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Paul S Smith - One of the best experts on this subject based on the ideXlab platform.
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the grothendieck group of non commutative non noetherian analogues of p1 and regular algebras of global dimension two
Journal of Algebra, 2015Co-Authors: Gautam Sisodia, Paul S SmithAbstract:Abstract Let V be a finite-dimensional positively-graded vector space. Let b ∈ V ⊗ V be a homogeneous element whose rank is dim ( V ) . Let A = T V / ( b ) , the quotient of the tensor algebra TV modulo the 2-sided ideal generated by b. Let gr ( A ) be the category of finitely presented graded left A-modules and fdim ( A ) its Full Subcategory of finite dimensional modules. Let qgr ( A ) be the quotient category gr ( A ) / fdim ( A ) . We compute the Grothendieck group K 0 ( qgr ( A ) ) . In particular, if the reciprocal of the Hilbert series of A, which is a polynomial, is irreducible, then K 0 ( qgr ( A ) ) ≅ Z [ θ ] ⊂ R as ordered abelian groups where θ is the smallest positive real root of that polynomial. When dim k ( V ) = 2 , qgr ( A ) is equivalent to the category of coherent sheaves on the projective line, P 1 , or a stacky P 1 if V is not concentrated in degree 1. If dim k ( V ) ≥ 3 , results of Piontkovski and Minamoto suggest that qgr ( A ) behaves as if it is the category of “coherent sheaves” on a non-commutative, non-noetherian analogue of P 1 .
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a non commutative homogeneous coordinate ring for the degree six del pezzo surface
Journal of Algebra, 2012Co-Authors: Paul S SmithAbstract:Abstract Let R be the free C -algebra on x and y modulo the relations x 5 = y x y and y 2 = x y x endowed with the Z -grading deg x = 1 and deg y = 2 . The ring R appears, in somewhat hidden guise, in a paper on quiver gauge theories. Let B 3 denote the blow up of CP 2 at three non-colinear points. The main result in this paper is that the category of quasi-coherent O B 3 -modules is equivalent to the quotient of the category of Z -graded R-modules modulo the Full Subcategory of modules that are the sum of their finite dimensional submodules. 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, Van den Bergh, and Stephenson implies that R is a noetherian domain of global dimension three.
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the space of penrose tilings and the non commutative curve with homogeneous coordinate ring k y 2
arXiv: Rings and Algebras, 2011Co-Authors: Paul S SmithAbstract: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.
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a non commutative homogeneous coordinate ring for the degree six del pezzo surface
arXiv: Rings and Algebras, 2009Co-Authors: Paul S SmithAbstract: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.
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a non commutative homogeneous coordinate ring for the third del pezzo surface
arXiv: Rings and Algebras, 2009Co-Authors: Paul S SmithAbstract:Let R be the free C-algebra on x and y modulo the relations x 5 = yxy and y 2 = xyx endowed with the Z-grading deg x = 1 and deg y = 2. Let B3 denote the blow up of CP 2 at three non-colliear points. The main result in this paper is that the category of quasi-coherent OB3-modules is equivalent to the quotient of the category of Z-graded R-modules modulo the Full Subcategory of modules M such that for each m ∈ M, (x, y) n m = 0 for n ≫ 0. This reduces almost all representation-theoretic questions about R to algebraic geometric questions about the del Pezzo surface B3. 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
Michael Muger - One of the best experts on this subject based on the ideXlab platform.
