Grothendieck

In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. Algebraic varieties are one of the central objects of study in classical (and to some extent, modern) algebraic geometry. Historically, the fundamental theorem of algebra established a link between algebra and geometry by saying that a polynomial in one variable over the complex numbers is determined by the set of its roots, which is an inherently geometric object. Building on this result, Hilbert's Nullstellensatz provides a fun ...

Including:

• Algebraic variety - Formal definitions
• Algebraic variety - Affine varieties
• Algebraic variety - Projective varieties
• Algebraic variety - Basic results
• Algebraic variety - Discussion and generalizations

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In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX-modules OXm → OXn. Here OX is the structure sheaf of local rings, given by definition on X. The form of the definition is a global (on X) way of carrying across the idea of a finitely-presented mo ...

Including:

• Coherent sheaf - Coherent cohomology

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List of mathematicians - Aa — Ac. Abakanowicz, Bruno (Poland, 1852 – 1900) Abdelkader, Mostafa A. (?, ? – ) Abel, Niels Henrik (Norway, 1802 – 1829) Abelson, Harold (USA, ? – ) Al-Abhari Athir al-Din Mufaddal ibn Umar (Persia, 1200 – 1265) Abhyankar, Shreeram Shankar (India/USA, 1930 – ) Abouabdillah, Driss (Morocco, 1948 – ) Abraham, Ralph (USA, 1936 – ) Abramowitz, Milton (?, ? – ) Achenwall, Gottfried (Germany, 1719 – 1772) Ackerm ...

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List of mathematicians, List of mathematicians - A, List of mathematicians - Aa — Ac, List of mathematicians - Ad, List of mathematicians - Ag — Ak, List of mathematicians - Al, List of mathematicians - Am — An, List of mathematicians - Ap — Ar, List of mathematicians - As — Az, List of mathematicians - B, List of mathematicians - Ba, List of mathematicians - Be, List of mathematicians - Bh — Bi, List of mathematicians - Bl, List of mathematicians - Bo, List of mathematicians - Br, List of mathematicians - Bu, List of mathematicians - By — Bz, List of mathematicians - C, List of mathematicians - Ca, List of mathematicians - Ce — Ch, List of mathematicians - Cl — Co, List of mathematicians - Cr — Cs, List of mathematicians - Ct — Cz, List of mathematicians - D, List of mathematicians - Da, List of mathematicians - De, List of mathematicians - Di, List of mathematicians - Do — Dr, List of mathematicians - Du, List of mathematicians - Dw — Dz, List of mathematicians - E, List of mathematicians - Ea — Ef, List of mathematicians - Eg — El, List of mathematicians - Em — Er, List of mathematicians - Es — Ez, List of mathematicians - F, List of mathematicians - Fa, List of mathematicians - Fe, List of mathematicians - Fi, List of mathematicians - Fl — Fo, List of mathematicians - Fr, List of mathematicians - Fu — Fz, List of mathematicians - G, List of mathematicians - Ga, List of mathematicians - Ge, List of mathematicians - Gh — Gi, List of mathematicians - Gl — Go, List of mathematicians - Gr, List of mathematicians - Gu, List of mathematicians - Gy — Gz, List of mathematicians - H, List of mathematicians - Ha, List of mathematicians - He, List of mathematicians - Hi, List of mathematicians - Hl — Ho, List of mathematicians - Hr — Hu, List of mathematicians - Hy — Hz, List of mathematicians - I, List of mathematicians - J, List of mathematicians - Ja, List of mathematicians - Je — Jo, List of mathematicians - Ju — Jz, List of mathematicians - K, List of mathematicians - Ka, List of mathematicians - Ke, List of mathematicians - Kh — Ki, List of mathematicians - Kl — Ko, List of mathematicians - Kr, List of mathematicians - Ku — Kz, List of mathematicians - L, List of mathematicians - La, List of mathematicians - Le, List of mathematicians - Li, List of mathematicians - Ll — Lo, List of mathematicians - Lu, List of mathematicians - Ly — Lz, List of mathematicians - M, List of mathematicians - Ma, List of mathematicians - Mc — Me, List of mathematicians - Mi, List of mathematicians - Mo, List of mathematicians - Mu — Mz, List of mathematicians - N, List of mathematicians - Na — Ne, List of mathematicians - Ni — No, List of mathematicians - Nu — Nz, List of mathematicians - O, List of mathematicians - Oa — Om, List of mathematicians - On — Oz, List of mathematicians - P, List of mathematicians - Pa, List of mathematicians - Pe, List of mathematicians - Pf — Pi, List of mathematicians - Pl — Po, List of mathematicians - Pr, List of mathematicians - Pt — Pz, List of mathematicians - Q, List of mathematicians - R, List of mathematicians - Ra, List of mathematicians - Re, List of mathematicians - Rh — Ri, List of mathematicians - Ro, List of mathematicians - Ru — Rz, List of mathematicians - S, List of mathematicians - Sa, List of mathematicians - Sc, List of mathematicians - Se, List of mathematicians - Sh, List of mathematicians - Si, List of mathematicians - Sk — Sm, List of mathematicians - Sn — So, List of mathematicians - Sp, List of mathematicians - Sr — St, List of mathematicians - Su — Sy, List of mathematicians - Sz, List of mathematicians - T, List of mathematicians - Ta, List of mathematicians - Te — Th, List of mathematicians - Ti — To, List of mathematicians - Tr — Ts, List of mathematicians - Tu, List of mathematicians - Tv — Tz, List of mathematicians - U, List of mathematicians - V, List of mathematicians - Va, List of mathematicians - Ve — Vi, List of mathematicians - Vo — Vz, List of mathematicians - W, List of mathematicians - Wa, List of mathematicians - We, List of mathematicians - Wh, List of mathematicians - Wi, List of mathematicians - Wo — Wz, List of mathematicians - X, List of mathematicians - Y, List of mathematicians - Ya — Yo, List of mathematicians - Yu — Yz, List of mathematicians - Z, List of mathematicians - Za, List of mathematicians - Ze — Zh, List of mathematicians - Zi — Zz

