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In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later ... |
closed field k (in classical algebraic geometry, k is usually the field of complex numbers). === Affine varieties === First, we define the topology on the affine space A n , {\displaystyle \mathbb {A} ^{n},} formed by the n-tuples of elements of k. The topology is defined by specifying its closed sets, rather than its ... |
^{n}} above, defines the Zariski topology on any affine variety. === Projective varieties === Recall that n {\displaystyle n} -dimensional projective space P n {\displaystyle \mathbb {P} ^{n}} is defined to be the set of equivalence classes of non-zero points in A n + 1 {\displaystyle \mathbb {A} ^{n+1}} by identifying... |
However, except for finite algebraic sets, no algebraic set is ever a Hausdorff space. In the old topological literature "compact" was taken to include the Hausdorff property, and this convention is still honored in algebraic geometry; therefore compactness in the modern sense is called "quasicompactness" in algebraic ... |
elements that are actually in P are precisely those whose reflection vanishes at P. So if we think of the map, associated to any element a of A: e a : ( P ∈ Spec A ) ↦ ( a mod P 1 ∈ Frac ( A / P ) ) {\displaystyle e_{a}\colon {\bigl (}P\in \operatorname {Spec} A{\bigr )}\mapsto \left({\frac {a\;{\bmod {P}}}{1}}\in ... |
a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal, and the set of the closed points is homeomorphic with the affine line k equipped with its Zariski topology. Because of this homeomorphism, some authors use the term af... |
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under ... |
question is to find a minimal basis, and ask whether the module of polynomial relations between the basis elements (known as the syzygies) is finitely generated over k [ V ] {\displaystyle k[V]} . Invariant theory of finite groups has intimate connections with Galois theory. One of the first major results was the main ... |
[ x 2 , x y , y 2 ] ≅ C [ a , b , c ] ( a c − b 2 ) {\displaystyle \mathbb {C} [x,y]^{\mathbb {Z} /2\mathbb {Z} }\cong \mathbb {C} [x^{2},xy,y^{2}]\cong {\frac {\mathbb {C} [a,b,c]}{(ac-b^{2})}}} This example forms the basis for doing many computations. == The nineteenth-century origins == Cayley first established inva... |
of invariants of G acting on the ring of polynomials R = S(V) is finitely generated. His proof used the Reynolds operator ρ from R to RG with the properties ρ(1) = 1 ρ(a + b) = ρ(a) + ρ(b) ρ(ab) = a ρ(b) whenever a is an invariant. Hilbert constructed the Reynolds operator explicitly using Cayley's omega process Ω, tho... |
x = ρ(a1)i1 + ... + ρ(an)in still holds for our modified ρ(ak), so we can again conclude that x lies in the R-algebra generated by i1,...,in. Hence, by induction on the degree, all elements of RG are in the R-algebra generated by i1,...,in. == Geometric invariant theory == The modern formulation of geometric invariant ... |
(1939), The Classical Groups. Their Invariants and Representations, Princeton University Press, ISBN 978-0-691-05756-9, MR 0000255 {{citation}}: ISBN / Date incompatibility (help) Weyl, Hermann (1939b), "Invariants", Duke Mathematical Journal, 5 (3): 489–502, doi:10.1215/S0012-7094-39-00540-5, ISSN 0012-7094, MR 000003... |
In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and its finitely generated modules. Informally, the lemma immediately gives a precise sen... |
( R ) M = M {\displaystyle J(R)M=M} , then M = 0 {\displaystyle M=0} . Proof: 1 − r {\displaystyle 1-r} (with r {\displaystyle r} as in Statement 1) is in the Jacobson radical so r {\displaystyle r} is invertible. More generally, one has that J ( R ) M {\displaystyle J(R)M} is a superfluous submodule of M {\displaystyl... |
