978:
2. An element of a ring is called clean if it is the sum of a unit and an idempotent, and is called almost clean if it is the sum of a regular element and an idempotent. A ring is called clean or almost clean if all its elements are clean or almost clean, and a module is called clean or
6039:
3. The rational or real rank of a valuation or place is the rational or real rank of its valuation group, which is the dimension of the corresponding rational or real vector space constructed by tensoring the valuation group with the rational or real
7757:
7481:
of rings is a homomorphism that is formally unramified and finitely presented. These are analogous to immersions in differential topology. An algebra over a ring is called unramified if the corresponding morphism is
3976:
6830:
of rings is a homomorphism that is formally smooth and finitely presented. These are analogous to submersions in differential topology. An algebra over a ring is called smooth if the corresponding morphism is
2536:
2455:
819:
The word "big" when applied to a module emphasizes that the module is not necessarily finitely generated. In particular a big CohenâMacaulay module is a module that has a system of parameters for which it is
4302:
if it cannot be written as an intersection of two larger ideals or submodules. If the ideal or submodule is the whole ring or module this is inconsistent with the definition of an irreducible ring or module.
1786:
1. A divisor of an integral domain is an equivalence class of non-zero fractional ideals, where two such ideals are called equivalent if they are contained in the same principal fractional ideals.
1683:
of a ring, often just called the dimension, is the maximal length of a chain of prime ideals, and the Krull dimension of a module is the maximal length of a chain of prime ideals containing its annihilator.
2328:
6339:
6257:
4294:
is a ring where the zero ideal is not an intersection of two non-zero ideals, and more generally an irreducible module is a module where the zero module cannot be written as an intersection of non-zero
2718:
if every element is a linear combination of a fixed finite number of elements. If the module happens to be an algebra this is much stronger than saying it is finitely generated as an algebra.
7485:
2. An ideal in a polynomial ring over a field is called unramified for some extension of the field if the corresponding extension of the ideal is an intersection of prime ideals.
2944:
3. Formally catenary rings are rings such that every quotient by a prime ideal is formally equidimensional. For
Noetherian local rings this is equivalent to the ring being
7606:
7538:
is a totally ordered abelian group. The valuation group of a valuation ring is the group of non-zero elements of the quotient field modulo the group of units of the valuation ring.
5565:
1067:
if it is
Noetherian and the Krull dimension is equal to the depth. A ring is called CohenâMacaulay if it is Noetherian and all localizations at maximal ideals are CohenâMacaulay.
7411:
2694:
A finite module (or algebra) over a ring usually means one that is finitely generated as a module. It may also mean one with a finite number of elements, especially in the term
5589:
3908:
6696:
579:
1. The analytic spread of an ideal of a local ring is the Krull dimension of the fiber at the special point of the local ring of the Rees algebra of the ideal.
7816:
is a complete
Noetherian topological ring with a basis of neighborhoods of 0 given by the powers of an ideal in the Jacobson radical (formerly called a semi-local ring).
2019:
5805:
2728:
3. An extension of fields is called finitely generated if elements of the larger field can all be expressed as rational functions of a finite generating set.
6766:
6733:
4957:
is the same as a
Japanese ring, in other words an integral domain whose integral closure in any finite extension of its quotient field is a finitely generated module.
5612:
1326:
of the spectrum is one that is a finite union of locally closed sets. For rings that are not
Noetherian the definition of a constructible subset is more complicated.
2086:
is a polynomial such that its leading term is 1, all other coefficients are divisible by a prime, and the constant term is not divisible by the square of the prime.
1754:
10. The dimension of a valuation ring over a field is the transcendence degree of its residue field; this is not usually the same as the Krull dimension.
4035:
7635:
2374:
2351:
7777:
7626:
3880:
2252:
2059:
2039:
8376:
8334:
8292:
8250:
8208:
8166:
8124:
8082:
6855:
of a ring, often just called the spectrum, is a locally ringed space whose underlying topological space is the set of prime ideals with the
Zariski topology.
3611:
2. The height of a valuation or place is the height of its valuation group, which is the number of proper convex subgroups of its valuation group.
3731:
states that the ring of polynomials over a field is
Noetherian, or more generally that any finitely generated algebra over a Noetherian ring is Noetherian.
1715:
of a vector space over a field is the minimal number of generators; this is unrelated to most other definitions of its dimension as a module over a field.
6484:. This is unrelated to the notion of a regular ring in commutative ring theory. In commutative algebra, commutative rings with this property are called
4684:
4. The localization of a ring at a prime ideal is the localization of the multiplicative subset given by the complement of the prime ideal.
2550:
is a universally catenary
Grothendieck ring such that for every finitely generated algebra the singular points of the spectrum form a closed subset.
404:
The word "absolutely" usually means "not relatively"; i.e., independent of the base field in some sense. It is often synonymous with "geometrically".
7425:
is an integral domain such that every element can be written as a product of primes in a way that is unique up to order and multiplication by units.
1120:
is a local ring that is complete in the topology (or rather uniformity) where the powers of the maximal ideal form a base of the neighborhoods at 0.
28:
7791:
is local ring that is
Henselian, pseudo-geometric, and such that any quotient ring by a prime ideal is a finite extension of a regular local ring.
7359:
if it is integral and its integral closure is a local ring. A local ring is called unibranch if the corresponding reduced local ring is unibranch.
5871:(or the field of fractions) of an integral domain is the localization at the prime ideal zero. This is sometimes confused with the first meaning.
3770:
states that a ring that is a finite module over a regular local ring or polynomial algebra is CohenâMacaulay if and only if it is a free module
3608:
of a prime ideal, also called its codimension or rank or altitude, is the supremum of the lengths of chains of prime ideals descending from it.
2577:
2. An extension of a module may mean either a module containing it as a submodule or a module mapping onto it as a quotient module.
8052:
8009:
7985:
7945:
1722:
of a module may refer to almost any of the various other dimensions, such as weak dimension, injective dimension, or projective dimension.
1071:
3913:
2460:
2379:
7014:
The superheight of an ideal is the supremum of the nonzero codimensions of the proper extensions of the ideal under ring homomorphisms.
6496:
4098:
is a group of automorphisms of a ring whose elements fix a given prime ideal and act trivially on the corresponding residue class ring.
8435:
8417:
4218:
The two different meanings of integral (no zero divisors, or every element being a root of a monic polynomial) are sometimes confused.
3071:
The word "geometrically" usually refers to properties that continue to hold after taking finite field extensions. For example, a ring
2261:
7834:
says that if a field is a finitely generated algebra over another field then it is a finite dimensional vector space over the field.
24:
6559:
6284:
6168:
5915:
4228:
2. An element is called integral over a subring if it is a root of a monic polynomial with coefficients in the subring.
4731:
4275:
An invertible fractional ideal is a fractional ideal that has an inverse in the monoid of fractional ideals under multiplication.
2191:
An algebra is said to be essentially of finite type over another algebra if it is a localization of a finitely generated algebra.
1110:
455:
1113:
is a
Noetherian local ring whose completion is the quotient of a regular local ring by an ideal generated by a regular sequence.
7846:
5833:
is a Grothendieck ring such that for every finitely generated algebra the singular points of the spectrum form a closed subset.
6554:) says that the depth of the localization at any prime ideal is the height of the prime ideal whenever the depth is less than
698:
32:
4838:
is a condition on an inverse system of modules that ensures the vanishing of the first derived functor of the inverse limit.
7504:
is a homomorphism from the non-zero elements of a field to a totally ordered abelian group, with properties similar to the
1094:
if it is finitely generated and every homomorphism to it from a finitely generated module has a finitely generated kernel.
8440:
3714:
1887:
is a duality between Artinian and Noetherian modules over a complete local ring that is finitely generated over a field.