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galois theory for braided tensor categories and the modular closure
Advances in Mathematics, 2000Co-Authors: Michael MugerAbstract:Abstract Given a braided tensor *-category C with conjugate (dual) objects and irreducible unit together with a Full symmetric Subcategory S we define a crossed product C ⋊ S . This construction yields a tensor *-category with conjugates and an irreducible unit. (A *-category is a category enriched over Vect C with positive *-operation.) A Galois correspondence is established between intermediate categories sitting between C and C ⋊ S and closed subgroups of the Galois group Gal( C ⋊ S / C )=Aut C ( C ⋊ S ) of C , the latter being isomorphic to the compact group associated with S by the duality theorem of Doplicher and Roberts. Denoting by D ⊂ C the Full Subcategory of degenerate objects, i.e., objects which have trivial monodromy with all objects of C , the braiding of C extends to a braiding of C ⋊ S iff S ⊂ D . Under this condition, C ⋊ S has no non-trivial degenerate objects iff S = D . If the original category C is rational (i.e., has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category C ≡ C ⋊ D is called the modular closure of C since in the rational case it is modular, i.e., gives rise to a unitary representation of the modular group SL(2, Z ). If all simple objects of S have dimension one the structure of the category C ⋊ S can be clarified quite explicitly in terms of group cohomology.
Evans, David E. - One of the best experts on this subject based on the ideXlab platform.
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Realizing the braided Temperley-Lieb-Jones C*-tensor categories as Hilbert C*-modules
'Springer Science and Business Media LLC', 2020Co-Authors: Aaserud, Andreas Naes, Evans, David E.Abstract:We associate to each Temperley-Lieb-Jones C*-tensor category TLJcat(delta) with parameter delta in the discrete range {2\cos(\pi/(k+2)),: ,k=1,2, ...} or 2 a certain C*-algebra B of compact operators. We use the unitary braiding on TLJcat(delta) to equip the category ModB of (right) Hilbert B-modules with the structure of a braided C*-tensor category. We show that TLJcat(delta) is equivalent, as a braided C*-tensor category, to the Full Subcategory of ModB whose objects are those modules which admit a finite orthonormal basis. Finally, we indicate how these considerations generalize to arbitrary finitely generated rigid braided C*-tensor categories
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Realizing the braided Temperley-Lieb-Jones C*-tensor categories as Hilbert C*-modules
'Springer Science and Business Media LLC', 2020Co-Authors: Aaserud, Andreas Naes, Evans, David E.Abstract:We associate to each Temperley-Lieb-Jones C*-tensor category $\mathcal{T}\!\mathcal{L}\mathcal{J}(\delta)$ with parameter $\delta$ in the discrete range $\{2\cos(\pi/(k+2))\,:\,k=1,2,\ldots\}\cup\{2\}$ a certain C*-algebra $\mathcal{B}$ of compact operators. We use the unitary braiding on $\mathcal{T}\!\mathcal{L}\mathcal{J}(\delta)$ to equip the category $\mathrm{Mod}_{\mathcal{B}}$ of (right) Hilbert $\mathcal{B}$-modules with the structure of a braided C*-tensor category. We show that $\mathcal{T}\!\mathcal{L}\mathcal{J}(\delta)$ is equivalent, as a braided C*-tensor category, to the Full Subcategory $\mathrm{Mod}_{\mathcal{B}}^f$ of $\mathrm{Mod}_{\mathcal{B}}$ whose objects are those modules which admit a finite orthonormal basis. Finally, we indicate how these considerations generalize to arbitrary finitely generated rigid braided C*-tensor categories.Comment: In the latest version, we corrected a couple of typos and reformatted the bibliography. The paper will appear in essentially this form in Communications in Mathematical Physics (2020
Positselski Leonid - One of the best experts on this subject based on the ideXlab platform.
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Unbounded derived categories of small and big modules: Is the natural functor Fully faithful?