Geometric interpretation ==== In this form, Nakayama's lemma takes on concrete geometrical significance. Local rings arise in geometry as the germs of functions at a point. Finitely generated modules over local rings arise quite often as germs of sections of vector bundles. Working at the level of germs rather than poi... |
. Then there is a prime ideal q {\displaystyle {\mathfrak {q}}} in S {\displaystyle S} such that q ∩ R = p {\displaystyle {\mathfrak {q}}\cap R={\mathfrak {p}}} . Moreover, q {\displaystyle {\mathfrak {q}}} can be chosen to contain any prime q 1 {\displaystyle {\mathfrak {q}}_{1}} of S {\displaystyle S} such that q 1 ∩... |
f : X → Y {\textstyle f:X\to Y} be a projective morphism between quasi-projective varieties. Then f {\textstyle f} is an isomorphism if and only if it is a bijection and the differential d f p {\textstyle df_{p}} is injective for all p ∈ X {\displaystyle p\in X} . == Proof == A standard proof of the Nakayama lemma uses... |
a subset of V, by the definition of J(R) and the fact that U/V is simple. Thus, if U contains at least one (proper) maximal submodule, U·J(R) is a proper submodule of U. However, this need not hold for arbitrary modules U over R, for U need not contain any maximal submodules. Naturally, if U is a Noetherian module, thi... |
( R ) {\displaystyle \sum _{s=1}^{m}r_{i,s}j_{s}\in \operatorname {J} (R)} for every i ∈ { 1 , … , n } {\displaystyle i\in \{1,\dots ,n\}} , and thus this becomes u = ∑ i = 1 n x i k i {\displaystyle u=\sum _{i=1}^{n}x_{i}k_{i}} for some k i ∈ J ( R ) {\displaystyle k_{i}\in \operatorname {J} (R)} , i = 1 , … , n {\d... |
i to be the least integer such that M i ≠ 0 {\displaystyle M_{i}\neq 0} , we see that M i {\displaystyle M_{i}} does not appear in R + M {\displaystyle R_{+}M} , so either M ≠ R + M {\displaystyle M\neq R_{+}M} , or such an i does not exist, i.e., M = 0 {\displaystyle M=0} . == See also == Module theory Serre–Swan theo... |
Introduction to Commutative Algebra is a well-known commutative algebra textbook written by Michael Atiyah and Ian G. Macdonald. It is on the list of 173 books essential for undergraduate math libraries. As of May 2025, Google Scholar lists over 8000 citations to this book. It deals with elementary concepts of commutat... |
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of fractions m s , {\displaystyle {\frac {m}{s}},} such that the denominator s belongs to ... |
form a s , {\displaystyle {\tfrac {a}{s}},} but also products of such fractions, such as a b s 2 . {\displaystyle {\tfrac {ab}{s^{2}}}.} So, the denominators will belong to the multiplicative set { 1 , s , s 2 , s 3 , … } {\displaystyle \{1,s,s^{2},s^{3},\ldots \}} of the powers of s. Therefore, one generally talks of ... |
case, a problem arises with zero divisors. Let S be a multiplicative set in a commutative ring R. Suppose that s ∈ S , {\displaystyle s\in S,} and 0 ≠ a ∈ R {\displaystyle 0\neq a\in R} is a zero divisor with a s = 0. {\displaystyle as=0.} Then a 1 {\displaystyle {\tfrac {a}{1}}} is the image in S − 1 R {\displaystyle ... |
, {\displaystyle 0\in S,} then S − 1 R {\displaystyle S^{-1}R} is the zero ring that has only one unique element 0. If S is the set of all regular elements of R (that is the elements that are not zero divisors), S − 1 R {\displaystyle S^{-1}R} is called the total ring of fractions of R. === Universal property === The (... |
{ 0 } , {\displaystyle S=\mathbb {Z} \setminus \{0\},} then S − 1 R {\displaystyle S^{-1}R} is the field Q {\displaystyle \mathbb {Q} } of the rational numbers. If R is an integral domain, and S = R ∖ { 0 } , {\displaystyle S=R\setminus \{0\},} then S − 1 R {\displaystyle S^{-1}R} is the field of fractions of R. The pr... |
