Searcharxiv⌕ Search

arXiv subjects

S. Paul Smith

Publications and source records attributed to S. Paul Smith.

At least 37 records · Page 2Linked to original sources
arXiv2013

Free algebras arising from positive-entropy automorphisms of surfaces

Let X be a smooth projective surface defined over an uncountable algebraically closed field k and let k(X) be its field of rational functions. Let s be an automorphism of X. This paper proves there is a non-negative integer n and elements a and b in k(X) such that the subalgebra of the skew Laurent extension k(X)[t, t^{-1};s] generated by at^n and bt^n is a free algebra if and only if the spectral radius for the action of s^* on the Neron-Severi group of X is >1. Thus, when s is an automorphism of a smooth complex projective surface X, C(X)[t,t^{-1};s] has a free subalgebra on >1 variables if and only if the topological entropy of s is positive. Furthermore, if s is an automorphism of k(X), then k(X)[t,t^{-1};s] contains a free subalgebra if and only if the dynamical degree of s is >1; in this situation, s might not be induced by an automorphism of any smooth projective surface. These results are used to show that certain twisted homogeneous coordinate rings of smooth projective surfaces contain free subalgebras.

math.RA↗
arXiv2013

Free subalgebras of the skew polynomial rings k[x,y][t;σ] and k[x,x^{-1},y,y^{-1}][t;σ]

Let σbe an automorphism of a commutative k-algebra R. The skew polynomial ring R[t;σ] is generated by R and an indeterminate t subject to the relations ta=σ(a)t for all a in R. For certain R and appropriate σthere are elements a and b in R such that the subalgebra of R[t;σ] generated by at and bt is a free algebra. If σis an automorphism of the polynomial ring k[x,y], then the subalgebra of k[x,y][t;σ] generated by xt and yt is free if and only if σis not conjugate to an elementary automorphism. If σis an automorphism of k[x,x^{-1},y,y^{- 1}] of the form σ(x)=x^ay^b and σ(y)=x^cy^d, then the subalgebra of k[x,x^{-1},y,y^{- 1}][t;σ] generated by xt and yt is free if the spectral radius of the 2x2 matrix {{a b} \\ {c d}} is >2; indeed, k[x,x^{-1},y,y^{- 1}][t;σ] contains a free subalgebra if and only if the spectral radius of 2x2 matrix {{a b} \\ {c d}} is >1.

math.RA↗
arXiv2013

Noncommutative complex differential geometry

This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a differential structure on a noncommutative algebra defined in terms of a differential graded algebra. This is compared to current ideas on noncommutative algebraic geometry.

math.AG↗
arXiv2012

The space of Penrose tilings and the non-commutative curve with homogeneous coordinate ring k /(y^2)

We construct a non-commutative scheme that behaves as if it is the space of Penrose tilings of the plane. Let k be a field and B=k (y^2). We consider B as the homogeneous coordinate ring of a non-commutative projective scheme. The category of "quasi-coherent sheaves" on it is, by fiat, the quotient category QGr(B):=Gr(B)/Fdim(B) and the category of coherent sheaves on it is qgr(B):=gr(B)/fdim(B), where gr(B) is the category of finitely presented graded modules and fdim(B) is the full subcategory of finite dimensional graded modules. We show that QGr B is equivalent to Mod S, the category of left modules over the ring S that is the direct limit of the directed system of finite dimensional semisimple algebras S_n=M_{f_n}(k) + M_{f_{n-1}}(k) where f_{n-1} and f_n$ are adjacent Fibonacci numbers and the maps S_n \to S_{n+1} are (a,b)--->(diag(a,b),a). When k is the complex numbers, the norm closure of S is the C^*-algebra Connes uses to view the space of Penrose tilings as a non-commutative space. Objects in QGr B have projective resolutions of length at most one so the non-commutative scheme is, in a certain sense, a smooth non-commutative curve. Penrose tilings of the plane are in bijection with infinite sequences z=z_0z_1 ... of 0s and 1s with no consecutive 1s. We associate to each such sequence a graded B-module, a "point module", that becomes a simple object O_z in QGr B that we think of as a "skyscraper sheaf" at a "point" on this non-commutative curve. Tilings T_z and T_{z'} determined by two such sequences are equivalent, i.e., the same up to a translation on R^2, if and only if O_z is isomorphic to O_{z'}. A result of Herbera shows that Ext^1(O_z,O_{z'}) is non-zero for all z and z'. This as an algebraic analogue of the fact that every equivalence class of tilings is dense in the set of all Penrose tilings.

