text stringlengths 14 4.79k | source stringlengths 13 304 | tokens float64 75 1.06k ⌀ | char_length float64 106 4.79k ⌀ | article_title stringlengths 16 300 ⌀ |
|---|---|---|---|---|
Let P be a principal H-bundle on M, equipped with a Cartan connection η: TP → g {\displaystyle {\mathfrak {g}}} . If g {\displaystyle {\mathfrak {g}}} is a reductive module for H, meaning that g {\displaystyle {\mathfrak {g}}} admits an Ad(H)-invariant splitting of vector spaces g = h ⊕ m {\displaystyle {\mathfrak {g}}... | Wikipedia - Cartan connection | null | null | null |
Rh*η m {\displaystyle {\mathfrak {m}}} = Ad(h−1)η m {\displaystyle {\mathfrak {m}}} for every h ∈ H. (η m {\displaystyle {\mathfrak {m}}} is equivariant under the right H-action. )In other words, η is a solder form for the bundle P. Hence, P equipped with the form η m {\displaystyle {\mathfrak {m}}} defines a (first or... | Wikipedia - Cartan connection | null | null | null |
Let P {\displaystyle P} be a smooth principal bundle with structure group G {\displaystyle G} and g = Lie ( G ) {\displaystyle {\mathfrak {g}}=\operatorname {Lie} (G)} . G {\displaystyle G} acts on g {\displaystyle {\mathfrak {g}}} via adjoint representation and so one can form the associated bundle: g P = P × Ad g .... | Wikipedia - Bracket of Lie algebra-valued forms | null | null | null |
Let P {\displaystyle {\mathcal {P}}} be a finite set of mutually independent random variables in the probability space Ω. Let A {\displaystyle {\mathcal {A}}} be a finite set of events determined by these variables. If there exists an assignment of reals x: A → ( 0 , 1 ) {\displaystyle x:{\mathcal {A}}\to (0,1)} to the... | Wikipedia - Algorithmic Lovász local lemma | null | null | null |
Let P(V) be a projective space, where V is a vector space over a field K, and be the canonical map that maps a nonzero vector to its equivalence class, which is the vector line containing p with the zero vector removed. Every linear subspace W of V is a union of lines. It follows that p(W) is a projective space, which ... | Wikipedia - Projective space | null | null | null |
Let P1 = (X1:Y1:Z1) and P2 = (X2:Y2:Z2) be two points distinct to θ. Assuming that Z1 = Z2 = 1 then the algorithm is given by: A = X1 Y2 B = Y1 X2 X3 = B Y1-Y2 A Y3 = X1 A-B X2 Z3 = Y2 X2-X1 Y1The cost needed is 8 multiplications and 3 additions readdition cost of 7 multiplications and 3 additions, depending on the fir... | Wikipedia - Hessian curves | null | null | null |
Let P1 and P2 be a pair of distinct points with homogeneous coordinates (x1, y1, z1) and (x2, y2, z2) respectively. These points determine a unique line l with an equation of the form ax + by + cz = 0 and must satisfy the equations: ax1 + by1 + cz1 = 0 and ax2 + by2 + cz2 = 0.In matrix form this system of simultaneous ... | Wikipedia - Incidence (geometry) | null | null | null |
Let P1, P2, …, Pm be given as foci of a curve C of class m. Let P be the product of the tangential equations of these points and Q the product of the tangential equations of the circular points at infinity. Then all the lines which are common tangents to both P = 0 and Q = 0 are tangent to C. So, by the AF+BG theorem, ... | Wikipedia - Focus (geometry) | null | null | null |
The tangential equations are X + 1 = 0 X − 1 = 0 {\displaystyle {\begin{aligned}X+1&=0\\X-1&=0\end{aligned}}} so P = X2 − 1 = 0. The tangential equations for the circular points at infinity are X + i Y = 0 X − i Y = 0 {\displaystyle {\begin{aligned}X+iY&=0\\X-iY&=0\end{aligned}}} so Q = X2 +Y2. | Wikipedia - Focus (geometry) | null | null | null |
Therefore, the tangential equation for a conic with the given foci is X 2 − 1 + c ( X 2 + Y 2 ) = 0 {\displaystyle X^{2}-1+c(X^{2}+Y^{2})=0} or ( 1 + c ) X 2 + c Y 2 = 1 , {\displaystyle (1+c)X^{2}+cY^{2}=1,} where c is an arbitrary constant. In point coordinates this becomes x 2 1 + c + y 2 c = 1. {\displaystyle {\fra... | Wikipedia - Focus (geometry) | null | null | null |
Let Person A (Alice) want to send a message to Person B (Bob). Hash algorithms won't be considered, so Alice has to sign every single bit of her message. Message-Bit b ∈ { 0 , 1 } {\displaystyle \in \{0,1\}} . Alice chooses M pairs of private keys { k 0 i , k 1 i } 1 ≤ i ≤ M {\displaystyle \{k_{0}^{i},k_{1}^{i}\}\quad ... | Wikipedia - Quantum digital signature | null | null | null |