that is not surjective in general. The ring S − 1 R {\displaystyle S^{-1}R} is a flat R-module (see § Localization of a module for details). If S = R ∖ p {\displaystyle S=R\setminus {\mathfrak {p}}} is the complement of a prime ideal p {\displaystyle {\mathfrak {p}}} , then S − 1 R , {\displaystyle S^{-1}R,} denoted R ... |
R {\displaystyle S^{-1}R} and T − 1 R {\displaystyle T^{-1}R} are isomorphic if and only if they have the same saturation, or, equivalently, if s belongs to one of the multiplicative sets, then there exists t ∈ R {\displaystyle t\in R} such that st belongs to the other. Saturated multiplicative sets are not widely used... |
p Z {\displaystyle p\mathbb {Z} } . This terminology can be explained by the fact that, if p is prime, the nonzero prime ideals of the localization of Z {\displaystyle \mathbb {Z} } are either the singleton set {p} or its complement in the set of prime numbers. == Localization and saturation of ideals == Let S be a mul... |
{\displaystyle S^{-1}(I+J)=S^{-1}I+S^{-1}J,\qquad \operatorname {sat} (I+J)=\operatorname {sat} (I)+\operatorname {sat} (J)} S − 1 ( I ⋅ J ) = S − 1 I ⋅ S − 1 J , sat ( I ⋅ J ) = sat ( I ) ⋅ sat ( J ) {\displaystyle S^{-1}(I\cdot J)=S^{-1}I\cdot S^{-1}J,\qquad \quad \operatorname {sat} (I\cdot J)=\operatorname {s... |
be equivalently defined by using tensor products: S − 1 M = S − 1 R ⊗ R M . {\displaystyle S^{-1}M=S^{-1}R\otimes _{R}M.} The proof of equivalence (up to a canonical isomorphism) can be done by showing that the two definitions satisfy the same universal property. === Module properties === If M is a submodule of an R-mo... |
a multiplicative set. In this case, the localization S − 1 R {\displaystyle S^{-1}R} is commonly denoted R p . {\displaystyle R_{\mathfrak {p}}.} The ring R p {\displaystyle R_{\mathfrak {p}}} is a local ring, that is called the local ring of R at p . {\displaystyle {\mathfrak {p}}.} This means that p R p = p ⊗ R R p {... |
properties. For example, an infinite direct product of fields is not an integral domain nor a Noetherian ring, while all its local rings are fields, and therefore Noetherian integral domains. == Non-commutative case == Localizing non-commutative rings is more difficult. While the localization exists for every set S of ... |
In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale topology was originally introduced by Alexander Grothendieck to define étale... |
whose objects are schemes U with a fixed étale morphism U → X. The morphisms are morphisms of schemes compatible with the fixed maps to X. The big étale site of X is the category Ét/X, that is, the category of schemes with a fixed map to X, considered with the étale topology. The étale topology can be defined using sli... |
U i ) {\displaystyle X=\bigcup _{i\in I}\varphi _{i}(U_{i})} . The category Ét(X) is the category of all étale schemes over X. The collection of all étale coverings of a étale scheme U over X i.e. an object in Ét(X) defines a Grothendieck pretopology on Ét(X) which in turn induces a Grothendieck topology, the étale top... |
In mathematics, ideal theory is the theory of ideals in commutative rings. While the notion of an ideal exists also for non-commutative rings, a much more substantial theory exists only for commutative rings (and this article therefore only considers ideals in commutative rings.) Throughout the articles, rings refer to... |
ring structure of Z {\displaystyle \mathbb {Z} } ; this ring is denoted as Z p {\displaystyle \mathbb {Z} _{p}} and is called the ring of p-adic integers. == Ideal class group == In a Dedekind domain A (e.g., a ring of integers in a number field or the coordinate ring of a smooth affine curve) with the field of fractio... |