1323:
1133:
4687:
5. A ring is called locally integral if it is reduced and the localization at every prime ideal is integral.
4260:
6. A ring is called locally integral if it is reduced and the localization at every prime ideal is integral.
7422:
7344:
4681:
at a (multiplicative) subset is the ring formed by forcing all elements of the mutliplicative subset to be invertible.
4702:
An extension of rings has the lying over property if the corresponding map between their prime spectra is surjective.
3721:
1055:
is a field or a complete discrete valuation ring of mixed characteristic (0,p) whose maximal ideal is generated by p.
4807:
3. A minimal primary decomposition is a primary decomposition with the smallest possible number of terms.
1964:
is a complex generalizing many of the properties of a dualizing module to rings that do not have a dualizing module.
8001:
6776:
An algebra over a field is called separable if its extension by any finite purely inseparable extension is reduced.
6152:
4404:
3575:
2722:
1870:
1659:
is generated by the entries of a matrix, with relations given by the determinants of the minors of some fixed size.
7445:
A universal field is an algebraically closed field with the uncountable transcendence degree over its prime field.
4257:
5. An algebra over a ring is called an integral algebra if all its elements are integral over the ring.
3742:
3735:
2725:
if it is finitely generated as an algebra, which is much weaker than saying it is finitely generated as a module.
2715:
1668:
643:
589:
is a quotient of a ring of convergent power series in a finite number of variables over a field with a valuation.
393:
40:
6944:
2612:
1992:
8412:, Interscience Tracts in Pure and Applied Mathematics, vol. 13, New York-London: Interscience Publishers,
8372:"ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique: IV. Ătude locale des schĂ©mas et des morphismes de schĂ©mas, QuatriĂšme partie"
8330:"ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique: IV. Ătude locale des schĂ©mas et des morphismes de schĂ©mas, TroisiĂšme partie"
7940:. Mathematics lecture note series (2. ed., 2. print ed.). Reading, Mass.: Benjamin/Cummings. p. 146.
6465:
4835:
4718:
4264:
928:
623:
412:
371:
329:
8246:"ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique: IV. Ătude locale des schĂ©mas et des morphismes de schĂ©mas, PremiĂšre partie"
7838:
5847:
was an old term for a (possibly non-Noetherian) local ring in books that assumed local rings to be Noetherian.
1064:
988:
396:
is the integral closure of an integral domain in an algebraic closure of the field of fractions of the domain.
8288:"ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique: IV. Ătude locale des schĂ©mas et des morphismes de schĂ©mas, Seconde partie"
7973:
7894:
7266:
6499:
is an invariant of a graded module over a graded ring related to the vanishing of various cohomology groups.
4819:
3767:
3054:
2658:
1973:
1811:
of an integral domain is a non-zero fractional ideal that is an intersection of principal fractional ideals.
1763:
1655:
This often refers to properties of an ideal generated by determinants of minors of a matrix. For example, a
1343:
1219:
614:
36:
7559:
6036:
2. The rank or height of a valuation is the Krull dimension of the corresponding valuation ring.
8363:
8321:
8279:
8237:
8195:
8153:
8111:
8069:
6516:
5475:
5294:
5142:
4678:
4490:
3760:
3562:
3541:
or module is one that is a direct sum of pieces indexed by an abelian group, often the group of integers.
2647:
1161:
900:
473:, with the conditions about regular rings in the definition replaced by conditions about Gorenstein rings.
6800:
is an archaic term for an algebraic number field whose ring of integers is a unique factorization domain.
5535:
5452:
of an ideal or submodule is an expression of it as a finite intersection of primary ideals or submodules.
1011:
The codepth of a finitely generated module over a Noetherian local ring is its dimension minus its depth.
7857:
7370:
7144:
6979:
6962:
A ring is called strictly local if it is a local Henselian ring whose residue field is separably closed.
6951:
5494:
5448:
5192:
5062:
4870:
4400:
4083:
3728:
3594:
3377:
3080:
2571:
2083:
1748:
1733:
1719:
1656:
1421:
444:
2223:
411:
is a ring such that all modules over it are flat. (Non-commutative rings with this property are called
6859:
4741:
is a special case of Matlis duality for local rings that are finitely generated algebras over a field.
4690:
6. A ring has some property locally if its spectrum is covered by spectra of localizations
2096:
4. An Eisenstein extension is an extension generated by a root of an Eisenstein polynomial.
8162:"ElĂ©ments de gĂ©omĂ©trie algĂ©brique: III. Ătude cohomologique des faisceaux cohĂ©rents, PremiĂšre partie"
7434:
7356:
5934:
5830:
5570:
5501:
5468:
5138:
5025:
4204:
3889:
3605:
2945:
2090:
1705:
1369:
2. Two elements of a ring are called coprime if the ideal they generate is the whole ring.
408:
8204:"ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique: III. Ătude cohomologique des faisceaux cohĂ©rents, Seconde partie"
5814:
2. A field extension is purely inseparable if it consists of purely inseparable elements.
5141:
states that the points of affine space correspond to maximal ideals of its coordinate ring, and the
5058:
is a ring that is a Noetherian module over itself, in other words every ideal is finitely generated.
4639:
is a ring with just one maximal ideal. In older books it is sometimes also assumed to be Noetherian.
4210:
4. The injective dimension of a module is the smallest length of an injective resolution.
8120:"ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique: II. Ătude globale Ă©lĂ©mentaire de quelques classes de morphismes"
7501:
7478:
7433:
A property is said to hold universally if it holds for various base changes. For example a ring is
7270:
7237:
7023:
6815:
5952:
4977:
4798:
4284:
4170:
4135:
3763:
is a representation of a ring as a finite free module over a polynomial ring or regular local ring.
3121:
2875:
2806:
2581:
2164:
2153:
2076:
2070:
1740:
1698:
1521:
1433:
1117:
646:
of a subset of a module is the ideal of elements whose product with any element of the subset is 0.
605:
253:
4945:
is an integral domain whose integral closure in its quotient field is a finitely generated module.
4810:
4. A minimal prime of a domain is a minimal element of the set of nonzero prime ideals.
7824:
7254:
6938:
6785:
6661:
6350:
6262:
5998:
5852:
5762:
4804:
2. A minimal resolution of a module is a resolution contained in any other resolution.
4573:
4569:
4193:
3707:
3395:
The various uses of the term "grade" are sometimes inconsistent and incompatible with each other.
3218:
2938:
2677:
2136:
A local ring is called equicharacteristic if it has the same characteristic as its residue field.
2125:
1644:
7831:
1998:
5776:
5065:
represents a finitely generated algebra over a field as a finite module over a polynomial ring.
4345:
is a ring such that the set of regular points of the spectrum contains a non-empty open subset.
8413:
8367:
8325:
8283:
8241:
8199:
8157:
8115:
8073:
8048:
8005:
7981:
7941:
6840:
6610:
The saturation of a subset of a ring or module is the smallest saturated subset containing it.
5986:
5837:
5487:
5048:
4866:
is an integral domain satisfying the ascending chain conditions on integral divisorial ideals.
4593:
4502:
4299:
4158:
3636:
3568:
3371:
3254:
3019:
2706:
An algebra over a ring is said to be of finite type if it is finitely generated as an algebra.
2200:
1961:
1793:
of a ring is an element of the free abelian group generated by the codimension 1 prime ideals.
6738:
6705:
8405:
8385:
8343:
8301:
8259:
8217:
8175:
8133:
8091:
8040:
7965:
7919:
7820:
7788:
6386:
6278:
5927:
5594:
5513:
5306:
5145:
states that closed subsets of a variety correspond to radical ideals of its coordinate ring.