math.RA↗
arXiv2012

The Grothendieck Group of a Quantum Projective Space Bundle

We compute the Grothendieck group K_0 of non-commutative analogues of quantum projective space bundles. Our results specialize to give the Grothendieck groups of non-commutative analogues of projective spaces, and specialize to recover the Grothendieck group of a usual projective space bundle over a regular noetherian separated scheme. As an application we develop an intersection theory for the quantum ruled surfaces defined by Van den Bergh.

math.QA↗
arXiv2012

Category equivalences involving graded modules over path algebras of quivers

Let kQ be the path algebra of a quiver Q with its standard grading. We show that the category of graded kQ-modules modulo those that are the sum of their finite dimensional submodules, QGr(kQ), is equivalent to several other categories: the graded modules over a suitable Leavitt path algebra, the modules over a certain direct limit of finite dimensional multi-matrix algebras, QGr(kQ') where Q' is the quiver whose incidence matrix is the n^{th} power of that for Q, and others. A relation with a suitable Cuntz-Krieger algebra is established. All short exact sequences in the full subcategory of finitely presented objects in QGr(kQ), split so that subcategory can be given the structure of a triangulated category with suspension functor the Serre degree twist (-1); it is shown that this triangulated category is equivalent to the "singularity category" for the radical square zero algebra kQ/kQ_{\ge 2}.

math.RA↗
arXiv2012

Corrigendum to "An equivalence of categories for graded modules over monomial algebras and path algebras of quivers" [J. Algebra, 353(1) (2012) 249-260]

Our published paper contains an incorrect statement of a result due to Artin and Zhang. This corrigendum gives the correct statement of their result and includes a new result that allows us to use their result to prove our main theorem. Thus the main theorem of our published paper is correct as stated but its proof must be modified.

math.RA↗
arXiv2012

A non-commutative homogeneous coordinate ring for the degree six del Pezzo surface

Let R be the free algebra on x and y modulo the relations x^5=yxy and y^2=xyx endowed with the grading deg x=1 and deg y=2. Let B_3 denote the blow up of the projective plane at three non-colliear points. The main result in this paper is that the category of quasi-coherent sheaves on B_3 is equivalent to the quotient of the category of graded R-modules modulo the full subcategory of modules M such that for each m in M, $(x,y)^nm=0$ for n sufficiently large. This is proved by showing the R is a twisted homogeneous coordinate ring (in the sense of Artin and Van den Bergh) for B_3. This reduces almost all representation-theoretic questions about R to algebraic geometric questions about the del Pezzo surface B_3. For example, the generic simple R-module has dimension six. Furthermore, the main result combined with results of Artin, Tate, and Van den Bergh, imply that R is a noetherian domain of global dimension three, and has other good homological properties.