also tight closure. == Reduction theory == == Local cohomology in ideal theory == Local cohomology can sometimes be used to obtain information on an ideal. This section assumes some familiarity with sheaf theory and scheme theory. Let M {\displaystyle M} be a module over a ring R {\displaystyle R} and I {\displaystyle ... |
In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number p, then this root can be lifted to a unique root modulo any higher power of p. More generally, if a polynomial... |
I, denoted f ≡ g ( mod I ) {\textstyle f\equiv g{\pmod {I}}} if they have the same coefficients modulo I, that is if f − g ∈ I R [ X ] . {\displaystyle f-g\in IR[X].} If h ∈ R [ X ] , {\displaystyle h\in R[X],} a factorization of h modulo I consists in two (or more) polynomials f, g in R [ X ] {\displaystyle R[X]} such... |
is a principal ideal domain, and, in particular, a unique factorization domain, which means that every nonzero polynomial in ( R / m ) [ X ] {\displaystyle (R/{\mathfrak {m}})[X]} can be factorized in a unique way as the product of a nonzero element of ( R / m ) {\displaystyle (R/{\mathfrak {m}})} and irreducible polyn... |
topology on R, which is called the m {\displaystyle {\mathfrak {m}}} -adic topology. The completion of this topology can be identified with the completion of the local ring R m , {\displaystyle R_{\mathfrak {m}},} and with the inverse limit lim ← R / m n . {\displaystyle \lim _{\leftarrow }R/{\mathfrak {m}}^{n}.} This ... |
. {\displaystyle A-B\in IR[X].} === Linear lifting === Let I be an ideal of a commutative ring R, and h ∈ R [ X ] {\displaystyle h\in R[X]} be a univariate polynomial with coefficients in R that has a leading coefficient α {\displaystyle \alpha } that is invertible modulo I (that is, the image of α {\displaystyle \alph... |
\delta _{h}\in I^{k}R[X],} of degree less than deg h , {\displaystyle \deg h,} such that δ h ≡ h − α f g ( mod I k + 1 ) . {\displaystyle \delta _{h}\equiv h-\alpha fg{\pmod {I^{k+1}}}.} (One may choose δ h = h − α f g , {\displaystyle \delta _{h}=h-\alpha fg,} but other choices may lead to simpler computations. For ... |
1 ) . {\displaystyle {\begin{aligned}\alpha (f+\beta d)&(g+\beta c)-h\\&\equiv \alpha fg-h+\alpha \beta (f(a\delta _{h}-qg)+g(b\delta _{h}-q'f))\\&\equiv \delta _{h}(-1+\alpha \beta (af+bg))-\alpha \beta fg(q+q')\\&\equiv 0{\pmod {I^{k+1}}}.\end{aligned}}} So, the existence assertion is verified with δ f = β d , δ g = ... |
g + δ g ′ ) = α ( f ( δ g − δ g ′ ) + g ( δ f − δ f ′ ) ) + α ( δ f ( δ g − δ g ′ ) − δ g ( δ f − δ f ′ ) ) ∈ I n R [ X ] . {\displaystyle {\begin{aligned}\alpha (f+\delta _{f})(g+\delta _{g})&-\alpha (f+\delta '_{f})(g+\delta '_{g})\\&=\alpha (f(\delta _{g}-\delta '_{g})+g(\delta _{f}-\delta '_{f}))+\alpha (\delta _{f... |
=== Linear lifting allows lifting a factorization modulo I n {\displaystyle I^{n}} to a factorization modulo I n + 1 . {\displaystyle I^{n+1}.} Quadratic lifting allows lifting directly to a factorization modulo I 2 n , {\displaystyle I^{2n},} at the cost of lifting also the Bézout's identity and of computing modulo I ... |
) + b ( g + δ g ) = 1 + Δ , {\displaystyle a(f+\delta _{f})+b(g+\delta _{g})=1+\Delta ,} where Δ = α + a δ f + b δ g ∈ I k R [ X ] . {\displaystyle \Delta =\alpha +a\delta _{f}+b\delta _{g}\in I^{k}R[X].} Setting δ a = − a Δ {\displaystyle \delta _{a}=-a\Delta } and δ b = − b Δ , {\displaystyle \delta _{b}=-b\Delta ,} ... |
( X − 119 ¯ ) ( X − 608 ¯ ) ( X − 611 ¯ ) ( X − 724 ¯ ) {\displaystyle {\bar {f}}(X)=X^{6}-{\overline {2}}=(X-{\overline {3}})\;(X-{\overline {116}})\;(X-{\overline {119}})\;(X-{\overline {608}})\;(X-{\overline {611}})\;(X-{\overline {724}})} with all factors relatively prime to each other, so that in Z 727 [ X ] {\dis... |