All the k 1 {\displaystyle k_{1}} keys will be used to sign the message-bit if b = 1.The function which maps k ↦ | f k ⟩ {\displaystyle k\mapsto |f_{k}\rangle } is known to all parties. Alice now computes the corresponding public keys { | f k 0 i ⟩ , | f k 1 i ⟩ } {\displaystyle \{|f_{k_{0}}^{i}\rangle ,|f_{k_{1}}^{i}\... | Wikipedia - Quantum digital signature | null | null | null |
Let Pi(1) be the prediction by the i'th model that the next bit will be a 1. Then the final prediction P(1) is calculated: xi = stretch(Pi(1)) P(1) = squash(Σi wi xi)where P(1) is the probability that the next bit will be a 1, Pi(1) is the probability estimated by the i'th model, and stretch(x) = ln(x / (1 - x)) squash... | Wikipedia - Context mixing | null | null | null |
Let Q = { 0 , 1 } ∪ { 2 ω 1 d , … , 2 ω N d } {\displaystyle Q=\{0,1\}\cup \{{2\omega _{1} \over d},\ldots ,{2\omega _{N} \over d}\}} . Since for each i , ω i = min ( Δ ( y i ′ , y i ) , d 2 ) {\displaystyle i,\omega _{i}=\min(\Delta (\mathbf {y_{i}'} ,\mathbf {y_{i}} ),{d \over 2})} , we have where q 1 < ⋯ < q m {\dis... | Wikipedia - Generalized minimum distance decoding | null | null | null |
Let Q and P be quasigroups. A quasigroup homotopy from Q to P is a triple (α, β, γ) of maps from Q to P such that α ( x ) β ( y ) = γ ( x y ) {\displaystyle \alpha (x)\beta (y)=\gamma (xy)\,} for all x, y in Q. A quasigroup homomorphism is just a homotopy for which the three maps are equal. An isotopy is a homotopy for... | Wikipedia - Loop (algebra) | null | null | null |
In terms of Latin squares, an isotopy (α, β, γ) is given by a permutation of rows α, a permutation of columns β, and a permutation on the underlying element set γ. An autotopy is an isotopy from a quasigroup to itself. The set of all autotopies of a quasigroup forms a group with the automorphism group as a subgroup. Ev... | Wikipedia - Loop (algebra) | null | null | null |
If a loop is isotopic to a group, then it is isomorphic to that group and thus is itself a group. However, a quasigroup that is isotopic to a group need not be a group. For example, the quasigroup on R with multiplication given by (x + y)/2 is isotopic to the additive group (R, +), but is not itself a group as it has n... | Wikipedia - Loop (algebra) | null | null | null |
Let Q be the 9x9 Sudoku matrix, N = {1, 2, 3, 4, 5, 6, 7, 8, 9}, and X represent a generic row, column, or block. N supplies symbols for filling Q as well as the index set for the 9 elements of any X. The given elements q in Q represent a partial function from Q to N. The solution R is a total relation and hence a func... | Wikipedia - Sudoku solving algorithms | null | null | null |
Let R ( X ) {\displaystyle {\mathcal {R}}(X)} be the set of all rational functions that are continuous on X {\displaystyle X} ; in other words functions that have no poles in X {\displaystyle X} . Then S = R ( X ) + R ( X ) ¯ {\displaystyle {\mathcal {S}}={\mathcal {R}}(X)+{\overline {{\mathcal {R}}(X)}}} is a *-subalg... | Wikipedia - Dirichlet algebra | null | null | null |
It can be shown that if an operator T {\displaystyle T} has X {\displaystyle X} as a spectral set, and R ( X ) {\displaystyle {\mathcal {R}}(X)} is a Dirichlet algebra, then T {\displaystyle T} has a normal boundary dilation. This generalises Sz.-Nagy's dilation theorem, which can be seen as a consequence of this by le... | Wikipedia - Dirichlet algebra | null | null | null |
Let R + {\displaystyle \mathbb {R} ^{+}} be the multiplicative group of positive real numbers, and let R {\displaystyle \mathbb {R} } be the additive group of real numbers. The logarithm function log: R + → R {\displaystyle \log :\mathbb {R} ^{+}\to \mathbb {R} } satisfies log ( x y ) = log x + log y {\displaysty... | Wikipedia - Isomorphism (category theory) | null | null | null |
Since log {\displaystyle \log } is a homomorphism that has an inverse that is also a homomorphism, log {\displaystyle \log } is an isomorphism of groups. The log {\displaystyle \log } function is an isomorphism which translates multiplication of positive real numbers into addition of real numbers. This facility makes i... | Wikipedia - Isomorphism (category theory) | null | null | null |
Let R = ( M , g ) {\displaystyle \ R=(M,g)\ } and R ′ = ( M ′ , g ′ ) {\displaystyle \ R'=(M',g')\ } be two (pseudo-)Riemannian manifolds, and let f: R → R ′ {\displaystyle \ f:R\to R'\ } be a diffeomorphism. Then f {\displaystyle \ f\ } is called an isometry (or isometric isomorphism) if g = f ∗ g ′ , {\displaystyle \... | Wikipedia - Orthonormal transformation | null | null | null |
