to a question of Rentschler. In the important case where R = D(X) and T = D(Y) are rings of differential operators on smooth irreducible algebraic affine varieties X and Y of the same dimension the result means that holonomicity is preserved by twisting (by an arbitrary algebra homomorphism). A short proof is given of the well-known fact that an affirmative solution to the Dixmier Problem (whether every algebra endomorphism φ of the Weyl algebra An is an automorphism) implies the Jacobian Conjecture. The Dixmier Problem has a positive answer if and only if the (twisted from both sides) A n -bimodule φAnφ is simple for each φ. (To begin with, it is not even clear whether it is finitely generated). The theorem implies that it has finite length (moreover, the bimodule is holonomic, thus simple subfactors of it have the least possible Gelfand-Kirillov dimension).

Irreducible Affine Varieties over a Free Group: II. Systems in Triangular Quasi-quadratic Form and Description of Residually Free Groups

We shall prove the conjecture of Myasnikov and Remeslennikov [4] which states that a finitely generated group is fully residually free (every finite set of nontrivial elements has nontrivial images under some homomorphism into a free group) if and only if it is embeddable in the Lyndon's exponential groupFZ[x], which is theZ[x]-completion of the free group. HereZ[x] is the ring of polynomials of one variable with integer coefficients. Historically, Lyndon's attempts to solve Tarski's famous problem concerning the elementary equivalence of free groups of different ranks led him to introduceFZ[x].An ∃-free group is a groupGsuch that the class of ∃-formulas, true inG, is the same as the class of ∃-formulas, true in a nonabelian free group. A finitely generated group is ∃-free if and only if it is fully residually free [22]. Our result gives an algebraic description of ∃-free groups.We shall give an algorithm to represent a solution set of an arbitrary system of equations overFas a union of finite number of irreducible components in the Zariski topology onFn. The solution set for every system is contained in the solution set of a finite number of systems in triangular form with quadratic words as leading terms. The possibility of such a decomposition for a solution set was conjectured by Razborov in [20] and also by Rips.We shall give a description of systems of equations determining irreducible components using methods developed in [13,19]; it is possible to find some of these methods in [18]. We are thankful to E. Rips for attracting our attention to these techniques.

Decidable regularly closed fields of algebraic numbers

A field K is regularly closed if every absolutely irreducible affine variety defined over K has K-rational points. This notion was first isolated by Ax [A] in his work on the elementary theory of finite fields. Later Jarden [J2] and Jarden and Kiehne [JK] extended this in different directions. One of the primary results in this area is that the elementary properties of a regularly closed field K with a free Galois group (on either finitely or countably many generators) are determined by the set of integer polynomials in one indeterminate with a zero in K. The method of proof employed in [J1], [J2] and [JK] is unusual for algebra since it is a measure-theoretic argument. In this brief summary we have not made any attempt at completeness. We refer the reader to the recent paper of Cherlin, van den Dries, and Macintyre [CDM] and to the forthcoming book by Fried and Jarden [FJ] for a more thorough discussion of the latest results. We would like to thank Moshe Jarden, Angus Macintyre, and Zoe Chatzidakis for their comments on an earlier version of this paper.A countable field K is ω-free if the absolute Galois group , where is the algebraic closure of K and is the free profinite group on ℵ0 generators.

Differential Ideals in Rings of Finitely Generated Type

Let 6 be a commutative Noetherian ring containing the rational numbers, T a family of derivations mapping 6 into itself, and f a T-differential 6ideal (i. e., DE C f for every D C T). Theorem 1 shows that the associated prime ideals of f are also T-differential and that f has an irredundant primary decomposition into T-differential primary ideals. Theorem 2 shows that in studying a differential (i. e., T-differential for maximal T) prime ideal p in a finitely generated extension =k [xr, , xn] of a base field k one can localize, i. e., that p is differential in J if and only if p 6p is differential in the local ring 6p. Already here one needs more than the Noetherian condition and the study becomes essentially one for rings of finitely generated type, i. e., quotient rings of finite extensions of a base field k (of characteristic 0). We describe the further results for the case that j = k[x] is a finite integral domain or, equivalently, is the ring of an irreducible affine variety V over k. Theorem 3 shows that the simple subvarieties of V yield nondifferential ideals . By a lemma of Zariski, if p 6p is not differential, then the completion 6p of 6p is of the form O [Em]], where u C p p and is analytically independent over E. We say that V is analytically a product at or along an irreducible subvariety 't if the completion of O., where p =c ('t), is of the form just mentioned. Intuitively we think of V as somewhat like a cylinder built on the section u 0 of V. Then a component of the singular locus of V could not very well yield a non-differential ideal in 6; and the facts correspond to this intuition: Theorems 4 and 5 give the proof. Theorem 12, Corollary shows that V is analytically a product along a subvariety St if and only if the ideal of ' in the ring of V is not differential. While this is in some ways a satisfactory criterion (for a prime ideal to be differential), we are further concerned with the question whether the condition