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Quillen was born in Orange, New Jersey. He entered Harvard University, where he earned both his BA (1961) and his PhD (1964), the latter of which was completed under the supervision of Raoul Bott with a thesis in partial differential equations. Quillen obtained a position at the Massachusetts Institute of Technology after completing his doctorate. However, he also spent a number of years at several other universities. This experience would prove to be important in influencing the direction of his research. He visited France twice: fir ...

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Daniel Quillen, Daniel Quillen - Biography, Daniel Quillen - Reference

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An algebraic space X is comprised of an affine scheme U and a closed subscheme R ⊂ U × U satisfying the following two conditions: 1. R is an equivalence relation as a subset of U × U 2. The projections pi: R → U onto each factor are étale maps. If a third condition 3. R is the trivial equivalence relation ov ...

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Algebraic space, Algebraic space - Definition, Algebraic space - Facts about algebraic spaces, Algebraic space - Applications, Algebraic space - Reference

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The algebraic geometers of the Italian school had often used the somewhat foggy concept of "generic point" when proving statements about algebraic varieties. What is true for the generic point is true for all points of the variety except a small number of special points. In the 1920s, Emmy Noether had first suggested a way to clarify the concept: start with the coordinate ring of the variety (the ring of all polynomial functions defined on the variety); the maximal ideals of this ring will correspond to ordinary points of the variety (under ...

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Scheme mathematics, Scheme mathematics - History and motivation, Scheme mathematics - Definitions, Scheme mathematics - The category of schemes, Scheme mathematics - Types of schemes, Scheme mathematics - O_X modules

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From a very young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and then commutative algebra and algebraic geometry, in the context of sheaf theory and homological algebra techniques. Serre's thesis refers to his dissertation on the Leray-Serre spectral sequence associated to a fibration. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl praised Serre in apparently extravagant terms, and also made the point that the award ...

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Jean-Pierre Serre, Jean-Pierre Serre - Life and career, Jean-Pierre Serre - Early work, Jean-Pierre Serre - Foundational work in algebraic geometry and the Weil conjectures, Jean-Pierre Serre - Other work, Jean-Pierre Serre - Awards, Jean-Pierre Serre - External link

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If F and G are an adjoint pair of functors, with F left adjoint to G, then the composition G o F will be a monad. Note that therefore a monad is a functor from a category to itself; and that if F and G were actually inverses as functors the corresponding monad would be the identity functor. In general adjunctions are not equivalences — they relate categories of different natures. The monad theory matters as part of the effort to capture what it is that adjunctions 'preserve'. The othe ...

