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Perfect Complex
In algebra, a perfect complex of modules over a commutative ring ''A'' is an object in the derived category of ''A''-modules that is quasi-isomorphic to a bounded complex of finite projective ''A''-modules. A perfect module is a module that is perfect when it is viewed as a complex concentrated at degree zero. For example, if ''A'' is Noetherian, a module over ''A'' is perfect if and only if it is finitely generated and of finite projective dimension. Other characterizations Perfect complexes are precisely the compact objects in the unbounded derived category D(A) of ''A''-modules. They are also precisely the dualizable objects in this category. A compact object in the ∞-category of (say right) module spectra over a ring spectrum is often called perfect; see also module spectrum. Pseudo-coherent sheaf When the structure sheaf \mathcal_X is not coherent, working with coherent sheaves has awkwardness (namely the kernel of a finite presentation can fail to be coherent). Beca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Spectrum
In stable homotopy theory, a ring spectrum is a spectrum ''E'' together with a multiplication map :''μ'': ''E'' ∧ ''E'' → ''E'' and a unit map : ''η'': ''S'' → ''E'', where ''S'' is the sphere spectrum. These maps have to satisfy associativity and unitality conditions up to homotopy, much in the same way as the multiplication of a ring is associative and unital. That is, : ''μ'' (id ∧ ''μ'') ~ ''μ'' (''μ'' ∧ id) and : ''μ'' (id ∧ ''η'') ~ id ~ ''μ''(''η'' ∧ id). Examples of ring spectra include singular homology with coefficients in a ring, complex cobordism, K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ..., and Morava K-theory. See also * Highly structured ring spectrum References * Algebraic topology Spectra (topology) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Complex
In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences. Elliptic complexes isolate those features common to the de Rham complex and the Dolbeault complex which are essential for performing Hodge theory. They also arise in connection with the Atiyah-Singer index theorem and Atiyah-Bott fixed point theorem. Definition If ''E''0, ''E''1, ..., ''E''''k'' are vector bundles on a smooth manifold ''M'' (usually taken to be compact), then a differential complex is a sequence :\Gamma(E_0) \stackrel \Gamma(E_1) \stackrel \ldots \stackrel \Gamma(E_k) of differential operators between the sheaves of sections of the ''E''''i'' such that ''P''''i''+1 \circ ''P''''i''=0. A differential complex with first order operators is elliptic if the sequence of symbols :0 \rightarrow \pi^*E_0 \stackrel \pi^*E_1 \stackrel \ldots \stackrel \pi^*E_k \rightarrow 0 is exact outside of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert–Burch Theorem
In mathematics, the Hilbert–Burch theorem describes the structure of some free resolutions of a quotient of a local or graded ring in the case that the quotient has projective dimension 2. proved a version of this theorem for polynomial rings, and proved a more general version. Several other authors later rediscovered and published variations of this theorem. gives a statement and proof. Statement If ''R'' is a local ring with an ideal ''I'' and : 0 \rightarrow R^m\stackrel R^n \rightarrow R \rightarrow R/I\rightarrow 0 is a free resolution of the ''R''- module ''R''/''I'', then ''m'' = ''n'' – 1 and the ideal ''I'' is ''aJ'' where ''a'' is a regular element of ''R'' and ''J'', a depth-2 ideal, is the first Fitting ideal \operatorname_1 I of ''I'', i.e., the ideal generated by the determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Presentation
In abstract algebra, algebra, a free presentation of a module (mathematics), module ''M'' over a commutative ring ''R'' is an exact sequence of ''R''-modules: :\bigoplus_ R \ \overset \to\ \bigoplus_ R \ \overset\to\ M \to 0. Note the image under ''g'' of the standard basis generating set of a module, generates ''M''. In particular, if ''J'' is finite, then ''M'' is a finitely generated module. If ''I'' and ''J'' are finite sets, then the presentation is called a finite presentation; a module is called finitely presented module, finitely presented if it admits a finite presentation. Since ''f'' is a module homomorphism between free modules, it can be visualized as an (infinite) matrix (mathematics), matrix with entries in ''R'' and ''M'' as its cokernel. A free presentation always exists: any module is a quotient module, quotient of a free module: F \ \overset\to\ M \to 0, but then the kernel (algebra), kernel of ''g'' is again a quotient of a free module: F' \ \overset \to\ \k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf Of Modules