math.RA↗
arXiv2011

Degenerate 3-dimensional Sklyanin algebras are monomial algebras

The 3-dimensional Sklyanin algebras, S(a,b,c), over a field k, form a flat family parametrized by points (a,b,c) lying in P^2-D, the complement of a set D of 12 points in the projective plane, P^2. When (a,b,c) is in D the algebras having the same defining relations as the 3-dimensional Sklyanin algebras are said to be "degenerate". Chelsea Walton showed the degenerate 3-dimensional Sklyanin algebras do not have the same properties as the non-degenerate ones. Here we prove that a degenerate Sklyanin algebra is isomorphic to the free algebra on u,v,w, modulo either the relations u^2=v^2=w^2=0 or the relations uv=vw=wu=0. These monomial algebras are Zhang twists of each other. Therefore all degenerate Sklyanin algebras have the same category of graded modules. A number of properties of the degenerate Sklyanin algebras follow from this observation. We exhibit a quiver Q and an ultramatricial algebra R such that if S is a degenerate Sklyanin algebra then the categories QGr(S), QGr(kQ), and Mod(R), are equivalent; neither Q nor R depends on S. Here QGr(-) denotes the category of graded right modules modulo the full subcategory of graded modules that are the sum of their finite dimensional submodules.

math.RA↗
arXiv2011

An equivalence of categories for graded modules over monomial algebras and path algebras of quivers

Let A be a finitely generated connected graded k-algebra defined by a finite number of monomial relations. Then there is a finite directed graph, Q, the Ufnarovskii graph of A, for which the categories QGr(A) and QGr(kQ) are equivalent: QGr(A) denotes the quotient category of graded A-modules modulo the subcategory consisting of those that are the sum of their finite dimensional submodules; QGr(kQ) has a similar definition. The proof makes use of an algebra homomorphism A--->kQ that may be of independent interest.

math.RA↗
arXiv2011

Shift equivalence and a category equivalence involving graded modules over path algebras of quivers

In this paper we associate an abelian category to a finite directed graph and prove the categories arising from two graphs are equivalent if the incidence matrices of the graphs are shift equivalent. The abelian category is the quotient of the category of graded vector space representations of the quiver obtained by making the graded representations that are the sum of their finite dimensional submodules isomorphic to zero. Actually, the main result in this paper is that the abelian categories are equivalent if the incidence matrices are strong shift equivalent. That result is combined with an earlier result of the author to prove that if the incidence matrices are shift equivalent, then the associated abelian categories are equivalent. Given William's Theorem that subshifts of finite type associated to two directed graphs are conjugate if and only if the graphs are strong shift equivalent, our main result can be reformulated as follows: if the subshifts associated to two directed graphs are conjugate, then the categories associated to those graphs are equivalent.

math.RA↗
arXiv2011

Non-commutative quadric surfaces

The 4-dimensional Sklyanin algebra is the homogeneous coordinate ring of a noncommutative analogue of projective 3-space. The degree-two component of the algebra contains a 2-dimensional subspace of central elements. The zero loci of those central elements, except 0, form a pencil of non-commutative quadric surfaces, We show that the behavior of this pencil is similar to that of a generic pencil of quadrics in the commutative projective 3-space. There are exactly four singular quadrics in the pencil. The singular and non-singular quadrics are characterized by whether they have one or two rulings by non-commutative lines. The Picard groups of the smooth quadrics are free abelian of rank two. The alternating sum of dimensions of Ext groups allows us to define an intersection pairing on the Picard group of the smooth noncommutative quadrics. A surprise is that a smooth noncommutative quadric can sometimes contain a "curve" having self-intersection number -2. Many of the methods used in our paper are noncommutative versions of methods developed by Buchweitz, Eisenbud and Herzog: in particular, the correspondence between the geometry of a quadric hypersurface and maximal Cohen-Macaulay modules over its homogeneous coordinate ring plays a key role. An important aspect of our work is to introduce definitions of non-commutative analogues of the familiar commutative terms used in this abstract. We expect the ideas we develop here for 2-dimensional non-commutative quadric hypersurfaces will apply to higher dimensional non-commutative quadric hypersurfaces and we develop them in sufficient generality to make such applications possible.

math.QA↗
arXiv2011

The non-commutative scheme having a free algebra as a homogeneous coordinate ring