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Monad category theory, Monad category theory - Introduction, Monad category theory - Formal definition, Monad category theory - Comonads and their importance, Monad category theory - Examples, Monad category theory - Uses

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From a very young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and then commutative algebra and algebraic geometry, in the context of sheaf theory and homological algebra techniques. Serre's thesis refers to his dissertation on the Leray-Serre spectral sequence associated to a fibration. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl praised Serre in apparently extravagant terms, and also made the point that the award ...

See also:

Jean-Pierre Serre, Jean-Pierre Serre - Life and career, Jean-Pierre Serre - Early work, Jean-Pierre Serre - Foundational work in algebraic geometry and the Weil conjectures, Jean-Pierre Serre - Other work, Jean-Pierre Serre - Awards, Jean-Pierre Serre - Works, Jean-Pierre Serre - External link

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Algebraic variety - Affine varieties. Let k be an algebraically closed field and let An be affine n-space over k. The polynomials f in the ring k[x1, ..., xn] can be viewed as k-valued functions on An by evaluating f at the points in An. For each subset S of k[x1, ..., xn], define the zero-locus of S to be the set of points in An on whic ...

See also:

Algebraic variety, Algebraic variety - Formal definitions, Algebraic variety - Affine varieties, Algebraic variety - Projective varieties, Algebraic variety - Basic results, Algebraic variety - Discussion and generalizations

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The crucial assumption we need to make about our abelian category A is that it have enough injectives, meaning that for every object A in A there exists a monomorphism A → I where I is an injective object in A. The right derived functors of the covariant left-exact functor F : A → B is then defined as follows. Start with an object X of A. Because there are enough injectives, we ...

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Derived functor, Derived functor - Motivation, Derived functor - Construction and first properties, Derived functor - Variations, Derived functor - Applications, Derived functor - Naturality, Derived functor - Generalization

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The case of the construction of vector bundles from data on a disjoint union of topological spaces is a straightforward place to start. Suppose X is a topological space covered by open sets Xi. Let Y to be the disjoint union of the Xi, so that there is a natural mapping p : Y → X. We think of Y as 'above' X, with the Xi projection 'down' onto X. With this language, descent ...

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Descent category theory, Descent category theory - Descent of vector bundles, Descent category theory - History

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A translation functor on a category D is an automorphism T from D to D. The image of X under Tn is usually written as X[n]. A triangle (X, Y, Z, u, v, w) is a set of 3 objects X, Y, and Z, together with morphisms u from X to Y, v from Y to Z and w from Z to X[1]. If (X, Y, Z, u, v, w) is a triangle then the rotated triangle is (Z[−1],XSee also:

Triangulated category, Triangulated category - Introduction, Triangulated category - Definition, Triangulated category - Comments on the axioms, Triangulated category - Examples, Triangulated category - t-structure and cores

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List of mathematicians - Va. Vacca, Giovanni (Italy, 1872 – 1953) Vailati, Giovanni (Italy, 1863 – 1909) de la Vallée-Poussin, Charles Jean (Belgium, 1866 – 1962) Vandermonde, Alexandre-Théophile (France, 1735 – 1796) VanLandingham, Wade (USA, ? – ) Vanstone, Scott (?, ? – ) Vapnik, Vladimir (Russia, ? – ) Varahamihira (India, 505 – 587) Varghese, Mathai (Australia, ? – ) Varignon, Pierre ...