Let R = F {\displaystyle R=F} be a polynomial ring over a field F. In this section, we suppose that an admissible monomial ordering has been fixed. Let G be a finite set of polynomials in R that generates an ideal I. The set G is a Gröbner basis (with respect to the monomial ordering), or, more precisely, a Gröbner ba... | Wikipedia - Gröbner bases | null | null | null |
The fact that so many characterizations are possible makes Gröbner bases very useful. For example, condition 3 provides an algorithm for testing ideal membership; condition 4 provides an algorithm for testing whether a set of polynomials is a Gröbner basis and forms the basis of Buchberger's algorithm for computing Grö... | Wikipedia - Gröbner bases | null | null | null |
For every admissible monomial ordering and every finite set G of polynomials, there is a Gröbner basis that contains G and generates the same ideal. Moreover, such a Gröbner basis may be computed with Buchberger's algorithm. This algorithm uses condition 4, and proceeds roughly as follows: add to G all nonzero results ... | Wikipedia - Gröbner bases | null | null | null |
Then a i {\displaystyle {\mathfrak {a}}_{i}} are ideals of R and as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to R. Equivalently, the above can be done through central ... | Wikipedia - Ring axioms | null | null | null |
Assume that R has the above decomposition. Then we can write By the conditions on a i , {\displaystyle {\mathfrak {a}}_{i},} one has that ei are central idempotents and eiej = 0, i ≠ j (orthogonal). Again, one can reverse the construction. | Wikipedia - Ring axioms | null | null | null |
Namely, if one is given a partition of 1 in orthogonal central idempotents, then let a i = R e i , {\displaystyle {\mathfrak {a}}_{i}=Re_{i},} which are two-sided ideals. If each ei is not a sum of orthogonal central idempotents, then their direct sum is isomorphic to R. An important application of an infinite direct p... | Wikipedia - Ring axioms | null | null | null |
Let R be a commutative ring (so R could be a field). An associative R-algebra (or more simply, an R-algebra) is a ring that is also an R-module in such a way that the two additions (the ring addition and the module addition) are the same operation, and scalar multiplication satisfies r ⋅ ( x y ) = ( r ⋅ x ) y = x ( r ⋅... | Wikipedia - Associative algebra | null | null | null |
Let R be a commutative ring and let A and B be R-algebras. Since A and B may both be regarded as R-modules, their tensor product A ⊗ R B {\displaystyle A\otimes _{R}B} is also an R-module. The tensor product can be given the structure of a ring by defining the product on elements of the form a ⊗ b by ( a 1 ⊗ b 1 ) ( a ... | Wikipedia - Tensor product of rings | null | null | null |
Let R be a commutative ring and let M be a finite free module over R. Then contraction operates on the full (mixed) tensor algebra of M in exactly the same way as it does in the case of vector spaces over a field. (The key fact is that the natural pairing is still perfect in this case.) More generally, let OX be a shea... | Wikipedia - Tensor contraction | null | null | null |
Let R be a commutative ring. The tensor product of R-modules applies, in particular, if A and B are R-algebras. In this case, the tensor product A ⊗ R B {\displaystyle A\otimes _{R}B} is an R-algebra itself by putting For example, A particular example is when A and B are fields containing a common subfield R. The tenso... | Wikipedia - Tensor products | null | null | null |
Let R be a noetherian ring or valuation ring. Then If R is noetherian, this follows from the fundamental theorem below (in particular, Krull's principal ideal theorem), but it is also a consequence of a more precise result. For any prime ideal p {\displaystyle {\mathfrak {p}}} in R, for any prime ideal q ⊋ p R {\displ... | Wikipedia - Dimension theory (algebra) | null | null | null |
This can be shown within basic ring theory (cf. Kaplansky, commutative rings). In addition, in each fiber of Spec R → Spec R {\displaystyle \operatorname {Spec} R\to \operatorname {Spec} R} , one cannot have a chain of primes ideals of length ≥ 2 {\displaystyle \geq 2} . Since an artinian ring (e.g., a field) has ... | Wikipedia - Dimension theory (algebra) | null | null | null |
Let R be a noetherian ring. The projective dimension of a finite R-module M is the shortest length of any projective resolution of M (possibly infinite) and is denoted by pd R M {\displaystyle \operatorname {pd} _{R}M} . We set g l . d i m R = sup { pd R M ∣ M is a finite module } {\displaystyle \operatorname {gl... | Wikipedia - Dimension theory (algebra) | null | null | null |