'Elsevier BV', 2021Co-Authors: Positselski Leonid, Schnürer, Olaf M.Abstract:Consider the obvious functor from the unbounded derived category of all finitely generated modules over a left noetherian ring $R$ to the unbounded derived category of all modules. We answer the natural question whether this functor defines an equivalence onto the Full Subcategory of complexes with finitely generated cohomology modules in two special cases. If $R$ is a quasi-Frobenius ring of infinite global dimension, then this functor is not Full. If $R$ has finite left global dimension, this functor is an equivalence. We also prove variants of the latter assertion for left coherent rings, for noetherian schemes and for locally noetherian Grothendieck categories.Comment: 23 pages, typo correcte
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Coherent analogues of matrix factorizations and relative singularity categories
'Mathematical Sciences Publishers', 2015Co-Authors: Efimov Alexander, Positselski LeonidAbstract:We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations is then identified with the triangulated category of relative singularities, while the similar exotic derived category of locally free matrix factorizations is its Full Subcategory. The latter category is identified with the kernel of the direct image functor corresponding to the closed embedding of the zero locus and acting between the conventional (absolute) triangulated categories of singularities. Similar results are obtained for matrix factorizations of infinite rank; and two different "large" versions of the triangulated category of relative singularities, corresponding to the approaches of Orlov and Krause, are identified in the case of a Cartier divisor. A version of the Thomason-Trobaugh-Neeman localization theory is proven for coherent matrix factorizations and disproven for locally free matrix factorizations of finite rank. Contravariant (coherent) and covariant (quasi-coherent) versions of the Serre-Grothendieck duality theorems for matrix factorizations are established, and pull-backs and push-forwards of matrix factorizations are discussed at length. A number of general results about derived categories of the second kind for CDG-modules over quasi-coherent CDG-algebras are proven on the way. Hochschild (co)homology of matrix factorization categories are discussed in an appendix.Comment: LaTeX 2e with pb-diagram and xy-pic; 114 pages, 13 commutative diagrams. v.8: new sections 2.10, 3.1 and 3.7 inserted; v.9: appendix B added, remarks inserted in sections 2.10 and 2.7, section 1.8 expanded; v.10: new section 3.3 inserted, the whole paper has two authors now; v.11: small corrections, additions, and improvements -- this is intended as the final versio
Aaserud, Andreas Naes - One of the best experts on this subject based on the ideXlab platform.
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Realizing the braided Temperley-Lieb-Jones C*-tensor categories as Hilbert C*-modules
'Springer Science and Business Media LLC', 2020Co-Authors: Aaserud, Andreas Naes, Evans, David E.Abstract:We associate to each Temperley-Lieb-Jones C*-tensor category TLJcat(delta) with parameter delta in the discrete range {2\cos(\pi/(k+2)),: ,k=1,2, ...} or 2 a certain C*-algebra B of compact operators. We use the unitary braiding on TLJcat(delta) to equip the category ModB of (right) Hilbert B-modules with the structure of a braided C*-tensor category. We show that TLJcat(delta) is equivalent, as a braided C*-tensor category, to the Full Subcategory of ModB whose objects are those modules which admit a finite orthonormal basis. Finally, we indicate how these considerations generalize to arbitrary finitely generated rigid braided C*-tensor categories
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Realizing the braided Temperley-Lieb-Jones C*-tensor categories as Hilbert C*-modules
'Springer Science and Business Media LLC', 2020Co-Authors: Aaserud, Andreas Naes, Evans, David E.Abstract:We associate to each Temperley-Lieb-Jones C*-tensor category $\mathcal{T}\!\mathcal{L}\mathcal{J}(\delta)$ with parameter $\delta$ in the discrete range $\{2\cos(\pi/(k+2))\,:\,k=1,2,\ldots\}\cup\{2\}$ a certain C*-algebra $\mathcal{B}$ of compact operators. We use the unitary braiding on $\mathcal{T}\!\mathcal{L}\mathcal{J}(\delta)$ to equip the category $\mathrm{Mod}_{\mathcal{B}}$ of (right) Hilbert $\mathcal{B}$-modules with the structure of a braided C*-tensor category. We show that $\mathcal{T}\!\mathcal{L}\mathcal{J}(\delta)$ is equivalent, as a braided C*-tensor category, to the Full Subcategory $\mathrm{Mod}_{\mathcal{B}}^f$ of $\mathrm{Mod}_{\mathcal{B}}$ whose objects are those modules which admit a finite orthonormal basis. Finally, we indicate how these considerations generalize to arbitrary finitely generated rigid braided C*-tensor categories.Comment: In the latest version, we corrected a couple of typos and reformatted the bibliography. The paper will appear in essentially this form in Communications in Mathematical Physics (2020