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List of mathematicians, List of mathematicians - A, List of mathematicians - Aa — Ac, List of mathematicians - Ad, List of mathematicians - Ag — Ak, List of mathematicians - Al, List of mathematicians - Am — An, List of mathematicians - Ap — Ar, List of mathematicians - As — Az, List of mathematicians - B, List of mathematicians - Ba, List of mathematicians - Be, List of mathematicians - Bh — Bi, List of mathematicians - Bl, List of mathematicians - Bo, List of mathematicians - Br, List of mathematicians - Bu, List of mathematicians - By — Bz, List of mathematicians - C, List of mathematicians - Ca, List of mathematicians - Ce — Ch, List of mathematicians - Cl — Co, List of mathematicians - Cr — Cs, List of mathematicians - Ct — Cz, List of mathematicians - D, List of mathematicians - Da, List of mathematicians - De, List of mathematicians - Di, List of mathematicians - Do — Dr, List of mathematicians - Du, List of mathematicians - Dw — Dz, List of mathematicians - E, List of mathematicians - Ea — Ef, List of mathematicians - Eg — El, List of mathematicians - Em — Er, List of mathematicians - Es — Ez, List of mathematicians - F, List of mathematicians - Fa, List of mathematicians - Fe, List of mathematicians - Fi, List of mathematicians - Fl — Fo, List of mathematicians - Fr, List of mathematicians - Fu — Fz, List of mathematicians - G, List of mathematicians - Ga, List of mathematicians - Ge, List of mathematicians - Gh — Gi, List of mathematicians - Gl — Go, List of mathematicians - Gr, List of mathematicians - Gu, List of mathematicians - Gy — Gz, List of mathematicians - H, List of mathematicians - Ha, List of mathematicians - He, List of mathematicians - Hi, List of mathematicians - Hl — Ho, List of mathematicians - Hr — Hu, List of mathematicians - Hy — Hz, List of mathematicians - I, List of mathematicians - J, List of mathematicians - Ja, List of mathematicians - Je — Jo, List of mathematicians - Ju — Jz, List of mathematicians - K, List of mathematicians - Ka, List of mathematicians - Ke, List of mathematicians - Kh — Ki, List of mathematicians - Kl — Ko, List of mathematicians - Kr, List of mathematicians - Ku — Kz, List of mathematicians - L, List of mathematicians - La, List of mathematicians - Le, List of mathematicians - Li, List of mathematicians - Ll — Lo, List of mathematicians - Lu, List of mathematicians - Ly — Lz, List of mathematicians - M, List of mathematicians - Ma, List of mathematicians - Mc — Me, List of mathematicians - Mi, List of mathematicians - Mo, List of mathematicians - Mu — Mz, List of mathematicians - N, List of mathematicians - Na — Ne, List of mathematicians - Ni — No, List of mathematicians - Nu — Nz, List of mathematicians - O, List of mathematicians - Oa — Om, List of mathematicians - On — Oz, List of mathematicians - P, List of mathematicians - Pa, List of mathematicians - Pe, List of mathematicians - Pf — Pi, List of mathematicians - Pl — Po, List of mathematicians - Pr, List of mathematicians - Pt — Pz, List of mathematicians - Q, List of mathematicians - R, List of mathematicians - Ra, List of mathematicians - Re, List of mathematicians - Rh — Ri, List of mathematicians - Ro, List of mathematicians - Ru — Rz, List of mathematicians - S, List of mathematicians - Sa, List of mathematicians - Sc, List of mathematicians - Se, List of mathematicians - Sh, List of mathematicians - Si, List of mathematicians - Sk — Sm, List of mathematicians - Sn — So, List of mathematicians - Sp, List of mathematicians - Sr — St, List of mathematicians - Su — Sy, List of mathematicians - Sz, List of mathematicians - T, List of mathematicians - Ta, List of mathematicians - Te — Th, List of mathematicians - Ti — To, List of mathematicians - Tr — Ts, List of mathematicians - Tu, List of mathematicians - Tv — Tz, List of mathematicians - U, List of mathematicians - V, List of mathematicians - Va, List of mathematicians - Ve — Vi, List of mathematicians - Vo — Vz, List of mathematicians - W, List of mathematicians - Wa, List of mathematicians - We, List of mathematicians - Wh, List of mathematicians - Wi, List of mathematicians - Wo — Wz, List of mathematicians - X, List of mathematicians - Y, List of mathematicians - Ya — Yo, List of mathematicians - Yu — Yz, List of mathematicians - Z, List of mathematicians - Za, List of mathematicians - Ze — Zh, List of mathematicians - Zi — Zz

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List of mathematicians - Wa. Wada, Hideo (Japan, ? – ) van der Waerden, Bartel Leendert (Netherlands, 1903 – 1996) Wagner, Manfred (Germany, 1948 – ) Wagstaff Jr., Samuel S. (?, ? – ) Wald, Abraham (Hungary, 1902 – 1950) Walker, Arthur Geoffrey (Britain, 1909 – 2001) Wall, C. T. C. (Britain, 1936 – ) Wall, D. D. (?, ? – ) Wall, Terry (Britain, 1936 – ) Wallace, David (Scotland, ? – ) ...