But then, by the local criterion for flatness, Tor 1 R ( M , k ) = 0 ⇒ M flat ⇒ M free ⇒ pd R ( M ) ≤ 0. {\displaystyle \operatorname {Tor} _{1}^{R}(M,k)=0\Rightarrow M{\text{ flat }}\Rightarrow M{\text{ free }}\Rightarrow \operatorname {pd} _{R}(M)\leq 0.} Now, completing the proof. | Wikipedia - Dimension theory (algebra) | null | null | null |
Q.E.D. Remark: The proof also shows that pd R K = pd R M − 1 {\displaystyle \operatorname {pd} _{R}K=\operatorname {pd} _{R}M-1} if M is not free and K {\displaystyle K} is the kernel of some surjection from a free module to M. Proof: If pd R M = 0 {\displaystyle \operatorname {pd} _{R}M=0} , then M is R-free and... | Wikipedia - Dimension theory (algebra) | null | null | null |
Then we have: pd R K = pd R M − 1 {\displaystyle \operatorname {pd} _{R}K=\operatorname {pd} _{R}M-1} as in the remark above. Thus, by induction, it is enough to consider the case pd R M = 1 {\displaystyle \operatorname {pd} _{R}M=1} . Then there is a projective resolution: 0 → P 1 → P 0 → M → 0 {\displaystyle 0\... | Wikipedia - Dimension theory (algebra) | null | null | null |
{\displaystyle \operatorname {Tor} _{1}^{R}(M,R_{1})={}_{f}M=\{m\in M\mid fm=0\}=0.} Hence, pd R ( M ⊗ R 1 ) {\displaystyle \operatorname {pd} _{R}(M\otimes R_{1})} is at most 1. Q.E.D. | Wikipedia - Dimension theory (algebra) | null | null | null |
Proof: If R is regular, we can write k = R / ( f 1 , … , f n ) {\displaystyle k=R/(f_{1},\dots ,f_{n})} , f i {\displaystyle f_{i}} a regular system of parameters. An exact sequence 0 → M → f M → M 1 → 0 {\displaystyle 0\to M{\overset {f}{\to }}M\to M_{1}\to 0} , some f in the maximal ideal, of finite modules, pd R M... | Wikipedia - Dimension theory (algebra) | null | null | null |
We begin with the inductive step. Set R 1 = R / f 1 R {\displaystyle R_{1}=R/f_{1}R} , f 1 {\displaystyle f_{1}} among a system of parameters. To show R is regular, it is enough to show R 1 {\displaystyle R_{1}} is regular. | Wikipedia - Dimension theory (algebra) | null | null | null |
But, since dim R 1 < dim R {\displaystyle \dim R_{1}<\dim R} , by inductive hypothesis and the preceding lemma with M = m {\displaystyle M={\mathfrak {m}}} , The basic step remains. Suppose dim R = 0 {\displaystyle \dim R=0} . We claim g l . | Wikipedia - Dimension theory (algebra) | null | null | null |
d i m R = 0 {\displaystyle \operatorname {gl.dim} R=0} if it is finite. (This would imply that R is a semisimple local ring; i.e., a field.) If that is not the case, then there is some finite module M {\displaystyle M} with 0 < pd R M < ∞ {\displaystyle 0<\operatorname {pd} _{R}M<\infty } and thus in fact we can fi... | Wikipedia - Dimension theory (algebra) | null | null | null |
By Nakayama's lemma, there is a surjection F → M {\displaystyle F\to M} from a free module F to M whose kernel K is contained in m F {\displaystyle {\mathfrak {m}}F} . Since dim R = 0 {\displaystyle \dim R=0} , the maximal ideal m {\displaystyle {\mathfrak {m}}} is an associated prime of R; i.e., m = ann ( s ) {\di... | Wikipedia - Dimension theory (algebra) | null | null | null |
Q.E.D. Proof: Let R be a regular local ring. Then gr R ≃ k {\displaystyle \operatorname {gr} R\simeq k} , which is an integrally closed domain. | Wikipedia - Dimension theory (algebra) | null | null | null |
It is a standard algebra exercise to show this implies that R is an integrally closed domain. Now, we need to show every divisorial ideal is principal; i.e., the divisor class group of R vanishes. But, according to Bourbaki, Algèbre commutative, chapitre 7, §. | Wikipedia - Dimension theory (algebra) | null | null | null |
4. Corollary 2 to Proposition 16, a divisorial ideal is principal if it admits a finite free resolution, which is indeed the case by the theorem. Q.E.D. | Wikipedia - Dimension theory (algebra) | null | null | null |
Let R be a quasi-coherent graded OX-algebra such that R0 = OX and R is locally generated as OX-algebra by R1. Then, by definition, the projective cone of R is: P ( C ) = Proj X R = lim → Proj ( R ( U ) ) {\displaystyle \mathbb {P} (C)=\operatorname {Proj} _{X}R=\varinjlim \operatorname {Proj} (R(U))} where the co... | Wikipedia - Projective completion of a cone | null | null | null |
