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Derived Tensor Product
In algebra, given a differential graded algebra ''A'' over a commutative ring ''R'', the derived tensor product functor is :- \otimes_A^ - : D(\mathsf_A) \times D(_A \mathsf) \to D(_R \mathsf) where \mathsf_A and _A \mathsf are the categories of right ''A''-modules and left ''A''-modules and ''D'' refers to the homotopy category (i.e., derived category). By definition, it is the left derived functor of the tensor product functor - \otimes_A - : \mathsf_A \times _A \mathsf \to _R \mathsf. Derived tensor product in derived ring theory If ''R'' is an ordinary ring and ''M'', ''N'' right and left modules over it, then, regarding them as discrete spectra, one can form the smash product of them: :M \otimes_R^L N whose ''i''-th homotopy is the ''i''-th Tor: :\pi_i (M \otimes_R^L N) = \operatorname^R_i(M, N). It is called the derived tensor product of ''M'' and ''N''. In particular, \pi_0 (M \otimes_R^L N) is the usual tensor product of modules ''M'' and ''N'' over ''R''. Geometrically, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Graded Algebra
In mathematics – particularly in homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often used to capture information about a topological or geometric space. Explicitly, a differential graded algebra is a graded associative algebra with a chain complex structure that is compatible with the algebra structure. In geometry, the de Rham algebra of differential forms on a manifold has the structure of a differential graded algebra, and it encodes the de Rham cohomology of the manifold. In algebraic topology, the singular cochains of a topological space form a DGA encoding the singular cohomology. Moreover, American mathematician Dennis Sullivan developed a DGA to encode the rational homotopy type of topological spaces. __TOC__ Definitions Let A_\bullet = \bigoplus\nolimits_ A_i be a \mathbb-graded algebra, with product \cdot, equipped with a map d\colon A_\bullet \to A_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Commutative rings appear in the following chain of subclass (set theory), class inclusions: Definition and first examples Definition A ''ring'' is a Set (mathematics), set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Modules
In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring of integers Z, it is the same thing as the category of abelian groups. The category of right modules is defined in a similar way. One can also define the category of bimodules over a ring ''R'' but that category is equivalent to the category of left (or right) modules over the enveloping algebra of ''R'' (or over the opposite of that). Note: Some authors use the term module category for the category of modules. This term can be ambiguous since it could also refer to a category with a monoidal-category action. Properties The categories of left and right modules are abelian categories. These categories have enough projectives and enough injectives. Mitchell's embedding theorem states every abelian category arises as a full sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proceeds on the basis that the objects of ''D''(''A'') should be chain complexes in ''A'', with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences. The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product Of Modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology, algebraic geometry, operator algebras and noncommutative geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection Product
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two Line (geometry), lines in a Plane (geometry), plane are not parallel, their intersection is the Point (geometry), point at which they meet. More generally, in set theory, the intersection of sets is defined to be the set of Element (mathematics), elements which belong to all of them. Unlike the Euclidean definition, this does not presume that the objects under consideration lie in a common space (mathematics), space. It simply means the overlapping area of two or more objects or geometries. Intersection is one of the basic concepts of geometry. An intersection can have various geometric shapes, but a point (geometry), point is the most common in a plane geometry. Incidence geometry defines an intersection (usually, of flat (geometry), flats) as an object of lower dimension (mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Scheme
In algebraic geometry, a derived scheme is a homotopy-theoretic generalization of a scheme in which classical commutative rings are replaced with derived versions such as differential graded algebras, commutative simplicial rings, or commutative ring spectra. From the functor of points point-of-view, a derived scheme is a sheaf ''X'' on the category of simplicial commutative rings which admits an open affine covering \. From the locally ringed space point-of-view, a derived scheme is a pair (X, \mathcal) consisting of a topological space ''X'' and a sheaf \mathcal either of simplicial commutative rings or of commutative ring spectra on ''X'' such that (1) the pair (X, \pi_0 \mathcal) is a scheme and (2) \pi_k \mathcal is a quasi-coherent \pi_0 \mathcal- module. A derived stack is a stacky generalization of a derived scheme. Differential graded scheme Over a field of characteristic zero, the theory is closely related to that of a differential graded scheme. By definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Commutative Ring
In algebra, a simplicial commutative ring is a monoid object, commutative monoid in the category (mathematics), category of simplicial abelian groups, or, equivalently, a simplicial object in the category of commutative rings. If ''A'' is a simplicial commutative ring, then it can be shown that \pi_0 A is a commutative ring, ring and \pi_i A are module (mathematics), modules over that ring (in fact, \pi_* A is a graded ring over \pi_0 A.) A topology-counterpart of this notion is a commutative ring spectrum. Examples *The ring of polynomial differential forms on simplexes. Graded ring structure Let ''A'' be a simplicial commutative ring. Then the ring structure of ''A'' gives \pi_* A = \oplus_ \pi_i A the structure of a graded-commutative graded ring as follows. By the Dold–Kan correspondence, \pi_* A is the homology of the chain complex corresponding to ''A''; in particular, it is a graded abelian group. Next, to multiply two elements, writing S^1 for the simplicial sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cotangent Complex
In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If f: X \to Y is a morphism of geometric or algebraic objects, the corresponding cotangent complex \mathbf_^\bullet can be thought of as a universal "linearization" of it, which serves to control the deformation theory of f. It is constructed as an object in a certain derived category of sheaves on X using the methods of homotopical algebra. Restricted versions of cotangent complexes were first defined in various cases by a number of authors in the early 1960s. In the late 1960s, Michel André and Daniel Quillen independently came up with the correct definition for a morphism of commutative rings, using simplicial methods to make precise the idea of the cotangent complex as given by taking the (non-abelian) left derived functor of Kähler differentials. Luc Illusie then globalized this d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme-theoretic Intersection
In algebraic geometry, the scheme-theoretic intersection of closed subschemes ''X'', ''Y'' of a scheme ''W'' is X \times_W Y, the fiber product of the closed immersions X \hookrightarrow W, Y \hookrightarrow W. It is denoted by X \cap Y. Locally, ''W'' is given as \operatorname R for some ring ''R'' and ''X'', ''Y'' as \operatorname(R/I), \operatorname(R/J) for some ideals ''I'', ''J''. Thus, locally, the intersection X \cap Y is given as :\operatorname(R/(I+J)). Here, we used R/I \otimes_R R/J \simeq R/(I + J) (for this identity, see tensor product of modules#Examples.) Example: Let X \subset \mathbb^n be a projective variety with the homogeneous coordinate ring ''S/I'', where ''S'' is a polynomial ring. If H = \ \subset \mathbb^n is a hypersurface defined by some homogeneous polynomial ''f'' in ''S'', then : X \cap H = \operatorname(S/(I, f)). If ''f'' is linear (deg = 1), it is called a hyperplane section. See also: Bertini's theorem. Now, a scheme-theoretic intersection ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