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List of mathematicians, List of mathematicians - A, List of mathematicians - Aa — Ac, List of mathematicians - Ad, List of mathematicians - Ag — Ak, List of mathematicians - Al, List of mathematicians - Am — An, List of mathematicians - Ap — Ar, List of mathematicians - As — Az, List of mathematicians - B, List of mathematicians - Ba, List of mathematicians - Be, List of mathematicians - Bh — Bi, List of mathematicians - Bl, List of mathematicians - Bo, List of mathematicians - Br, List of mathematicians - Bu, List of mathematicians - By — Bz, List of mathematicians - C, List of mathematicians - Ca, List of mathematicians - Ce — Ch, List of mathematicians - Cl — Co, List of mathematicians - Cr — Cs, List of mathematicians - Ct — Cz, List of mathematicians - D, List of mathematicians - Da, List of mathematicians - De, List of mathematicians - Di, List of mathematicians - Do — Dr, List of mathematicians - Du, List of mathematicians - Dw — Dz, List of mathematicians - E, List of mathematicians - Ea — Ef, List of mathematicians - Eg — El, List of mathematicians - Em — Er, List of mathematicians - Es — Ez, List of mathematicians - F, List of mathematicians - Fa, List of mathematicians - Fe, List of mathematicians - Fi, List of mathematicians - Fl — Fo, List of mathematicians - Fr, List of mathematicians - Fu — Fz, List of mathematicians - G, List of mathematicians - Ga, List of mathematicians - Ge, List of mathematicians - Gh — Gi, List of mathematicians - Gl — Go, List of mathematicians - Gr, List of mathematicians - Gu, List of mathematicians - Gy — Gz, List of mathematicians - H, List of mathematicians - Ha, List of mathematicians - He, List of mathematicians - Hi, List of mathematicians - Hl — Ho, List of mathematicians - Hr — Hu, List of mathematicians - Hy — Hz, List of mathematicians - I, List of mathematicians - J, List of mathematicians - Ja, List of mathematicians - Je — Jo, List of mathematicians - Ju — Jz, List of mathematicians - K, List of mathematicians - Ka, List of mathematicians - Ke, List of mathematicians - Kh — Ki, List of mathematicians - Kl — Ko, List of mathematicians - Kr, List of mathematicians - Ku — Kz, List of mathematicians - L, List of mathematicians - La, List of mathematicians - Le, List of mathematicians - Li, List of mathematicians - Ll — Lo, List of mathematicians - Lu, List of mathematicians - Ly — Lz, List of mathematicians - M, List of mathematicians - Ma, List of mathematicians - Mc — Me, List of mathematicians - Mi, List of mathematicians - Mo, List of mathematicians - Mu — Mz, List of mathematicians - N, List of mathematicians - Na — Ne, List of mathematicians - Ni — No, List of mathematicians - Nu — Nz, List of mathematicians - O, List of mathematicians - Oa — Om, List of mathematicians - On — Oz, List of mathematicians - P, List of mathematicians - Pa, List of mathematicians - Pe, List of mathematicians - Pf — Pi, List of mathematicians - Pl — Po, List of mathematicians - Pr, List of mathematicians - Pt — Pz, List of mathematicians - Q, List of mathematicians - R, List of mathematicians - Ra, List of mathematicians - Re, List of mathematicians - Rh — Ri, List of mathematicians - Ro, List of mathematicians - Ru — Rz, List of mathematicians - S, List of mathematicians - Sa, List of mathematicians - Sc, List of mathematicians - Se, List of mathematicians - Sh, List of mathematicians - Si, List of mathematicians - Sk — Sm, List of mathematicians - Sn — So, List of mathematicians - Sp, List of mathematicians - Sr — St, List of mathematicians - Su — Sy, List of mathematicians - Sz, List of mathematicians - T, List of mathematicians - Ta, List of mathematicians - Te — Th, List of mathematicians - Ti — To, List of mathematicians - Tr — Ts, List of mathematicians - Tu, List of mathematicians - Tv — Tz, List of mathematicians - U, List of mathematicians - V, List of mathematicians - Va, List of mathematicians - Ve — Vi, List of mathematicians - Vo — Vz, List of mathematicians - W, List of mathematicians - Wa, List of mathematicians - We, List of mathematicians - Wh, List of mathematicians - Wi, List of mathematicians - Wo — Wz, List of mathematicians - X, List of mathematicians - Y, List of mathematicians - Ya — Yo, List of mathematicians - Yu — Yz, List of mathematicians - Z, List of mathematicians - Za, List of mathematicians - Ze — Zh, List of mathematicians - Zi — Zz