4738:
4643:
4422:
4311:
4291:
4251:
4186:
4162:
4146:
3754:
3550:
3351:
3035:
3001:
2965:
2219:
2207:
1891:
1884:
1808:
1775:
1726:
1355:
1255:
888:
808:
710:
686:
505:(often a field) is a ring (or sometimes an integral domain) that is finitely generated over
47:
8397:
8355:
8313:
8271:
8229:
8187:
8145:
8103:
8062:
8019:
7752:{\displaystyle m\cdot ((a_{1},\cdots ,a_{i-1})\colon a_{i})\subset (a_{1},\cdots ,a_{i-1})}
5238:
1. A module is called perfect if its projective dimension is equal to its grade.
4759:
is a duality between Artinian and Noetherian modules over a complete Noetherian local ring.
4189:
is one with the property that maps from submodules to it can be extended to larger modules.
8393:
8351:
8309:
8267:
8225:
8183:
8141:
8099:
8058:
8036:
8015:
7977:
7923:
7247:
7227:
7215:
6827:
6643:
6353:
is a Noetherian local ring whose dimension is equal to the dimension of its tangent space.
5461:
5055:
4452:
4445:
4416:
4378:
4222:
3717:
identifies irreducible subsets of affine space with radical ideals of the coordinate ring.
3384:
2871:
2784:
2635:
1820:
1797:
1680:
1445:
1378:
1091:
672:
470:
466:
310:
226:
582:
2. The analytic deviation of an ideal is its analytic spread minus its height.
4969:
is a Noetherian universally Japanese ring. These are also called pseudo-geometric rings.
3498:. This is not consistent with the other definition of the grade of a module given above.
2356:
2333:
8026:
7762:
7611:
7512:
6852:
6624:
5868:
5644:
4823:
4756:
4557:
4514:
4385:
4323:
4197:
3865:
3798:
3780:
3675:
3553:
is a set of generators for an ideal of a polynomial ring satisfying certain conditions.
3538:
3380:
is a Noetherian local ring that has finite injective dimension as a module over itself.
2879:
2686:
4. A quotient field may mean either a residue field of a field of fractions.
2623:
2547:
2237:
2044:
2024:
1877:
1691:
1687:
869:
485:-adic topology on a ring has a base of neighborhoods of 0 given by powers of the ideal
276:
8371:
8329:
8287:
7230:
of a module over a ring is an element annihilated by some regular element of the ring.
6089:>0, a polynomial in several variables is called reduced if it has degree less than
3738:
describes a free resolution of a quotient of a local ring with projective dimension 2.
3063:
is an integral domain such that any two elements have a greatest common divisor (GCD).
1778:
is a module such that multiplication by any regular element of the ring is surjective.
8429:
7807:
7629:
7233:
2. The torsion submodule of a module is the submodule of torsion elements.
7156:
7050:
5371:
5348:
5260:
3. A field is called perfect if all finite extension fields are separable.
5188:
5074:
4775:
4763:
4724:
4441:
4429:
4095:
3991:
3786:
3694:
3682:
3031:
2737:
1098:
951:
is called clean if it has a finite filtration all of whose quotients are of the form
679:
666:
7918:(Corrected reprint of the 2nd ed.), New York: Dover Publications, p. 119,
1346:
is the ideal given by the inverse image of some ideal under a homomorphism of rings.
811:
is an integral domain in which the sum of two principal ideals is a principal ideal.
7881:
7813:
7553:
1. Weak dimension is an alternative name for flat dimension of a module.
6647:
6631:
6363:
3. A regular element of a ring is an element that is not a zero divisor.
6357:
6158:
6079:
5274:
5232:
5196:
5042:
4526:
4407:
if and only if the rank of a corresponding Jacobian matrix is the maximum possible.
3700:
3667:) into a product of coprime monic polynomials can be lifted to a factorization in
2695:
1790:
872:
is a Noetherian local ring such that every system of parameters is a weak sequence.
829:
8245:
8203:
8161:
5012:
Some power is zero. Can be applied to elements of a ring or ideals of a ring. See
3387:
is a ring all of whose localizations at prime ideals are Gorenstein local rings.
8119:
8077:
8030:
7995:
7243:
4. A torsion module is one all of whose elements are torsion elements.
4235:
of a ring is called almost integral over a subring if there is a regular element
529:
of a ring is called almost integral over a subring if there is a regular element
6914:
6574:
of a ring or module is called saturated with respect to a multiplicative subset
5632:
5341:
5081:
4966:
4863:
4750:
4357:
is a ring such that the set of regular points of the spectrum is an open subset.
4119:
3630:
3452:), which for a finitely generated module over a Noetherian ring is the smallest
2777:
2559:
1880:
is a duality between Artinian and Noetherian modules over a complete local ring.
1046:
911:
The center of a valuation (or place) is the ideal of elements of positive order.
769:
517:
A local ring that is a localization of a finitely-generated domain over a field.
6870:
A decreasing filtration of a module is called stable (with respect to an ideal
2680:, or fraction field, of an integral domain is the smallest field containing it.
2673:
1. A commutative ring such that every nonzero element has an inverse
8044:
5864:
1. A quotient of a ring by an ideal, or of a module by a submodule.
4636:
4130:
is the set of all homogeneous components of minimal degree of the elements in
3983:
3060:
2839:
1744:
1052:
660:
7128:
An element of the kernel of one of the maps in a free resolution of a module.
5361:
4. "Prime sequence" is an alternative name for a regular sequence.
4254:
of a subring of a ring is the ring of all elements that are integral over it.
1436:
is a group of automorphisms of a ring whose elements fix a given prime ideal.
1334:
The content of a polynomial is a greatest common divisor of its coefficients.
552:
if its field of quotients is a finite extension of the field of quotients of
5164:
is an isomorphism from the highest non-zero exterior power of the module to
5013:
4822:, which says that a local ring that is finite over a regular local ring is
3789:
is an algebra with a special basis similar to a basis of standard monomials.
1712:
1405:
is a sort of universal homogeneous coordinate ring for a projective variety.
6265:
of an algebra is a way of writing in it in terms of polynomial subalgebras.
5285:
is the group of isomorphism classes of finite projective modules of rank 1.
4781:
2. A maximal CohenâMacaulay module over a Noetherian local ring
4432:
is a ring such that every prime ideal is an intersection of maximal ideals.
2683:
3. A residue field is the quotient of a ring by a maximal ideal.
2654:
is a flat module whose tensor product with any non-zero module is non-zero.
1366:
1. Two ideals are called coprime if their sum is the whole ring.
1082:-th local cohomology of the ring along the maximal ideal has finite length.
4801:
of an ideal is a minimal element of the set of prime ideals containing it.
4381:
is a matrix whose entries are the partial derivatives of some polynomials.
2760:
elements is the ideal generated by the determinants of the minors of size
4954:
4942:
4366:
4354:
4342:
3710:
measures the rate of growth of a module over a graded ring or local ring.
2930:
is then called a formally smooth, formally unramified, or formally etale
1402:
7827:
whose closed sets are the sets of prime ideals containing a given ideal.
4493:
of a ring is the universal module with a derivation from the ring to it.
1034:
2. A complete Noetherian local ring with finite residue field
8389:
8347:
8305:
8263:
8221:
8179:
8137:
8095:
1729:
of a ring is the supremum of the projective dimensions of its modules.
597:
This often refers to properties of the completion of a local ring; cf.
7273:
of a ring is formed by forcing all non zero divisors to have inverses.
6947:
is a quotient of a polynomial algebra by a square-free monomial ideal.
5077:
is an integral domain that is integrally closed in its quotient field.
3801:(or envelope) of a module is a minimal injective module containing it.
1571:
is the maximal ideal of a local ring this is just called the depth of
571:
2. The altitude of an ideal is another name for its height.
7884:
in a ring is an element whose product with some nonzero element is 0.
4505:
are the integers of the imaginary quadratic field of discriminant â7.
4174:
4139:
1671:
is an invariant that measures how far the ring is from being regular.
6519:
is a chain complex whose only non-zero homology group is the module.