∑ n = 0 N c n ( s − r ) n , c n = f ( n ) ( r ) / n ! . {\displaystyle f(s)=\sum _{n=0}^{N}c_{n}(s-r)^{n},\qquad c_{n}=f^{(n)}(r)/n!.} From r ≡ s mod p k , {\displaystyle r\equiv s{\bmod {p}}^{k},} we see that s − r = tpk for some integer t. Let f ( s ) = ∑ n = 0 N c n ( t p k ) n = f ( r ) + t p k f ′ ( r ) + ∑ n = 2 ... |
1 {\displaystyle f(X)=X^{6}+10X-1} , we find in Q 2 [ X ] {\displaystyle \mathbb {Q} _{2}[X]} | f ( X ) | = max { | a 0 | , … , | a n | } = max { 0 , 1 , 0 } = 1 {\displaystyle {\begin{aligned}|f(X)|&=\max\{|a_{0}|,\ldots ,|a_{n}|\}\\&=\max\{0,1,0\}=1\end{aligned}}} but max { | a 0 | , | a n | } = 0 {\displaystyle \max... |
f ′ ( r ) ≢ 0 mod p . {\displaystyle f'(s)\equiv f'(r)\not \equiv 0{\bmod {p}}.} So the lifting can be repeated, and starting from a solution rk of f ( x ) ≡ 0 mod p k {\displaystyle f(x)\equiv 0{\bmod {p}}^{k}} we can derive a sequence of solutions rk+1, rk+2, ... of the same congruence for successively higher powers ... |
a priori we don't know whether we can lift them to modulo 8, but in fact we can, since g(1) is 0 mod 8 and g(3) is 0 mod 8, giving solutions at 1, 3, 5, and 7 mod 8. Since of these only g(1) and g(7) are 0 mod 16 we can lift only 1 and 7 to modulo 16, giving 1, 7, 9, and 15 mod 16. Of these, only 7 and 9 give g(x) = 0 ... |
0 and | b − a | p < | f ′ ( a ) | p . {\displaystyle |b-a|_{p}<|f'(a)|_{p}.} The construction of b amounts to showing that the recursion from Newton's method with initial value a converges in the p-adics and we let b be the limit. The uniqueness of b as a root fitting the condition | b − a | p < | f ′ ( a ) | p {\displ... |
the discussion above more explicit, let us find a "square root of 2" (the solution to x 2 − 2 = 0 {\displaystyle x^{2}-2=0} ) in the 7-adic integers. Modulo 7 one solution is 3 (we could also take 4), so we set r 1 = 3 {\displaystyle r_{1}=3} . Hensel's lemma then allows us to find r 2 {\displaystyle r_{2}} as follows:... |
valid but the more general one is, let f ( x ) = x 2 − 17 {\displaystyle f(x)=x^{2}-17} and a = 1. {\displaystyle a=1.} Then f ( a ) = − 16 {\displaystyle f(a)=-16} and f ′ ( a ) = 2 , {\displaystyle f'(a)=2,} so | f ( a ) | 2 < | f ′ ( a ) | 2 2 , {\displaystyle |f(a)|_{2}<|f'(a)|_{2}^{2},} which implies there is a un... |
2 + 2 4 + 2 8 + 2 11 + ⋯ {\displaystyle 1+2+2^{2}+2^{4}+2^{8}+2^{11}+\cdots } Another example where we can use the more general version of Hensel's lemma but not the basic version is a proof that any 3-adic integer c ≡ 1 mod 9 is a cube in Z 3 . {\displaystyle \mathbb {Z} _{3}.} Let f ( x ) = x 3 − c {\displaystyle f(x... |
a zero-divisor then b is unique. This result can be generalized to several variables as follows: Theorem. Let A be a commutative ring that is complete with respect to ideal m ⊂ A . {\displaystyle {\mathfrak {m}}\subset A.} Let f 1 , … , f n ∈ A [ x 1 , … , x n ] {\displaystyle f_{1},\ldots ,f_{n}\in A[x_{1},\ldots ,x_{... |
of A. If A is noetherian, Ah will also be noetherian, and Ah is manifestly algebraic as it is constructed as a limit of étale neighbourhoods. This means that Ah is usually much smaller than the completion  while still retaining the Henselian property and remaining in the same category. == See also == Hasse–Minkowski t... |
In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have a simpler structure tha... |
any r ∈ R are given by cosets r + In.) The (I-adic) completion is the inverse limit of the factor rings, R ^ I = lim ← ( R / I n ) {\displaystyle {\widehat {R}}_{I}=\varprojlim (R/I^{n})} pronounced "R I hat". The kernel of the canonical map π from the ring to its completion is the intersection of the powers of I. Th... |
x ) ) {\displaystyle \mathbb {C} [x,y]/(y^{2}-x^{2}(1+x))} have similar looking singularities at the origin when viewing their graphs (both look like a plus sign). Notice that in the second case, any Zariski neighborhood of the origin is still an irreducible curve. If we use completions, then we are looking at a "small... |