Then Proj ( R ( U ) ) {\displaystyle \operatorname {Proj} (R(U))} has the line bundle O(1) given by the hyperplane bundle O P r ( 1 ) {\displaystyle {\mathcal {O}}_{\mathbb {P} ^{r}}(1)} of P r {\displaystyle \mathbb {P} ^{r}} ; gluing such local O(1)'s, which agree locally, gives the line bundle O(1) on P ( C ) {\di... | Wikipedia - Projective completion of a cone | null | null | null |
Let R be a ring and M a module over it. A sequence of elements x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} in R {\displaystyle R} is called an M-regular sequence if x 1 {\displaystyle x_{1}} is not a zero-divisor on M {\displaystyle M} and x i {\displaystyle x_{i}} is not a zero divisor on M / ( x 1 , … , x i − 1 ... | Wikipedia - Dimension theory (algebra) | null | null | null |
Then, by definition, the depth of a finite R-module M is the supremum of the lengths of all M-regular sequences in m {\displaystyle {\mathfrak {m}}} . For example, we have depth M = 0 ⇔ m {\displaystyle \operatorname {depth} M=0\Leftrightarrow {\mathfrak {m}}} consists of zerodivisors on M ⇔ m {\displaystyle \Leftrig... | Wikipedia - Dimension theory (algebra) | null | null | null |
Example: A regular Noetherian local ring is Cohen–Macaulay (since a regular system of parameters is an R-regular sequence.) In general, a Noetherian ring is called a Cohen–Macaulay ring if the localizations at all maximal ideals are Cohen–Macaulay. We note that a Cohen–Macaulay ring is universally catenary. | Wikipedia - Dimension theory (algebra) | null | null | null |
This implies for example that a polynomial ring k {\displaystyle k} is universally catenary since it is regular and thus Cohen–Macaulay. Proof: We first prove by induction on n the following statement: for every R-module M and every M-regular sequence x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} in m {\displaystyl... | Wikipedia - Dimension theory (algebra) | null | null | null |
But the latter is zero since the annihilator of N contains some power of x n {\displaystyle x_{n}} . Thus, from the exact sequence 0 → M → x 1 M → M 1 → 0 {\displaystyle 0\to M{\overset {x_{1}}{\to }}M\to M_{1}\to 0} and the fact that x 1 {\displaystyle x_{1}} kills N, using the inductive hypothesis again, we get provi... | Wikipedia - Dimension theory (algebra) | null | null | null |
It remains to show Ext R n ( N , M ) ≠ 0 {\displaystyle \operatorname {Ext} _{R}^{n}(N,M)\neq 0} if n = depth M {\displaystyle n=\operatorname {depth} M} . By (⁎) we can assume n = 0. Then m {\displaystyle {\mathfrak {m}}} is associated with M; thus is in the support of M. On the other hand, m ∈ Supp ( N ) . | Wikipedia - Dimension theory (algebra) | null | null | null |
{\displaystyle {\mathfrak {m}}\in \operatorname {Supp} (N).} It follows by linear algebra that there is a nonzero homomorphism from N to M modulo m {\displaystyle {\mathfrak {m}}} ; hence, one from N to M by Nakayama's lemma. | Wikipedia - Dimension theory (algebra) | null | null | null |
Q.E.D. The Auslander–Buchsbaum formula relates depth and projective dimension. Proof: We argue by induction on pd R M {\displaystyle \operatorname {pd} _{R}M} , the basic case (i.e., M free) being trivial. | Wikipedia - Dimension theory (algebra) | null | null | null |
By Nakayama's lemma, we have the exact sequence 0 → K → f F → M → 0 {\displaystyle 0\to K{\overset {f}{\to }}F\to M\to 0} where F is free and the image of f is contained in m F {\displaystyle {\mathfrak {m}}F} . Since pd R K = pd R M − 1 , {\displaystyle \operatorname {pd} _{R}K=\operatorname {pd} _{R}M-1,} what we... | Wikipedia - Dimension theory (algebra) | null | null | null |
If i < depth K − 1 {\displaystyle i<\operatorname {depth} K-1} , then since depth K ≤ depth R {\displaystyle \operatorname {depth} K\leq \operatorname {depth} R} by inductive hypothesis, we see Ext R i ( k , M ) = 0. {\displaystyle \operatorname {Ext} _{R}^{i}(k,M)=0.} If i = depth K − 1 {\displaystyle i=\ope... | Wikipedia - Dimension theory (algebra) | null | null | null |
{\displaystyle \operatorname {Ext} _{R}^{i}(k,M)\neq 0.} Q.E.D. | Wikipedia - Dimension theory (algebra) | null | null | null |
As a matter of notation, for any R-module M, we let One sees without difficulty that Γ m {\displaystyle \Gamma _{\mathfrak {m}}} is a left-exact functor and then let H m j = R j Γ m {\displaystyle H_{\mathfrak {m}}^{j}=R^{j}\Gamma _{\mathfrak {m}}} be its j-th right derived functor, called the local cohomology of R. Si... | Wikipedia - Dimension theory (algebra) | null | null | null |