6507:
The quotient of a ring, especially a local ring, by a maximal ideal.
5840:
is a morphism between complexes inducing an isomorphism on homology.
4314:
of a graded algebra is generated by all elements of positive degree.
3745:
measures the severity of singularities in a positive characteristic.
1635:
of an integral domain is its integral closure in its quotient field.
903:
if all maximal chains between two prime ideals have the same length.
675:
is a module satisfying the descending chain condition on submodules.
4388:
of a quotient of a polynomial ring by an ideal of pure codimension
3971:{\displaystyle \bigcap _{n\geq 1}I^{n}({\mathfrak {a}}\otimes M)=0}
3571:
is a Noetherian ring whose formal fibers are geometrically regular.
856:
is an ideal isomorphic (as a module) to a torsion-free quotient of
425:
3. An ideal in a polynomial ring over a field is called
418:
2. An ideal in a polynomial ring over a field is called
8035:, Graduate Texts in Mathematics, vol. 150, Berlin, New York:
5344:
is a proper ideal whose complement is closed under multiplication.
4785:
is a CohenâMacaulay module whose dimension is the same as that of
2574:
is the ideal generated by the image under a homomorphism of rings.
2531:{\displaystyle \operatorname {Ann} _{R}(y)=\{r\in R\mid ry=0\}=xR}
2450:{\displaystyle \operatorname {Ann} _{R}(x)=\{r\in R\mid rx=0\}=yR}
2128:(or hull) of a module is a minimal injective module containing it.
1000:
6862:
of a ring is the set of maximal ideals with the Zariski topology.
6033:
1. Another older name for the height of a prime ideal.
5084:
is a ring whose localizations at prime ideals are normal domains.
4369:
is a ring such that any finitely generated algebra is a J-1 ring.
3724:
gives a finite free resolution of modules over a polynomial ring.
1736:
of a ring is the supremum of the flat dimensions of its modules.
6043:
3. The minimum number of generators of a free module.
3096:
is normal, regular, or reduced for every finite extension field
2210:
over a field is a finite product of finite separable extensions.
2104:
An embedded prime of a module is a non-minimal associated prime.
1448:
is a Noetherian integrally closed domain of dimension at most 1.
1424:
if it can be written as a direct sum of two non-zero submodules.
422:
if its extension remains prime for every extension of the field.
5910:", says that localization at any prime ideal of height at most
5332:
A presentable ring is one that is a quotient of a regular ring.
2323:{\displaystyle \operatorname {Ann} _{R}(x)=\{r\in R\mid rx=0\}}
1708:
of a module is the shortest length of a projective resolution.
1701:
of a module is the shortest length of an injective resolution.
1492:
is the number of extensions of the valuation to a larger field
1003:
computer algebra system for computations in commutative algebra
979:
almost clean if its endomorphism ring is clean or almost clean.
967:. A stronger variation of this definition says that the primes
6360:
is a ring whose localizations at all prime ideals are regular.
5504:
of a module is the smallest length of a projective resolution.
4766:
is an injective envelope of the residue field of a local ring.
4529:) is a ring with a well behaved theory of prime factorization.
682:
is a ring satisfying the descending chain condition on ideals.
8078:"ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique: I. Le langage des schĂ©mas"
6334:{\displaystyle M\to M^{**},m\mapsto \langle \cdot ,m\rangle }
6252:{\displaystyle \oplus _{n=0}^{\infty }t^{n}I^{n}=R\subset R.}
5765:
over a field if either the field has characteristic zero and
1766:
is an integrally closed Noetherian local ring of dimension 1.
919:
A strictly increasing or decreasing sequence of prime ideals.
53:
7853:-th powers of the maximal ideals containing the prime ideal.
5989:
of a ring is an extension generated by radicals of elements.
5051:
is a module such that every submodule is finitely generated.
4267:
if it is its own integral closure in the field of fractions.
4225:
or integral ring is a nontrivial ring without zero-divisors.
2780:
is a module such that tensoring with it preserves exactness.
1140:
of the quotient field are contained in a finitely generated
8000:, Cambridge Studies in Advanced Mathematics, vol. 39,
6954:
is a way of writing a ring in terms of polynomial subrings.
4778:
is a maximal element of the set of proper ideals of a ring.
1358:
is a module with exactly one associated prime..
7523:
is in its quotient field and if it is nonzero then either
4560:
is a ring in which any ideal has a primary decomposition.
689:
establishes a certain stability of filtration by an ideal.
6630:
2. "Semi-local ring" is an archaic term for a
5944:
of a ring is an element such that some positive power is
5478:
is an integral domain such that every ideal is principal.
4517:
is a free resolution constructed from a regular sequence.
4200:
of a module is a smallest injective module containing it.
2937:
2. A Noetherian local ring is called formally
2626:
is an alternative name for a unique factorization domain.
1873:
is a duality for cohomology of modules over a local ring.
1694:
of a module is the shortest length of a flat resolution.
1019:
The codimension of a prime ideal is another name for its
7437:
if all finitely generated algebras over it are catenary.
3685:
of a local ring is a Henselian ring constructed from it.
2941:(or quasi-unmixed) if its completion is equidimensional.
1302:
The conormal module of a quotient of a ring by an ideal
7257:
is a module isomorphic to a submodule of a free module.
2203:
if it is formally etale and locally finitely presented.
1894:(also called a canonical module) for a Noetherian ring
6627:
is a ring with only a finite number of maximal ideals.
4287:
if it cannot be written as a product of two non-units.
2926:) is surjective, injective, or bijective. The algebra
1599:
from a ring to a module that satisfies Leibniz's rule
7765:
7638:
7614:
7562:
7373:
7240:
is a module with no torsion elements other than zero.
6741:
6708:
6664:
6287:
6171:
5779:
5597:
5573:
5538:
5490:
is a module such that every epimorphism to it splits.
5351:
of a ring is an element that generates a prime ideal.
4086:
if it is not the direct sum of two proper submodules.
3916:
3892:
3868:
2463:
2382:
2359:
2336:
2264:
2240:
2047:
2027:
2001:
429:
if it is unramified for every extension of the field.
6788:, usually applied to a topology on a ring or module.
5631:"Pseudogeometric ring" is an alternative name for a
4425:
of a ring is the intersection of its maximal ideals.
2093:
states that an Eisenstein polynomial is irreducible.
2079:
is the ring generated by a primitive cube root of 1.
1647:
of a module is the top exterior power of the module.
618:
if its completion has no nonzero nilpotent elements.
7250:
Tor are the derived functors of the tensor product.
7147:
of a local ring is the dual of its cotangent space.
5358:
is a localization of the integers at a prime ideal.
4908:, or more precisely the length of the localization
4818:1. Miracle flatness is another name for
3525:is usually not the same as the grade of the module
7771:
7751:
7620:
7600:
7405:
6760:
6727:
6690:
6333:
6251:
5799:
5606:
5583:
5559:
4596:if the natural map from their tensor product over
3970:
3902:
3874:
3578:is a duality theorem for modules over local rings.
3414:over a Noetherian ring is the length of a maximal
2665:is an algebra that is faithfully flat as a module.
2530:
2449:
2368:
2345:
2322:
2246:
2053:
2033:
2013:
1037:3. An alternative name for a Cohen ring
6378:that does not annihilate any non-zero element of
5407:. More generally a primary submodule of a module
4727:is an alternative name for a CohenâMacaulay ring.
3237:is integrally closed has the going down property.
1136:if, whenever all positive powers of some element
609:if its completion is an integrally closed domain.
7849:in a polynomial ring is the intersection of the
6464:6. In non-commutative ring theory, a
5769:is in the field or the field has characteristic
5324:âȘâ preserving addition and multiplication and 1.
3482:over a Noetherian local ring with maximal ideal
2768:of the matrix of relations defining the module.