R-modules is exact: it preserves short exact sequences. In particular, taking quotients of rings commutes with completion, meaning that for any quotient R-algebra R / I {\displaystyle R/I} , there is an isomorphism R / I ^ ≅ R ^ / I ^ . {\displaystyle {\widehat {R/I}}\cong {\widehat {R}}/{\widehat {I}}.} Cohen structur... |
Combinatorial commutative algebra is a relatively new, rapidly developing mathematical discipline. As the name implies, it lies at the intersection of two more established fields, commutative algebra and combinatorics, and frequently uses methods of one to address problems arising in the other. Less obviously, polyhedr... |
contains exposition of current research topics: Miller, Ezra; Sturmfels, Bernd (2005). Combinatorial commutative algebra. Graduate Texts in Mathematics. Vol. 227. Springer. ISBN 0-387-22356-8. Zbl 1066.13001. Herzog, Jürgen; Hibi, Takayuki (2011). Monomial Ideals. Graduate Texts in Mathematics. Vol. 260. Springer. ISBN... |
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two mo... |
reverse morphism in the opposite category Cop (composed by reversing all morphisms in C) is an epimorphism. An example comes from reversing the direction of inequalities in a partial order. So, if X is a set and ≤ a partial order relation, we can define a new partial order relation ≤new by x ≤new y if and only if y ≤ x... |
In algebra, the kernel of a homomorphism is the relation describing how elements in the domain of the homomorphism become related in the image. A homomorphism is a function that preserves the underlying algebraic structure in the domain to its image. When the algebraic structures involved have an underlying group struc... |
it is a normal subgroup. Thus, there is a corresponding quotient group G / (ker f). This is isomorphic to f(G), the image of G under f (which is a subgroup of H also), by the first isomorphism theorem for groups. === Ring homomorphisms === Let R and S be rings (assumed unital) and let f be a ring homomorphism from R to... |
theorem for vector spaces states that this quotient space is naturally isomorphic to the image of T (which is a subspace of W). As a consequence, the dimension of V equals the dimension of the kernel plus the dimension of the image. === Module homomorphisms === Let R {\displaystyle R} be a ring, and let M {\displaystyl... |
the Klein 4-group. Define a mapping φ : Q 8 → V 4 {\displaystyle \varphi :Q_{8}\to V_{4}} to be: φ ( ± 1 ) = 1 {\displaystyle \varphi (\pm 1)=1} φ ( ± i ) = a {\displaystyle \varphi (\pm i)=a} φ ( ± j ) = b {\displaystyle \varphi (\pm j)=b} φ ( ± k ) = c {\displaystyle \varphi (\pm k)=c} Then this mapping is a homomorp... |
R {\displaystyle x\in \mathbb {R} } . == Quotient algebras == The kernel of a homomorphism can be used to define a quotient algebra. For instance, if φ : G → H {\displaystyle \varphi :G\to H} denotes a group homomorphism, and denote K = ker φ {\displaystyle K=\ker \varphi } , then consider G / K {\displaystyle G/K} t... |
(and be well-defined). For a ring R {\displaystyle R} (possibly a field when describing vector spaces) and a module homomorphism φ : M → N {\displaystyle \varphi :M\to N} with kernel K = ker φ {\displaystyle K=\ker \varphi } , one can define scalar multiplication on G / K {\displaystyle G/K} by r ( x + K ) = r x + K ... |
∈ I {\displaystyle \alpha ,\beta \in I} , yields: ( r + α ) ( s + β ) + I = r s + I {\displaystyle (r+\alpha )(s+\beta )+I=rs+I} Setting r = s = 0 {\displaystyle r=s=0} implies that I {\displaystyle I} is closed under multiplication, while setting α = s = 0 {\displaystyle \alpha =s=0} shows that r β ∈ I {\displaystyle ... |