Let R be a ring and x an element in it. We form the chain complex K(x) given by K ( x ) i = R {\displaystyle K(x)_{i}=R} for i = 0, 1 and K ( x ) i = 0 {\displaystyle K(x)_{i}=0} for any other i with the differential For any R-module M, we then get the complex K ( x , M ) = K ( x ) ⊗ R M {\displaystyle K(x,M)=K(x)\otim... | Wikipedia - Dimension theory (algebra) | null | null | null |
A Koszul complex is a powerful computational tool. For instance, it follows from the theorem and the corollary (Here, one uses the self-duality of a Koszul complex; see Proposition 17.15. of Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry.) Another instance would be Remark: The theorem can be used t... | Wikipedia - Dimension theory (algebra) | null | null | null |
Indeed, by the above theorem, Tor s R ( k , k ) ≠ 0 {\displaystyle \operatorname {Tor} _{s}^{R}(k,k)\neq 0} and thus g l . d i m R ≥ s {\displaystyle \operatorname {gl.dim} R\geq s} . On the other hand, as g l . | Wikipedia - Dimension theory (algebra) | null | null | null |
d i m R = pd R k {\displaystyle \operatorname {gl.dim} R=\operatorname {pd} _{R}k} , the Auslander–Buchsbaum formula gives g l . d i m R = dim R {\displaystyle \operatorname {gl.dim} R=\dim R} . Hence, dim R ≤ s ≤ g l . | Wikipedia - Dimension theory (algebra) | null | null | null |
d i m R = dim R {\displaystyle \dim R\leq s\leq \operatorname {gl.dim} R=\dim R} . We next use a Koszul homology to define and study complete intersection rings. Let R be a Noetherian local ring. | Wikipedia - Dimension theory (algebra) | null | null | null |
By definition, the first deviation of R is the vector space dimension where x _ = ( x 1 , … , x d ) {\displaystyle {\underline {x}}=(x_{1},\dots ,x_{d})} is a system of parameters. By definition, R is a complete intersection ring if dim R + ϵ 1 ( R ) {\displaystyle \dim R+\epsilon _{1}(R)} is the dimension of the tan... | Wikipedia - Dimension theory (algebra) | null | null | null |
Let R be a ring. The injective dimension of an R-module M denoted by id R M {\displaystyle \operatorname {id} _{R}M} is defined just like a projective dimension: it is the minimal length of an injective resolution of M. Let Mod R {\displaystyle \operatorname {Mod} _{R}} be the category of R-modules. Proof: Suppose g ... | Wikipedia - Dimension theory (algebra) | null | null | null |
Let M be an R-module and consider a resolution where I i {\displaystyle I_{i}} are injective modules. For any ideal I, which is zero since Ext R n + 1 ( R / I , − ) {\displaystyle \operatorname {Ext} _{R}^{n+1}(R/I,-)} is computed via a projective resolution of R / I {\displaystyle R/I} . Thus, by Baer's criterion, N... | Wikipedia - Dimension theory (algebra) | null | null | null |
We conclude that sup { id R M | M } ≤ n {\displaystyle \sup\{\operatorname {id} _{R}M|M\}\leq n} . Essentially by reversing the arrows, one can also prove the implication in the other way. Q.E.D. | Wikipedia - Dimension theory (algebra) | null | null | null |
The theorem suggests that we consider a sort of a dual of a global dimension: It was originally called the weak global dimension of R but today it is more commonly called the Tor dimension of R. Remark: for any ring R, w . g l . d i m R ≤ g l . d i m R {\displaystyle \operatorname {w.gl.dim} R\leq \operatorname {gl... | Wikipedia - Dimension theory (algebra) | null | null | null |
Let R denote the region enclosed by a curve r(φ) and the rays φ = a and φ = b, where 0 < b − a ≤ 2π. Then, the area of R is This result can be found as follows. First, the interval is divided into n subintervals, where n is some positive integer. Thus Δφ, the angle measure of each subinterval, is equal to b − a (the t... | Wikipedia - Polar plot | null | null | null |
For each subinterval i = 1, 2, ..., n, let φi be the midpoint of the subinterval, and construct a sector with the center at the pole, radius r(φi), central angle Δφ and arc length r(φi)Δφ. The area of each constructed sector is therefore equal to Hence, the total area of all of the sectors is As the number of subinterv... | Wikipedia - Polar plot | null | null | null |
Let R {\displaystyle R} and S {\displaystyle S} be given binary relations. Then the following concepts are useful: The converse of R {\displaystyle R} is the relation { ( y , x ): x R y } {\displaystyle \left\{\left(y,x\right):xRy\right\}} . The domain of R {\displaystyle R} is the set { x: ∃ y ( x R y ) } {\displaysty... | Wikipedia - Implementation of mathematics in set theory | null | null | null |