2721:2. An algebra over a ring is called
1194:2. The composite of a valuation ring
1101:is a ring that is a coherent module over itself.
887:"Canonical module" is an alternative term for a
436:is an alternative term for geometrically normal.
7281:A trivial ring is a ring with only one element.
4854:A product of powers of generators of an algebra
4617:A relation between ideals in a Gorenstein ring.
2960:is the ring of fractions of an integral domain
568:of a ring is an archaic name for its dimension.
7461:is called unmixed if all associated primes of
4876:A subset of a ring closed under multiplication
4650:is given by the derived functors of direct-lim
4298:3. An ideal or submodule is called
3678:is a local ring in which Hensel's lemma holds.
2592:such that every non-zero submodule intersects
2179:such that every non-zero submodule intersects
1823:is a ring with no zero-divisors and where 1â 0.
1031:1. A complete Noetherian local ring
7218:, the derived functors of the tensor product.
5471:is a ring such that every ideal is principal.
5028:of a ring is the ideal of nilpotent elements.
4283:1. An element of a ring is called
3816:A submodule of a ring. Special cases include:
2714:1. A module over a ring is called
2199:1. A morphism of rings is called
1202:of its residue field is the inverse image of
46:In this article, all rings are assumed to be
8:
6328:
6316:
6085:2. Over a ring of characteristic
4403:is a criterion stating that a local ring is
4243:is in the subring for all positive integers
3643:is a complete local ring with maximal ideal
2516:
2489:
2435:
2408:
2317:
2290:
2156:if it intersects every nonzero submodule of
537:is in the subring for all positive integers
355:,...} is a ring of formal power series over
6370:-regular element of a ring for some module
6082:is one with no non-zero nilpotent elements.
4980:states that if a finitely generated module
3703:is an alternative term for a Jacobson ring.
1290:is connected if its zeroth degree piece is
5524:1. A finitely generated module
4326:of a module is a minimal associated prime.
2562:, the derived functors of the Hom functor.
1078:< the Krull dimension of the ring, the
943:1. A finitely generated module
852:A Bourbaki ideal of a torsion-free module
7764:
7734:
7715:
7696:
7674:
7655:
7637:
7613:
7589:
7570:
7561:
7397:
7378:
7372:
7178:not in any minimal prime ideal such that
6746:
6740:
6713:
6707:
6682:
6669:
6663:
6298:
6286:
6207:
6197:
6187:
6176:
6170:
5958:5. The radical of a submodule
5955:is the ideal of radicals of its elements.
5789:
5784:
5778:
5596:
5575:
5574:
5572:
5544:
5543:
5537:
5220:
3947:
3946:
3937:
3921:
3915:
3894:
3893:
3891:
3867:
3253:of commutative rings is said to have the
3120:of commutative rings is said to have the
2468:
2462:
2387:
2381:
2358:
2335:
2269:
2263:
2239:
2046:
2026:
2000:
1074:is a Noetherian local ring such that for
971:must be minimal primes of the support of
931:is a non-negative integer generating the
630:4. Two local rings are called
7994:Bruns, Winfried; Herzog, JĂŒrgen (1993),
7508:-adic valuation of the rational numbers.
7289:The type of a finitely generated module
7166:of a ring with positive characteristic
5207:) of its spectrum with the closed point
1270:of commutative rings is the image under
29:glossary of classical algebraic geometry
7906:
7413:in a ring that generate the unit ideal.
5624:3. A morphism of modules is
5617:2. A morphism of modules is
5160:An orientation of a module over a ring
5100:is called normally flat along an ideal
5004:Occasionally used to mean "unramified".
4107:
3505:) of an ideal is given the grade grade(
935:-ideal of multiples of 1 that are zero.
627:if its completion has no zero divisors.
548:is called almost finite over a subring
6550:on a ring (for a non-negative integer
6468:is a ring such that for every element
6001:is a group of automorphisms of a ring
5940:3. A radical of an element
5902:on a ring (for a non-negative integer
5497:is a resolution by projective modules.
3655:, then any factorization of its image
2790:3. For flat dimension, see
1995:states: given a finite ring extension
1512:is called the ramification deficiency.
359:satisfying some convergence condition.
7601:{\displaystyle (a_{1},\cdots ,a_{r})}
7100:(if finite) elements of a local ring
6054:is the dimension of the vector space
5464:is an ideal generated by one element.
5137:Over algebraically closed field, the
4207:is a resolution by injective modules.
2222:is an integral domain with a form of
1583:this is called the depth of the ring
1540:is an ideal, is the smallest integer
1063:1. A local ring is called
621:3. A local ring is called
612:2. A local ring is called
603:1. A local ring is called
7:
8377:Publications MathĂ©matiques de l'IHĂS
8335:Publications MathĂ©matiques de l'IHĂS
8293:Publications MathĂ©matiques de l'IHĂS
8251:Publications MathĂ©matiques de l'IHĂS
8209:Publications MathĂ©matiques de l'IHĂS
8167:Publications MathĂ©matiques de l'IHĂS
8125:Publications MathĂ©matiques de l'IHĂS
8083:Publications MathĂ©matiques de l'IHĂS
7799:
7546:
7493:
7336:
7136:
6843:is the sum of its simple submodules.
6527:
5879:
5822:
5176:
5153:
4934:
4710:
4549:
4482:
4334:
3978:(for example, this is the case when
3882:is idealwise separated for an ideal
3809:
3586:
3478:3. The grade of a module
3011:
2604:
2041:is a Noetherian ring if and only if
1984:
1413:
1274:of the annihilator of the kernel of
1172:is the inverse limit of the modules
880:
745:
634:if their completions are isomorphic.
385:
7860:of a local ring with maximal ideal
6046:4. The rank of a module
6025:=1 it is called the inertia group.)
5576:
5560:{\displaystyle M_{\mathfrak {p}}=0}
5545:
5516:is a semiherediary integral domain.
4826:if and only if it is a flat module.
4584:Two subfields of a field extension
4392:is the ideal generated by the size
3948:
3895:
3494:. This is also called the depth of
3422:. This is also called the depth of
2638:is a module whose annihilator is 0.
1381:of a local ring with maximal ideal
7841:on holomorphic functions says the
7421:Also called a factorial domain. A
7406:{\displaystyle v_{1},\dots ,v_{n}}
7186:for all sufficiently large powers
6188:
3354:states that an integral extension
3221:states that an integral extension
14:
7970:Commutative algebra. Chapters 1â7
6811:2. Special in some way
4263:7. A domain is called
1835:is said to dominate a local ring
1090:1. A module is called
544:2. An integral domain
25:list of algebraic geometry topics
7864:is the dual of the vector space
6925:is free for some natural number
6816:singular computer algebra system
6102:
5231:In non-commutative ring theory,
5134:German for "zero locus theorem".
4732:Macaulay computer algebra system
4604:they generate is an isomorphism.
4145:2. In the context of
3079:is called geometrically normal,
2787:is a resolution by flat modules.
1902:such that for any maximal ideal
1111:local complete intersection ring
513:algebraic-geometrical local ring
456:geometrically regular local ring
7847:symbolic power of a prime ideal
7608:of elements of a maximal ideal
7355:A reduced local ring is called
7170:>0 consists of the elements
5856:; see formally equidimensional.
5628:if the cokernel is pseudo-zero.
5584:{\displaystyle {\mathfrak {p}}}
3903:{\displaystyle {\mathfrak {a}}}
3410:on a finitely-generated module
3053:1. Abbreviation for
2846:is a map from a flat module to
1898:is a finitely-generated module
1072:generalized CohenâMacaulay ring
193:
18:glossary of commutative algebra
7916:Algebraic extensions of fields
7746:
7708:
7702:
7686:
7648:
7645:
7595:
7563:
7198:is the ideal generated by all
7033:such that the localization of
6560:Serre's criterion on normality
6497:CastelnuovoâMumford regularity
6313:
6291:
6243:
6237:
6228:
6219:
6113:A reduction ideal of an ideal
6005:fixing some given prime ideal
5916:Serre's criterion on normality
4992:is the Jacobson radical, then
4538:
4396:minors of the Jacobian matrix.