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List of mathematicians - Za. Zadeh, Lotfi (Azerbaijan, 1921 – ) Zadeh, Qazi (Persia, 1364 – 1436) Zagier, Don (USA, 1951 – ) Zahradník, Miloš (Czech Republic, ? – ) Zakrajšek, Egon (Slovenia, 1941 – 2002) Zarankiewicz, Kazimierz (Poland, 1902 – 1959) Zaremba, Stanislaw (Poland 1863 - 1942) Zariski, Oscar (USA/Russia, 1899 – 1986) Al-Zarqali, Abu Ishaq Ibrahim ibn Yahya (Arabia/Spain, 1028 – 1087) Zassenhaus, ...

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List of mathematicians, List of mathematicians - A, List of mathematicians - Aa — Ac, List of mathematicians - Ad, List of mathematicians - Ag — Ak, List of mathematicians - Al, List of mathematicians - Am — An, List of mathematicians - Ap — Ar, List of mathematicians - As — Az, List of mathematicians - B, List of mathematicians - Ba, List of mathematicians - Be, List of mathematicians - Bh — Bi, List of mathematicians - Bl, List of mathematicians - Bo, List of mathematicians - Br, List of mathematicians - Bu, List of mathematicians - By — Bz, List of mathematicians - C, List of mathematicians - Ca, List of mathematicians - Ce — Ch, List of mathematicians - Cl — Co, List of mathematicians - Cr — Cs, List of mathematicians - Ct — Cz, List of mathematicians - D, List of mathematicians - Da, List of mathematicians - De, List of mathematicians - Di, List of mathematicians - Do — Dr, List of mathematicians - Du, List of mathematicians - Dw — Dz, List of mathematicians - E, List of mathematicians - Ea — Ef, List of mathematicians - Eg — El, List of mathematicians - Em — Er, List of mathematicians - Es — Ez, List of mathematicians - F, List of mathematicians - Fa, List of mathematicians - Fe, List of mathematicians - Fi, List of mathematicians - Fl — Fo, List of mathematicians - Fr, List of mathematicians - Fu — Fz, List of mathematicians - G, List of mathematicians - Ga, List of mathematicians - Ge, List of mathematicians - Gh — Gi, List of mathematicians - Gl — Go, List of mathematicians - Gr, List of mathematicians - Gu, List of mathematicians - Gy — Gz, List of mathematicians - H, List of mathematicians - Ha, List of mathematicians - He, List of mathematicians - Hi, List of mathematicians - Hl — Ho, List of mathematicians - Hr — Hu, List of mathematicians - Hy — Hz, List of mathematicians - I, List of mathematicians - J, List of mathematicians - Ja, List of mathematicians - Je — Jo, List of mathematicians - Ju — Jz, List of mathematicians - K, List of mathematicians - Ka, List of mathematicians - Ke, List of mathematicians - Kh — Ki, List of mathematicians - Kl — Ko, List of mathematicians - Kr, List of mathematicians - Ku — Kz, List of mathematicians - L, List of mathematicians - La, List of mathematicians - Le, List of mathematicians - Li, List of mathematicians - Ll — Lo, List of mathematicians - Lu, List of mathematicians - Ly — Lz, List of mathematicians - M, List of mathematicians - Ma, List of mathematicians - Mc — Me, List of mathematicians - Mi, List of mathematicians - Mo, List of mathematicians - Mu — Mz, List of mathematicians - N, List of mathematicians - Na — Ne, List of mathematicians - Ni — No, List of mathematicians - Nu — Nz, List of mathematicians - O, List of mathematicians - Oa — Om, List of mathematicians - On — Oz, List of mathematicians - P, List of mathematicians - Pa, List of mathematicians - Pe, List of mathematicians - Pf — Pi, List of mathematicians - Pl — Po, List of mathematicians - Pr, List of mathematicians - Pt — Pz, List of mathematicians - Q, List of mathematicians - R, List of mathematicians - Ra, List of mathematicians - Re, List of mathematicians - Rh — Ri, List of mathematicians - Ro, List of mathematicians - Ru — Rz, List of mathematicians - S, List of mathematicians - Sa, List of mathematicians - Sc, List of mathematicians - Se, List of mathematicians - Sh, List of mathematicians - Si, List of mathematicians - Sk — Sm, List of mathematicians - Sn — So, List of mathematicians - Sp, List of mathematicians - Sr — St, List of mathematicians - Su — Sy, List of mathematicians - Sz, List of mathematicians - T, List of mathematicians - Ta, List of mathematicians - Te — Th, List of mathematicians - Ti — To, List of mathematicians - Tr — Ts, List of mathematicians - Tu, List of mathematicians - Tv — Tz, List of mathematicians - U, List of mathematicians - V, List of mathematicians - Va, List of mathematicians - Ve — Vi, List of mathematicians - Vo — Vz, List of mathematicians - W, List of mathematicians - Wa, List of mathematicians - We, List of mathematicians - Wh, List of mathematicians - Wi, List of mathematicians - Wo — Wz, List of mathematicians - X, List of mathematicians - Y, List of mathematicians - Ya — Yo, List of mathematicians - Yu — Yz, List of mathematicians - Z, List of mathematicians - Za, List of mathematicians - Ze — Zh, List of mathematicians - Zi — Zz