(which is a subalgebra of B). == See also == Kernel (linear algebra) Kernel (category theory) Kernel of a function Equalizer (mathematics) Zero set == Notes == == References == Axler, Sheldon. Linear Algebra Done Right (4th ed.). Springer. Burris, Stanley; Sankappanavar, H.P. (2012). A Course in Universal Algebra (Mill... |
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ... |
in the work of Jean-Victor Poncelet and Bernhard Riemann) that algebraic geometry over the real numbers is simplified by working over the field of complex numbers, which has the advantage of being algebraically closed. The early 20th century saw analogies between algebraic geometry and number theory, suggesting the que... |
constructed by taking points in a very large algebraically closed field, called a universal domain. This worked awkwardly: there were many different generic points for the same variety. (In the later theory of schemes, each algebraic variety has a single generic point.) In the 1950s, Claude Chevalley, Masayoshi Nagata ... |
{\displaystyle {\mathcal {O}}_{X}(U)} called the ring of regular functions on U {\displaystyle U} . One can think of a scheme as being covered by "coordinate charts" that are affine schemes. The definition means exactly that schemes are obtained by gluing together affine schemes using the Zariski topology. In the early... |
solutions of the defining equations of X with values in R. When R is a field k, X(k) is also called the set of k-rational points of X. More generally, for a scheme X over a commutative ring R and any commutative R-algebra S, an S-point of X means a morphism Spec(S) → X over R. One writes X(S) for the set of S-points of... |
V ( p ) = { q ∈ X with p ⊂ q } {\displaystyle V({\mathfrak {p}})=\{{\mathfrak {q}}\in X\ \ {\text{with}}\ \ {\mathfrak {p}}\subset {\mathfrak {q}}\}} , specially including all the closed points of the subvariety, i.e. m a {\displaystyle {\mathfrak {m}}_{a}} with a ∈ V ¯ {\displaystyle a\in {\bar {V}}} , or equivalently... |
the equation x 2 = y 2 ( y + 1 ) {\displaystyle x^{2}=y^{2}(y+1)} defines a nodal cubic curve in the affine plane A k 2 {\displaystyle \mathbb {A} _{k}^{2}} , corresponding to the scheme V = Spec k [ x , y ] / ( x 2 − y 2 ( y + 1 ) ) {\displaystyle V=\operatorname {Spec} k[x,y]/(x^{2}-y^{2}(y+1))} . === Spec of the i... |
kind of "regular function" on the closed points, a very special type among the arbitrary functions f {\displaystyle f} with f ( m p ) ∈ F p {\displaystyle f({\mathfrak {m}}_{p})\in \mathbb {F} _{p}} . Note that the point m p {\displaystyle {\mathfrak {m}}_{p}} is the vanishing locus of the function n = p {\displaystyle... |
non-constant polynomial with no integer factor and which is irreducible modulo p {\displaystyle p} . Thus, we may picture Y {\displaystyle Y} as two-dimensional, with a "characteristic direction" measured by the coordinate p {\displaystyle p} , and a "spatial direction" with coordinate x {\displaystyle x} . A given pri... |
at m = ( p , f ( x ) ) {\displaystyle {\mathfrak {m}}=(p,f(x))} is k ( m ) = Z [ x ] / m = F p [ x ] / ( f ( x ) ) ≅ F p ( α ) {\displaystyle k({\mathfrak {m}})=\mathbb {Z} [x]/{\mathfrak {m}}=\mathbb {F} _{p}[x]/(f(x))\cong \mathbb {F} _{p}(\alpha )} , a field extension of F p {\displaystyle \mathbb {F} _{p}} adjoinin... |
x + c {\displaystyle f(x,y)=y^{2}-x^{3}+ax^{2}+bx+c} is an elliptic curve, then the fibers over its discriminant locus, where Δ f = − 4 a 3 c + a 2 b 2 + 18 a b c − 4 b 3 − 27 c 2 = 0 mod p , {\displaystyle \Delta _{f}=-4a^{3}c+a^{2}b^{2}+18abc-4b^{3}-27c^{2}=0\ {\text{mod}}\ p,} are all singular schemes. For example, ... |