3959:
3943:
3501:4. The grade grade(
3432:2. The grade grade(
3398:1. The grade grade(
3271:is a chain of prime ideals in
3138:is a chain of prime ideals in
3083:, or geometrically reduced if
2791:
2483:
2477:
2402:
2396:
2284:
2278:
2113:
1847:contains the maximal ideal of
1508:, and sometimes ÎŽ rather than
1286:A graded algebra over a field
756:is a module over a local ring
729:has a submodule isomorphic to
699:algebra with straightening law
33:glossary of algebraic geometry
1:
7297:over a Noetherian local ring
5741:) for every associated prime
5677:2. A pure subring
5621:if the kernel is pseudo-zero.
4884:The multiplicity of a module
4444:(also called N-2 ring) is an
4142:of the homogeneous elements.)
2858:1. A homomorphism
1258:of a surjective homomorphism
598:
287:, consisting of all elements
188:
7976:(Berlin), Berlin, New York:
7936:Matsumura, Hideyuki (1981).
7114:regular system of parameters
6389:with respect to some module
5721:3. A pure module
4846:An archaic term for an ideal
3521:. So the grade of the ideal
1134:completely integrally closed
1124:completely integrally closed
7423:unique factorization domain
7417:unique factorization domain
7345:unique factorization domain
7029:is the set of prime ideals
6893:for all sufficiently large
6691:{\displaystyle x^{3}=y^{2}}
5906:), "regular in codimension
3245:1. An extension
3112:1. An extension
1488:is the inertia degree, and
1484:is the ramification index,
1242:is the integral closure of
1020:
443:is an alternative term for
8457:
8002:Cambridge University Press
7914:McCarthy, Paul J. (1991),
7806:1. Named after
7084:-primary ideal containing
6937:1. Named after
6430:is regular for the module
6393:is a sequence of elements
6151:1. Named after
6139:for some positive integer
5729:is a module such that dim(
4462:, the integral closure of
4231:3. An element
3779:1. Named after
3753:1. Named after
3576:Grothendieck local duality
3362:has the going up property.
3018:An alternative name for a
2894:, the natural map from Hom
2611:An alternative name for a
2187:essentially of finite type
2144:1. A submodule
2075:1. The ring of
2014:{\displaystyle A\subset B}
1949:) and is 1-dimensional if
1871:Grothendieck local duality
1472:of a valuation of a field
1226:is the annihilator of the
8436:Glossaries of mathematics
8045:10.1007/978-1-4612-5350-1
7556:2. A sequence
7202:th powers of elements of
7116:if it actually generates
6808:1. Not regular
6117:with respect to a module
5966:is the ideal of elements
5807:is in the field for some
5800:{\displaystyle x^{p^{r}}}
5757:1. An element
5654:is a submodule such that
5235:has an unrelated meaning.
4470:is a finitely generated
3715:Hilbert's Nullstellensatz
3651:is a monic polynomial in
3304:, there is a prime ideal
3171:, there is a prime ideal
1843:and the maximal ideal of
1669:deviation of a local ring
1595:An additive homomorphism
1322:For a Noetherian ring, a
525:1. An element
413:von Neumann regular rings
394:absolute integral closure
388:absolute integral closure
63:
41:glossary of module theory
7174:such that there is some
7112:-primary ideal. It is a
6784:An alternative term for
6466:von Neumann regular ring
6062:over the quotient field
6050:over an integral domain
6009:and acting trivially on
5970:such that some power of
5183:A Noetherian local ring
4836:Mittag-Leffler condition
4830:Mittag-Leffler condition
4719:Francis Sowerby Macaulay
4036:incomparability property
4022:incomparability property
3722:Hilbert's syzygy theorem
3000:An alternative name for
2850:with superfluous kernel.
1747:is the dimension of its
929:characteristic of a ring
624:analytically irreducible
330:formal power series ring
183:
8364:Grothendieck, Alexandre
8322:Grothendieck, Alexandre
8280:Grothendieck, Alexandre
8238:Grothendieck, Alexandre
8196:Grothendieck, Alexandre
8154:Grothendieck, Alexandre
8112:Grothendieck, Alexandre
8070:Grothendieck, Alexandre
7974:Elements of Mathematics
7895:Glossary of ring theory
7823:is the topology on the
7367:A sequence of elements
7305:is the dimension (over
7267:total ring of fractions
7060:is the set of elements
6818:for commutative algebra
6761:{\displaystyle s^{3}=y}
6728:{\displaystyle s^{2}=x}
5685:is a subring such that
5312:with values in a field
5241:2. An ideal
4896:is the number of times
4239:of the subring so that
4046:are distinct primes of
4034:is said to satisfy the
3055:greatest common divisor
2890:with a nilpotent ideal
2797:4. A module
2659:faithfully flat algebra
2588:is a module containing
2330:, is a principal ideal
1974:discrete valuation ring
1764:discrete valuation ring
1758:discrete valuation ring
1467:ramification deficiency
1393:over the residue field.
1344:contraction of an ideal
1191:1. Not prime
963:an associated prime of
947:over a Noetherian ring
632:analytically isomorphic
615:analytically unramified
533:of the subring so that
469:are generalizations of
452:absolutely simple point
37:glossary of ring theory
7773:
7753:
7622:
7602:
7515:is an integral domain
7407:
6762:
6729:
6692:
6517:resolution of a module
6335:
6253:
5801:
5608:
5607:{\displaystyle \leq 1}
5585:
5561:
5476:principal ideal domain
5295:principal ideal domain
5143:strong Nullstellensatz
4679:localization of a ring
4458:of its quotient field
4173:of the multiplicative
4138:of the multiplicative
3972:
3904:
3876:
3761:Hironaka decomposition
3563:Alexander Grothendieck
2972:is a submodule of the
2648:faithfully flat module
2532:
2451:
2370:
2347:
2324:
2248:
2055:
2035:
2015:
1504:of the characteristic
1246:in its quotient field.
1222:of an integral domain
1162:completion of a module
178:
173:
168:
163:
158:
153:
148:
143:
138:
133:
128:
123:
118:
113:
108:
103:
98:
93:
88:
83:
78:
73:
68:
7858:Zariski tangent space
7774:
7754:
7623:
7603:
7527:or its inverse is in
7469:have the same height.
7408:
7145:Zariski tangent space
7080:. It is the smallest
6952:Stanley decomposition
6763:
6730:
6693:
6336:
6281:if the canonical map
6254:
5802:
5609:
5586:
5567:for all prime ideals
5562:
5495:projective resolution
5449:primary decomposition
5249:is called perfect if
5221:#system of parameters
5063:Noether normalization
4871:multiplicative subset
4572:is the length of any
4405:geometrically regular
3973:
3905:
3877:
3743:HilbertâKunz function
3736:HilbertâBurch theorem
3729:Hilbert basis theorem
3595:highest common factor
3418:-regular sequence in
3378:Gorenstein local ring
3081:geometrically regular
2572:extension of an ideal
2533:
2452:
2371:
2353:whose annihilator is
2348:
2325:
2249:
2084:Eisenstein polynomial
2061:is a Noetherian ring.
2056:
2036:
2016:
1749:Zariski tangent space
1734:weak global dimension
1720:homological dimension
1575:, and if in addition
1198:and a valuation ring
445:geometrically regular
427:absolutely unramified
7997:Cohen-Macaulay rings
7839:Zariski's main lemma
7763:
7636:
7612:
7560:
7435:universally catenary
7371:
7092:system of parameters
6945:StanleyâReisner ring
6739:
6706:
6662:
6472:there is an element
6285:
6169:
5831:quasi-excellent ring
5777:
5595:
5571:
5536:
5502:projective dimension
5469:principal ideal ring
5257:is a perfect module.