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List of mathematicians - Ya — Yo. Yasuaki, Aida (Japan, 1747 – 1817) Yates, Frank (?, 1902 – 1994) Yates, Samuel (?, ? – ) Yau, Shing-Tung (China/Hong Kong, 1949 – ) Yildirim, Cem (Turkey, ? – ) Yoccoz, Jean-Christophe (France, 1957 – ) Yorke, James A. (?, 1941 – ) Yosida, Kosaku (Japan, 1909 – ) Young, Alfred (Britain, 1873 – 1940) Young, Arthur (?, 1741 – 1820) Young, Grace Chisholm (Britain, 1868 – 1944) Young, W ...

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List of mathematicians, List of mathematicians - A, List of mathematicians - Aa — Ac, List of mathematicians - Ad, List of mathematicians - Ag — Ak, List of mathematicians - Al, List of mathematicians - Am — An, List of mathematicians - Ap — Ar, List of mathematicians - As — Az, List of mathematicians - B, List of mathematicians - Ba, List of mathematicians - Be, List of mathematicians - Bh — Bi, List of mathematicians - Bl, List of mathematicians - Bo, List of mathematicians - Br, List of mathematicians - Bu, List of mathematicians - By — Bz, List of mathematicians - C, List of mathematicians - Ca, List of mathematicians - Ce — Ch, List of mathematicians - Cl — Co, List of mathematicians - Cr — Cs, List of mathematicians - Ct — Cz, List of mathematicians - D, List of mathematicians - Da, List of mathematicians - De, List of mathematicians - Di, List of mathematicians - Do — Dr, List of mathematicians - Du, List of mathematicians - Dw — Dz, List of mathematicians - E, List of mathematicians - Ea — Ef, List of mathematicians - Eg — El, List of mathematicians - Em — Er, List of mathematicians - Es — Ez, List of mathematicians - F, List of mathematicians - Fa, List of mathematicians - Fe, List of mathematicians - Fi, List of mathematicians - Fl — Fo, List of mathematicians - Fr, List of mathematicians - Fu — Fz, List of mathematicians - G, List of mathematicians - Ga, List of mathematicians - Ge, List of mathematicians - Gh — Gi, List of mathematicians - Gl — Go, List of mathematicians - Gr, List of mathematicians - Gu, List of mathematicians - Gy — Gz, List of mathematicians - H, List of mathematicians - Ha, List of mathematicians - He, List of mathematicians - Hi, List of mathematicians - Hl — Ho, List of mathematicians - Hr — Hu, List of mathematicians - Hy — Hz, List of mathematicians - I, List of mathematicians - J, List of mathematicians - Ja, List of mathematicians - Je — Jo, List of mathematicians - Ju — Jz, List of mathematicians - K, List of mathematicians - Ka, List of mathematicians - Ke, List of mathematicians - Kh — Ki, List of mathematicians - Kl — Ko, List of mathematicians - Kr, List of mathematicians - Ku — Kz, List of mathematicians - L, List of mathematicians - La, List of mathematicians - Le, List of mathematicians - Li, List of mathematicians - Ll — Lo, List of mathematicians - Lu, List of mathematicians - Ly — Lz, List of mathematicians - M, List of mathematicians - Ma, List of mathematicians - Mc — Me, List of mathematicians - Mi, List of mathematicians - Mo, List of mathematicians - Mu — Mz, List of mathematicians - N, List of mathematicians - Na — Ne, List of mathematicians - Ni — No, List of mathematicians - Nu — Nz, List of mathematicians - O, List of mathematicians - Oa — Om, List of mathematicians - On — Oz, List of mathematicians - P, List of mathematicians - Pa, List of mathematicians - Pe, List of mathematicians - Pf — Pi, List of mathematicians - Pl — Po, List of mathematicians - Pr, List of mathematicians - Pt — Pz, List of mathematicians - Q, List