In mathematics, a sheaf of ''O''-modules or simply an ''O''-module over a ringed space (''X'', ''O'') is a sheaf ''F'' such that, for any open subset ''U'' of ''X'', ''F''(''U'') is an ''O''(''U'')-module and the restriction maps ''F''(''U'') → ''F''(''V'') are compatible with the restriction maps ''O''(''U'') → ''O''(''V''): the restriction of ''fs'' is the restriction of ''f'' times the restriction of ''s'' for any ''f'' in ''O''(''U'') and ''s'' in ''F''(''U''). The standard case is when ''X'' is a scheme and ''O'' its structure sheaf. If ''O'' is the constant sheaf \underline, then a sheaf of ''O''-modules is the same as a sheaf of abelian groups (i.e., an abelian sheaf). If ''X'' is the prime spectrum of a ring ''R'', then any ''R''-module defines an ''O''''X''-module (called an associated sheaf) in a natural way. Similarly, if ''R'' is a graded ring and ''X'' is the Proj of ''R'', then any graded module defines an ''O''''X''-module in a natural way. ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ringed Space
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of rings called a structure sheaf. It is an abstraction of the concept of the rings of continuous (scalar-valued) functions on open subsets. Among ringed spaces, especially important and prominent is a locally ringed space: a ringed space in which the analogy between the stalk at a point and the ring of germs of functions at a point is valid. Ringed spaces appear in analysis as well as complex algebraic geometry and the scheme theory of algebraic geometry. Note: In the definition of a ringed space, most expositions tend to restrict the rings to be commutative rings, including Hartshorne and Wikipedia. ''Éléments de géométrie algébrique'', on the other hand, does not impose the commutativity assumption, although the book mostly co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Séminaire De Géométrie Algébrique Du Bois Marie
In mathematics, the (''SGA''; from French: "Seminar on Algebraic Geometry of Bois Marie") was an influential seminar run by French mathematician Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the (IHÉS) near Paris. (The name came from the small wood on the estate in Bures-sur-Yvette where the IHÉS was located from 1962.) The seminar notes were eventually published in twelve volumes, all except one in the Springer Lecture Notes in Mathematics series. Style The material has a reputation of being hard to read for a number of reasons. More elementary or foundational parts were relegated to the EGA series of Grothendieck and Jean Dieudonné, causing long strings of logical dependencies in the statements. The style is very abstract and makes heavy use of category theory. Moreover, an attempt was made to achieve maximally general statements, while assuming that the reader is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module Spectrum
In algebra, a module spectrum is a spectrum (topology), spectrum with an action of a ring spectrum; it generalizes a module (mathematics), module in abstract algebra. The ∞-category of (say right) module spectra is stable ∞-category, stable; hence, it can be considered as either analog or generalization of the derived category of modules over a ring. K-theory Lurie defines the K-theory of a ring spectrum ''R'' to be the K-theory of the ∞-category of perfect modules over ''R'' (a perfect module being defined as a compact object in the ∞-category of module spectra.) See also *G-spectrum References *J. LurieLecture 19: Algebraic K-theory of Ring Spectra Spectra (topology) {{algebra-stub ... [...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] |
Dualizable Object
In category theory, a branch of mathematics, a dual object is an analogue of a dual vector space from linear algebra for objects in arbitrary monoidal categories. It is only a partial generalization, based upon the categorical properties of duality for finite-dimensional vector spaces. An object admitting a dual is called a dualizable object. In this formalism, infinite-dimensional vector spaces are not dualizable, since the dual vector space ''V''∗ doesn't satisfy the axioms. Often, an object is dualizable only when it satisfies some finiteness or compactness property. A category in which each object has a dual is called autonomous or rigid. The category of finite-dimensional vector spaces with the standard tensor product is rigid, while the category of all vector spaces is not. Motivation Let ''V'' be a finite-dimensional vector space over some field ''K''. The standard notion of a dual vector space ''V''∗ has the following property: for any ''K''-vector spaces ''U'' and ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Object (mathematics)
In mathematics, compact objects, also referred to as finitely presented objects, or objects of finite presentation, are objects in a category satisfying a certain finiteness condition. Definition An object ''X'' in a category ''C'' which admits all filtered colimits (also known as direct limits) is called ''compact'' if the functor :\operatorname_C(X, \cdot) : C \to \mathrm, Y \mapsto \operatorname_C(X, Y) commutes with filtered colimits, i.e., if the natural map :\operatorname \operatorname_C(X, Y_i) \to \operatorname_C(X, \operatorname_i Y_i) is a bijection for any filtered system of objects Y_i in ''C''. Since elements in the filtered colimit at the left are represented by maps X \to Y_i, for some ''i'', the surjectivity of the above map amounts to requiring that a map X \to \operatorname_i Y_i factors over some Y_i. The terminology is motivated by an example arising from topology mentioned below. Several authors also use a terminology which is more closely related to algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