x − 1 ] {\displaystyle \mathrm {Spec} \,\mathbb {C} [x,x^{-1}]} . To show X is not affine, one computes that every regular function on X extends to a regular function on A n {\displaystyle \mathbb {A} ^{n}} when n ≥ 2: this is analogous to Hartogs's lemma in complex analysis, though easier to prove. That is, the inclus... |
(In fact, X(C) can be identified with C − 0.) By contrast, a scheme X over a field k has enough information to determine the set X(E) of E-rational points for every extension field E of k. (In particular, the closed subscheme of A2R defined by x2 + y2 = −1 is a nonempty topological space.) Generic point. The points of ... |
a field k {\displaystyle k} , with coordinate ring k [ x , y ] {\displaystyle k[x,y]} , consider the x-axis, which is the variety V ( y ) {\displaystyle V(y)} , and the parabola y = x 2 {\displaystyle y=x^{2}} , which is V ( x 2 − y ) {\displaystyle V(x^{2}-y)} . Their scheme-theoretic intersection is defined by the id... |
for example, a vector bundle on a closed subscheme Y of X can be viewed as a coherent sheaf on X that is zero outside Y (by the direct image construction). In this way, coherent sheaves on a scheme X include information about all closed subschemes of X. Moreover, sheaf cohomology has good properties for coherent (and q... |
leads to a theory that can remember higher information, in the same way that derived functors in homological algebra yield higher information about operations such as tensor product and the Hom functor on modules. == See also == Flat morphism, Smooth morphism, Proper morphism, Finite morphism, Étale morphism Stable cur... |
In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are 3 x x 2 + 2 x − 3 {\displaystyle {\frac {3x}{x^{2}+2x-3}}} and x + 2 x 2 − 3 {\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}} . Algebraic fractions are subject to the same law... |
reverse process of expressing a proper rational fraction as the sum of two or more fractions is called resolving it into partial fractions. For example, 2 x x 2 − 1 = 1 x − 1 + 1 x + 1 . {\displaystyle {\frac {2x}{x^{2}-1}}={\frac {1}{x-1}}+{\frac {1}{x+1}}.} Here, the two terms on the right are called partial fraction... |
In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of a... |
of R-modules). By definition, a ring is a monoid object in the category of abelian groups; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the category of modules. Pushing this idea further, some authors have introduced a "generalized ring" as a monoid object in s... |
) + φ ( y ) φ ( x y ) = φ ( x ) φ ( y ) φ ( 1 ) = 1 {\displaystyle {\begin{aligned}\varphi (r\cdot x)&=r\cdot \varphi (x)\\\varphi (x+y)&=\varphi (x)+\varphi (y)\\\varphi (xy)&=\varphi (x)\varphi (y)\\\varphi (1)&=1\end{aligned}}} The class of all R-algebras together with algebra homomorphisms between them form a categ... |
A quasi-free algebra, introduced by Cuntz and Quillen, is a sort of generalization of a free algebra and a semisimple algebra over an algebraically closed field. === Representation theory === The universal enveloping algebra of a Lie algebra is an associative algebra that can be used to study the given Lie algebra. If ... |
∈ a {\displaystyle f,g\in {\mathfrak {a}}} , f ∗ g = f g − 1 2 { f , g } u + ⋯ , {\displaystyle f*g=fg-{\frac {1}{2}}\{f,g\}u+\cdots ,} then a [ [ u ] ] {\displaystyle {\mathfrak {a}}[\![u]\!]} is called a deformation quantization of a {\displaystyle {\mathfrak {a}}} . A quantized enveloping algebra. The dual of such a... |
co-multiplication Δ(f)(g, h) = f(gh) and co-unit ε(f) = f(1). The "co-" refers to the fact that they satisfy the dual of the usual multiplication and unit in the algebra axiom. Hence, the dual A* is an associative algebra. The co-multiplication and co-unit are also important in order to form a tensor product of represe... |
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