That is, the set { y: ∃ x ( x R y ) } {\displaystyle \left\{y:\exists x\left(xRy\right)\right\}} . The field of R {\displaystyle R} is the union of the domain and range of R {\displaystyle R} . The preimage of a member x {\displaystyle x} of the field of R {\displaystyle R} is the set { y: y R x } {\displaystyle \left\... | Wikipedia - Implementation of mathematics in set theory | null | null | null |
The downward closure of a member x {\displaystyle x} of the field of R {\displaystyle R} is the smallest set D {\displaystyle D} containing x {\displaystyle x} , and containing each z R y {\displaystyle zRy} for each y ∈ D {\displaystyle y\in D} (i.e., including the preimage of each of its elements with respect to R {\... | Wikipedia - Implementation of mathematics in set theory | null | null | null |
This could be done by representing a relation R {\displaystyle R} with codomain B {\displaystyle B} as ( R , B ) {\displaystyle \left(R,B\right)} , but our development will not require this. In ZFC, any relation whose domain is a subset of a set A {\displaystyle A} and whose range is a subset of a set B {\displaystyle ... | Wikipedia - Implementation of mathematics in set theory | null | null | null |
( n + 1 ) ! {\displaystyle \dim(TL_{n}(\delta ))={\frac {(2n)!}{n!(n+1)!}}} The Temperley–Lieb algebra T L n ( δ ) {\displaystyle TL_{n}(\delta )} is a subalgebra of the Brauer algebra B n ( δ ) {\displaystyle {\mathfrak {B}}_{n}(\delta )} , and therefore also of the partition algebra P n ( δ ) {\displaystyle P_{n}(\de... | Wikipedia - Temperley-Lieb algebra | null | null | null |
Let R {\displaystyle R} be a commutative ring, I {\displaystyle I} an ideal of R {\displaystyle R} and M {\displaystyle M} a finitely generated R {\displaystyle R} -module with the property that I M {\displaystyle IM} is properly contained in M {\displaystyle M} . (That is, some elements of M {\displaystyle M} are not ... | Wikipedia - Depth (algebra) | null | null | null |
By definition, the depth of a local ring R {\displaystyle R} with a maximal ideal m {\displaystyle {\mathfrak {m}}} is its m {\displaystyle {\mathfrak {m}}} -depth as a module over itself. If R {\displaystyle R} is a Cohen-Macaulay local ring, then depth of R {\displaystyle R} is equal to the dimension of R {\displayst... | Wikipedia - Depth (algebra) | null | null | null |
Let R {\displaystyle R} be a fixed commutative ring with unit. In most applications this is a field, but this is not needed for the definitions. Let also A {\displaystyle A} be an R {\displaystyle R} -algebra. | Wikipedia - Cellular algebra | null | null | null |
Let R {\displaystyle R} be an algebra over C {\displaystyle \mathbb {C} } , as is the case for applications to vertex algebras. An R {\displaystyle R} -valued formal distribution in n {\displaystyle n} variables z 1 , ⋯ , z n {\displaystyle z_{1},\cdots ,z_{n}} is an arbitrary series with each A i 1 , ⋯ , i n ∈ R {\dis... | Wikipedia - Formal distribution | null | null | null |
Let R {\displaystyle R} be the rate of the linear code, which is equal to K / N {\displaystyle K/N} Let there are S {\displaystyle S} subcode nodes in the graph. If the degree of the subcode is n {\displaystyle n} , then the code must have ( n m ) S {\displaystyle \left({\dfrac {n}{m}}\right)S} digits, as each digit no... | Wikipedia - Zemor's decoding algorithm | null | null | null |
Let R {\displaystyle \mathbb {R} } be the field of real numbers, C {\displaystyle \mathbb {C} } be the field of complex numbers, and H {\displaystyle \mathbb {H} } the quaternions. A central simple algebra (sometimes called a Brauer algebra) is a simple finite-dimensional algebra over a field F {\displaystyle F} whose ... | Wikipedia - Simple algebra | null | null | null |
These results follow from the Frobenius theorem. Every finite-dimensional simple algebra over C {\displaystyle \mathbb {C} } is a central simple algebra, and is isomorphic to a matrix ring over C {\displaystyle \mathbb {C} } . | Wikipedia - Simple algebra | null | null | null |
Every finite-dimensional central simple algebra over a finite field is isomorphic to a matrix ring over that field. The algebra of all linear transformations of an infinite-dimensional vector space over a field k {\displaystyle k} is a simple ring that is not a semisimple ring. It is also a simple algebra over k {\disp... | Wikipedia - Simple algebra | null | null | null |
Let R(x) be a polynomial of degree m − 1 where m ≥ 2 and the coefficient of the highest-order term be a ≠ 0. Assuming the following holds true for all polynomials of degree m − 1: Δ h m − 1 ( x ) = a h m − 1 ( m − 1 ) ! {\displaystyle \Delta _{h}^{m-1}(x)=ah^{m-1}(m-1)!} | Wikipedia - Newtonian series | null | null | null |