5139:weak Nullstellensatz
4820:Hironaka's criterion
4694:having the property.
4491:KĂ€hler differentials
4451:such that for every
4205:injective resolution
4102:infinitely generated
3998:finitely generated).
3914:
3890:
3886:if for every ideal,
3866:
3768:Hironaka's criterion
3282:is a prime ideal of
3149:is a prime ideal of
2946:universally catenary
2613:StanleyâReisner ring
2461:
2380:
2357:
2334:
2262:
2258:if its annihilator,
2238:
2091:Eisenstein criterion
2045:
2025:
1999:
1993:EakinâNagata theorem
1706:projective dimension
1385:is the vector space
1324:constructible subset
860:by a free submodule.
832:is a ring such that
409:absolutely flat ring
8441:Commutative algebra
8032:Commutative algebra
7938:Commutative algebra
7479:unramified morphism
7301:with residue field
7271:total quotient ring
7238:torsion-free module
7104:with maximal ideal
7024:support of a module
6650:in which, whenever
6192:
5953:radical of an ideal
5211:removed is trivial.
5195:at least 2 and the
4730:2. The
4698:lying over property
4600:to the subfield of
4485:KĂ€hler differential
4165:of the elements in
4118:1. In a
4082:A module is called
3858:idealwise separated
3831:with maximal ideal
3821:ideal of definition
3122:going down property
2876:formally unramified
2870:of rings is called
2582:essential extension
2165:essential extension
2154:essential submodule
2108:embedding dimension
2077:Eisenstein integers
2071:Gotthold Eisenstein
1741:embedding dimension
1732:8. The
1725:7. The
1718:6. The
1711:5. The
1704:4. The
1699:injective dimension
1697:3. The
1686:2. The
1633:derived normal ring
1567:) is nonzero. When
1463:ramification defect
1434:decomposition group
1428:decomposition group
1420:A module is called
1118:complete local ring
760:with residue field
606:analytically normal
57:Contents:
8390:10.1007/bf02732123
8348:10.1007/bf02684343
8306:10.1007/bf02684322
8264:10.1007/bf02684747
8222:10.1007/bf02684890
8180:10.1007/bf02684274
8138:10.1007/bf02699291
8096:10.1007/bf02684778
7959:General references
7856:6. The
7825:spectrum of a ring
7819:3. The
7769:
7749:
7618:
7598:
7403:
7255:torsionless module
7246:5. The
7108:that generates an
6939:Richard P. Stanley
6858:2. The
6851:1. The
6814:3. The
6758:
6725:
6688:
6351:regular local ring
6341:is an isomorphism.
6331:
6263:Rees decomposition
6249:
6172:
6157:2. The
5999:ramification group
5993:ramification group
5951:4. The
5933:2. The
5926:1. The
5797:
5763:purely inseparable
5697:is a submodule of
5662:is a submodule of
5604:
5581:
5557:
5500:3. The
5375:is a proper ideal
4677:3. The
4642:2. The
4574:composition series
4570:length of a module
4421:1. The
4401:Jacobian criterion
4399:3. The
4384:2. The
4377:1. The
4250:4. The
4194:injective envelope
4177:of the monomials).
4161:is the set of all
4108:finitely generated
3968:
3932:
3900:
3872:
3835:is a proper ideal
3827:over a local ring
3741:7. The
3734:6. The
3727:5. The
3708:Hilbert polynomial
3681:3. The
3604:1. The
3350:2. The
3219:going down theorem
3217:2. The
2723:finitely generated
2716:finitely generated
2710:finitely generated
2678:field of fractions
2676:2. The
2528:
2447:
2369:{\displaystyle xR}
2366:
2346:{\displaystyle yR}
2343:
2320:
2256:exact zero divisor
2244:
2230:exact zero divisor
2224:Euclid's algorithm
2132:equicharacteristic
2126:injective envelope
2089:3. The
2051:
2031:
2011:
1739:9. The
1679:1. The
1657:determinantal ring
1645:determinant module
1639:determinant module
1579:is the local ring
685:5. The
564:1. The
501:over another ring
441:Absolutely regular
8406:Nagata, Masayoshi
8054:978-0-387-94268-1
8011:978-0-521-41068-7
7987:978-3-540-64239-8
7966:Bourbaki, Nicolas
7947:978-0-8053-7026-3
7772:{\displaystyle i}
7621:{\displaystyle m}
7477:1. An
7343:Abbreviation for
7056:of a prime ideal
6841:socle of a module
6646:is a commutative
6374:is an element of
6366:4. An
6093:in each variable.
6017:for some integer
5987:radical extension
5914:is regular. (cf.
5838:quasi-isomorphism
5626:pseudo-surjective
5488:projective module
5435:or some power of
5399:or some power of
5293:Abbreviation for
5049:Noetherian module
4917:as a module over
4888:at a prime ideal
4594:linearly disjoint
4580:linearly disjoint
4525:A Krull ring (or
4503:Kleinian integers
4290:2. An
4265:integrally closed
4221:1. An
4203:3. An
4192:2. An
4185:1. An
4163:leading monomials
4159:monomial ordering
4050:lying over prime
3917:
3875:{\displaystyle M}
3819:1. An
3593:Abbreviation for
3569:Grothendieck ring
3372:Daniel Gorenstein
3255:going up property
3036:Gaussian integers
3020:Grothendieck ring
3002:fractional ideals
2996:fractionary ideal
2580:3. An
2570:1. An
2254:is said to be an
2247:{\displaystyle x}
2206:2. An
2163:2. An
2082:2. An
2054:{\displaystyle B}
2034:{\displaystyle A}
1972:Abbreviation for
1962:dualizing complex
987:Abbreviation for
899:A ring is called
780:-dimension of Ext
721:is a prime ideal
678:4. An
671:3. An
585:3. An
450:6. An
434:Absolutely normal
407:1. An
50:with identity 1.