Let k be a field and TV the tensor algebra on a k-vector space V of dimension n>1. This paper proves that the quotient category QGr(TV) := Gr(TV)/Fdim of graded TV-modules modulo those that are unions of finite dimensional modules is equivalent to the category of modules over the direct limit of matrix algebras, M_n(k)^{\otimes r}. QGr(TV) is viewed as the category of "quasi-coherent sheaves" on the non-commutative scheme Proj(TV). The subcategory qgr(TV) consisting of the finitely presented objects is viewed as the category of coherent sheaves on Proj(TV). We show qgr(TV) has no indecomposable objects, no noetherian objects, and no simple objects. Moreover, every short exact sequence in qgr(TV) splits. The equivalence of categories result can be interpreted as saying that Proj(TV) is an "affine non-commutative scheme".

math.RA↗
arXiv2011

A quotient stack related to the Weyl algebra

Let A denote the ring of differential operators on the affine line with its two usual generators t and d/dt given degrees +1 and -1 respectively. Let X be the stack having coarse moduli space the affine line Spec k[z] and isotropy groups Z/2 at each integer point. Then the category of graded A-modules is equivalent to the category of quasi-coherent sheaves on X. Version 2: corrected typos and deleted appendix at referee's suggestion.

math.RA↗
arXiv2011

A 3-Calabi-Yau algebra with G_2 symmetry constructed from the octonions

This paper concerns an associative graded algebra A that is the enveloping algebra of a Lie algebra with exponential growth. The algebra A is 3-Calabi-Yau. There is a Z-form of A so for every field k there is an algebra A_k. An algebraic group of type G_2 acts as degree-preserving automorphisms of A. The algebra A is generated by 7 elements modulo 7 homogeneous quadratic relations. It can be constructed from the octonions; the same construction applied to the quaternions produces the commutative polynomial ring in 3 variables. If V is the 7-dimensional irreducible representation of the complex semisimple Lie algebra of type G_2, then A is isomorphic to the tensor algebra TV modulo the ideal generated by the submodule of V \otimes V that is isomorphic to V. A can be defined as a superpotential algebra derived from a 3-form on R^7 having an open GL(7) orbit and compact isotropy group. Classification of the finite-dimensional representations of A is equivalent to classifying square matrices Y with purely imaginary octonion entries such that the imaginary part of Y^2 is zero. We show that A is a Koszul algebra. Its Koszul dual is a quotient of the exterior algebra on 7 variables, has Hilbert series 1+7t+7t^2+t^3, and is a symmetric Frobenius algebra. Although not noetherian, A is graded coherent (proved by Piontkovski).

math.RA↗
arXiv2010

A Derived Equivalence For A Del Pezzo Surface Of Degree 6 Over An Arbitrary Field

Let $S$ be a degree six del Pezzo surface over an arbitrary field $F$. Motivated by the first author's classification of all such $S$ up to isomorphism in terms of a separable $F$-algebra $B \times Q \times F$, and by his K-theory isomorphism $K_n(S) \cong K_n(B \times Q \times F)$ for $n \ge 0$, we prove an equivalence of derived categories $$ \sD^b(\coh S) \equiv \sD^b(\mod A) $$ where $A$ is an explicitly given finite dimensional $F$-algebra whose semisimple part is $B \times Q \times F$. Submitted to the Journal of K-theory

math.AG↗
arXiv2008

A generalization of Watts's Theorem: Right exact functors on module categories

Watts's Theorem says that a right exact functor F:Mod R-->Mod S that commutes with direct sums is isomorphic to -\otimes_R B where B is the R-S-bimodule FR. The main result in this paper is the following: if A is a cocomplete abelian category and F:Mod R --> A is a right exact functor commuting with direct sums, then F is isomorphic to - \otimes_R B where B is a suitable R-module in A, i.e., a pair (B,f) consisting of an object B in A and a ring homomorphism f:R --> Hom_A(B,B). Part of the point is to give meaning to the notation -\otimes_R B. That is done in the paper by Artin and Zhang on Abstract Hilbert Schemes. The present paper is a natural extension of some of the ideas in the first part of their paper.

math.RA↗