Let S(x) be a polynomial of degree m. With one pairwise difference: As ahm ≠ 0, this results in a polynomial T(x) of degree m − 1, with ahm as the coefficient of the highest-order term. Given the assumption above and m − 1 pairwise differences (resulting in a total of m pairwise differences for S(x)), it can be found t... | Wikipedia - Newtonian series | null | null | null |
Given two sequences S 1 {\displaystyle S_{1}} and S 2 {\displaystyle S_{2}} , the idea is to measure the cost of obtaining S 2 {\displaystyle S_{2}} from S 1 {\displaystyle S_{1}} using operators from the algebra. Let A = a 1 , a 2 , … a n {\displaystyle A={a_{1},a_{2},\ldots a_{n}}} be a sequence of operators such tha... | Wikipedia - Optimal matching | null | null | null |
One should consider at this point that there might exist different such sequences A {\displaystyle A} that transform S 1 {\displaystyle S_{1}} into S 2 {\displaystyle S_{2}} ; a reasonable choice is to select the cheapest of such sequences. We thus call distance d ( S 1 , S 2 ) = min A { c ( A ) s u c h t h a t S 2 = A... | Wikipedia - Optimal matching | null | null | null |
The distance function is symmetric if insertion and deletion costs are equal c ( a I n s ) = c ( a D e l ) {\displaystyle c(a^{\rm {Ins}})=c(a^{\rm {Del}})} ; the term indel cost usually refers to the common cost of insertion and deletion. Considering a set composed of only the three basic operations described above, t... | Wikipedia - Optimal matching | null | null | null |
Let S = { f λ | λ ∈ Λ } {\displaystyle S=\{f_{\lambda }|\lambda \in \Lambda \}} be the set of all monic irreducible polynomials in K. For each f λ ∈ S {\displaystyle f_{\lambda }\in S} , introduce new variables u λ , 1 , … , u λ , d {\displaystyle u_{\lambda ,1},\ldots ,u_{\lambda ,d}} where d = d e g r e e ( f λ ) {\d... | Wikipedia - Separably closed field | null | null | null |
Since I is strictly smaller than R, Zorn's lemma implies that there exists a maximal ideal M in R that contains I. The field K1=R/M has the property that every polynomial f λ {\displaystyle f_{\lambda }} with coefficients in K splits as the product of x − ( u λ , i + M ) , {\displaystyle x-(u_{\lambda ,i}+M),} and henc... | Wikipedia - Separably closed field | null | null | null |
Let S = ⟨ 5, 7, 9 ⟩. Then we have: The set of elements in S: S = {0, 5, 7, 9, 10, 12, 14, ...}. The minimal set of generators of S: {5, 7, 9}. The embedding dimension of S: e(S) = 3. | Wikipedia - Numerical monoid | null | null | null |
The multiplicity of S: m(S) = 5. The set of gaps in S: G(S) = {1, 2, 3, 4, 6, 8, 11, 13}. The Frobenius number of S is F(S) = 13, and its conductor is 14. The genus of S: g(S) = 8.Numerical semigroups with small Frobenius number or genus | Wikipedia - Numerical monoid | null | null | null |
Let S be a complex surface (in particular a 4-dimensional manifold) and let C → S {\displaystyle C\to S} be a smooth (non-singular) connected complex curve. Then 2 g ( C ) − 2 = 2 − c 1 ( S ) {\displaystyle 2g(C)-2=^{2}-c_{1}(S)} where g ( C ) {\displaystyle g(C)} is the genus of C, 2 {\displaystyle ^{2}} denotes th... | Wikipedia - Adjunction formula (algebraic geometry) | null | null | null |
Let S be a multiplicative set in a commutative ring R, and j: R → S − 1 R {\displaystyle j\colon R\to S^{-1}R} be the canonical ring homomorphism. Given an ideal I in R, let S − 1 I {\displaystyle S^{-1}I} the set of the fractions in S − 1 R {\displaystyle S^{-1}R} whose numerator is in I. This is an ideal of S − 1 R ,... | Wikipedia - Localization of a module | null | null | null |
{\displaystyle {\begin{aligned}{\frac {\partial ^{2}f}{\partial u^{2}}}&=\Gamma _{11}^{1}{\frac {\partial f}{\partial u}}+\Gamma _{11}^{2}{\frac {\partial f}{\partial v}}+Ln\\{\frac {\partial ^{2}f}{\partial u\partial v}}&=\Gamma _{12}^{1}{\frac {\partial f}{\partial u}}+\Gamma _{12}^{2}{\frac {\partial f}{\partial v}}... | Wikipedia - Differentiable surface | null | null | null |
{\displaystyle K={\frac {1}{(EG-F^{2})^{2}}}\det {\begin{pmatrix}-{1 \over 2}{\frac {\partial ^{2}E}{\partial v^{2}}}+{\frac {\partial ^{2}F}{\partial u\partial v}}-{1 \over 2}{\frac {\partial ^{2}G}{\partial u^{2}}}&{1 \over 2}{\frac {\partial E}{\partial u}}&{\frac {\partial F}{\partial u}}-{1 \over 2}{\frac {\partia... | Wikipedia - Differentiable surface | null | null | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.