of mathematicians - R, List of mathematicians - Ra, List of mathematicians - Re, List of mathematicians - Rh — Ri, List of mathematicians - Ro, List of mathematicians - Ru — Rz, List of mathematicians - S, List of mathematicians - Sa, List of mathematicians - Sc, List of mathematicians - Se, List of mathematicians - Sh, List of mathematicians - Si, List of mathematicians - Sk — Sm, List of mathematicians - Sn — So, List of mathematicians - Sp, List of mathematicians - Sr — St, List of mathematicians - Su — Sy, List of mathematicians - Sz, List of mathematicians - T, List of mathematicians - Ta, List of mathematicians - Te — Th, List of mathematicians - Ti — To, List of mathematicians - Tr — Ts, List of mathematicians - Tu, List of mathematicians - Tv — Tz, List of mathematicians - U, List of mathematicians - V, List of mathematicians - Va, List of mathematicians - Ve — Vi, List of mathematicians - Vo — Vz, List of mathematicians - W, List of mathematicians - Wa, List of mathematicians - We, List of mathematicians - Wh, List of mathematicians - Wi, List of mathematicians - Wo — Wz, List of mathematicians - X, List of mathematicians - Y, List of mathematicians - Ya — Yo, List of mathematicians - Yu — Yz, List of mathematicians - Z, List of mathematicians - Za, List of mathematicians - Ze — Zh, List of mathematicians - Zi — Zz

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The categorical dual definition is a formal definition of a comonad; this can be said quickly in the terms that a comonad for a category C is a monad for the opposite category Cop. It is therefore a functor U from C to itself, with a set of axioms for counit and comultiplication that come from reversing the arrows everywhere in the definition just given. Since a comonoid is not a basic structure in abstract algebra, this is less familiar at a ...

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Monad category theory, Monad category theory - Introduction, Monad category theory - Formal definition, Monad category theory - Comonads and their importance, Monad category theory - Examples, Monad category theory - Uses

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If C is a category, a monad on C consists of a functor T : C → C together with two natural transformations: η : 1C → T (where 1C denotes the identity functor on C) and μ : T2 → T (where T2 is the functor T o T from C to C). These are required to fulfill the following axioms: μ o Tμ = μ o μT (as natural transformations T3See also:

Monad category theory, Monad category theory - Introduction, Monad category theory - Formal definition, Monad category theory - Comonads and their importance, Monad category theory - Examples, Monad category theory - Uses

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There are many ways one can qualify a scheme. According to a basic idea of Grothendieck, conditions should be applied to a morphism of schemes. Any scheme S has a unique morphism to Spec(Z), so this attitude, part of the relative point of view, doesn't lose anything. For detail on the development of scheme theory, which quickly becomes technically demanding, see first glossary of scheme theory. ...

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Scheme mathematics, Scheme mathematics - History and motivation, Scheme mathematics - Definitions, Scheme mathematics - The category of schemes, Scheme mathematics - Types of schemes, Scheme mathematics - O_X modules

Read more here: » Scheme mathematics: Encyclopedia II - Scheme mathematics - Types of schemes