8448:
8422:
8401:
8359:
8317:
8275:
8233:
8191:
8149:
8107:
8065:
8022:
7990:
7952:
7951:
7933:
7927:
7926:
7911:
7821:Zariski topology
7812:2. A
7789:Weierstrass ring
7783:Weierstrass ring
7778:
7776:
7775:
7770:
7758:
7756:
7755:
7750:
7745:
7744:
7720:
7719:
7701:
7700:
7685:
7684:
7660:
7659:
7627:
7625:
7624:
7619:
7607:
7605:
7604:
7599:
7594:
7593:
7575:
7574:
7534:3. A
7511:2. A
7500:1. A
7412:
7410:
7409:
7404:
7402:
7401:
7383:
7382:
7319:
7318:
7253:6. A
7248:torsion functors
7236:3. A
7226:1. A
7216:Torsion functors
7002:(for submodules
6950:3. A
6943:2. A
6860:maximal spectrum
6767:
6765:
6764:
6759:
6751:
6750:
6734:
6732:
6731:
6726:
6718:
6717:
6697:
6695:
6694:
6689:
6687:
6686:
6674:
6673:
6623:1. A
6387:regular sequence
6385:5. A
6356:2. A
6349:1. A
6340:
6338:
6337:
6332:
6306:
6305:
6261:3. A
6258:
6256:
6255:
6250:
6212:
6211:
6202:
6201:
6191:
6186:
6078:1. A
5985:6. A
5928:Jacobson radical
5867:2. A
5845:Quasi-local ring
5836:2. A
5829:1. A
5806:
5804:
5803:
5798:
5796:
5795:
5794:
5793:
5670:for all modules
5643:1. A
5619:pseudo-injective
5613:
5611:
5610:
5605:
5590:
5588:
5587:
5582:
5580:
5579:
5566:
5564:
5563:
5558:
5550:
5549:
5548:
5493:2. A
5486:1. A
5474:3. A
5467:2. A
5460:1. A
5446:2. A
5369:1. A
5356:prime local ring
5354:3. A
5347:2. A
5340:1. A
5307:place of a field
5054:2. A
5047:1. A
4978:Nakayama's lemma
4973:Nakayama's lemma
4797:1. A
4774:1. A
4762:2. A
4739:Macaulay duality
4723:1. A
4644:local cohomology
4635:1. A
4497:Kleinian integer
4453:finite extension
4428:2. A
4423:Jacobson radical
4312:irrelevant ideal
4292:irreducible ring
4252:integral closure
4187:injective module
4168:
4156:
4133:
4129:
3982:is a Noetherian
3977:
3975:
3974:
3969:
3952:
3951:
3942:
3941:
3931:
3909:
3907:
3906:
3901:
3899:
3898:
3881:
3879:
3878:
3873:
3846:is contained in
3785:2. A
3759:2. A
3755:Heisuke Hironaka
3706:2. A
3674:2. A
3567:1. A
3513:) of the module
3486:is the grade of
3466:
3465:
3383:3. A
3376:2. A
3352:going up theorem
3233:is a domain and
3059:2. A
2966:fractional ideal
2952:fractional ideal
2838:5. A
2783:2. A
2776:1. A
2657:2. A
2646:1. A
2634:1. A
2537:
2535:
2534:
2529:
2473:
2472:
2456:
2454:
2453:
2448:
2392:
2391:
2375:
2373:
2372:
2367:
2352:
2350:
2349:
2344:
2329:
2327:
2326:
2321:
2274:
2273:
2253:
2251:
2250:
2245:
2220:Euclidean domain
2214:Euclidean domain
2060:
2058:
2057:
2052:
2040:
2038:
2037:
2032:
2020:
2018:
2017:
2012:
1960:5. A
1940:
1926:
1925:
1892:dualizing module
1890:4. A
1885:Macaulay duality
1809:divisorial ideal
1803:divisorial ideal
1789:2. A
1776:divisible module
1727:global dimension
1554:
1553:
1356:coprimary module
1256:congruence ideal
1250:congruence ideal
1116:2. A
1109:1. A
1070:2. A
1051:2. A
1027:coefficient ring
889:dualizing module
790:
789:
711:associated prime
687:Artin-Rees lemma
467:Acceptable rings
420:absolutely prime
58:
8456:
8455:
8451:
8450:
8449:
8447:
8446:
8445:
8426:
8425:
8420:
8404:
8368:Dieudonné, Jean
8362:
8326:Dieudonné, Jean
8320:
8284:Dieudonné, Jean
8278:
8242:Dieudonné, Jean
8236:
8200:Dieudonné, Jean
8194:
8158:Dieudonné, Jean
8152:
8116:Dieudonné, Jean
8110:
8074:Dieudonné, Jean
8068:
8055:
8037:Springer-Verlag
8027:Eisenbud, David
8025:
8012:
7993:
7988:
7978:Springer-Verlag
7964:
7961:
7956:
7955:
7948:
7935:
7934:
7930:
7913:
7912:
7908:
7903:
7891:
7877:
7832:Zariski's lemma
7803:
7798:
7784:
7761:
7760:
7730:
7711:
7692:
7670:
7651:
7634:
7633:
7610:
7609:
7585:
7566:
7558:
7557:
7550:
7545:
7536:valuation group
7497:
7492:
7474:
7450:
7442:
7430:
7418:
7393:
7374:
7369:
7368:
7364:
7352:
7340:
7335:
7317:
7312:
7311:
7310:
7286:
7278:
7262:
7228:torsion element
7223:
7211:
7152:
7140:
7135:
7125:
7093:
7046:
7019:
7011:
6967:
6959:
6934:
6902:
6892:
6883:
6867:
6848:
6836:
6828:smooth morphism
6823:
6805:
6793:
6781:
6773:
6742:
6737:
6736:
6709:
6704:
6703:
6678:
6665:
6660:
6659:
6644:seminormal ring
6639:
6620:
6615:
6607:
6567:
6549:
6538:
6537:
6526:
6512:
6504:
6493:
6486:absolutely flat
6456:
6447:
6440:
6429:
6420:such that each
6415:
6406:
6399:
6346:
6294:
6283:
6282:
6270:
6203:
6193:
6167:
6166:
6148:
6110:
6098:
6075:
6030:
5994:
5923:
5901:
5890:
5889:
5878:
5861:
5826:
5821:
5785:
5780:
5775:
5774:
5754:
5706:
5640:
5593:
5592:
5569:
5568:
5539:
5534:
5533:
5521:
5509:
5483:
5462:principal ideal
5457:
5411:is a submodule
5366:
5337:
5329:
5302:
5290:
5270:
5265:
5228:
5216:
5180:
5175:
5157:
5152:
5131:
5130:Nullstellensatz
5112:-module ⊕
5089:
5070:
5056:Noetherian ring
5038:
5033:
5021:
5009:
5001:
4974:
4962:
4950:
4938:
4933:
4925:
4916:
4881:
4873:
4859:
4851:
4843:
4831:
4815:
4794:
4771:
4746:
4714:
4709:
4699:
4661:
4655:
4632:
4627:
4622:
4614:
4609:
4581:
4565:
4553:
4548:
4534:
4533:Krull dimension
4522:
4510:
4498:
4486:
4481:
4446:integral domain
4437:
4417:Nathan Jacobson
4412:
4379:Jacobian matrix
4374:
4362:
4350:
4338:
4333:
4319:
4307:
4280:
4272:
4223:integral domain
4215:
4182:
4166:
4154:
4131:
4127:
4115:
4103:
4091:
4079:
4023:
4003:
3933:
3912:
3911:
3888:
3887:
3864:
3863:
3859:
3813:
3808:
3794:
3776:
3750:
3690:
3639:states that if
3626:
3621:
3616:
3601:
3590:
3585:
3558:
3546:
3534:
3464:
3459:
3458:
3457:
3392:
3385:Gorenstein ring
3367:
3346:
3335:
3328:
3321:
3310:
3303:
3292:
3281:
3270:
3263:
3242:
3213:
3202:
3195:
3188:
3177:
3170:
3159:
3148:
3137:
3130:
3109:
3092:
3068:
3050:
3034:is the ring of
3027:
3015:
3010:
2997:
2953:
2939:equidimensional
2917:
2899:
2872:formally smooth
2855:
2821:-module ⊕
2809:along an ideal
2785:flat resolution
2773:
2747:
2733:
2711:
2703:
2691:
2670:
2643:
2636:faithful module
2631:
2620:
2608:
2603:
2567:
2555:
2543:
2464:
2459:
2458:
2383:
2378:
2377:
2355:
2354:
2332:
2331:
2265:
2260:
2259:
2236:
2235:
2234:A zero divisor
2231:
2215:
2196:
2188:
2141:
2133:
2121:
2109:
2101:
2066:
2043:
2042:
2023:
2022:
1997:
1996:
1988:
1983:
1969:
1924:
1919:
1918:
1917:
1915:
1866:
1861:
1856:
1839:if it contains
1828:
1821:integral domain
1816:
1804:
1798:Cartier divisor
1783:
1771:
1759:
1681:Krull dimension
1676:
1664:
1652:
1640:
1628:
1592:
1552:
1547:
1546:
1545:
1517:
1458:
1453:
1446:Dedekind domain
1441:
1440:Dedekind domain
1429:
1417:
1412:
1398:
1379:cotangent space
1374:
1363:
1351:
1339:
1331:
1319:
1299:
1283:
1251:
1215:
1188:
1157:
1125:
1106:
1087:
1060:
1042:
1028:
1016:
1008:
996:
984:
940:
924:
916:
908:
896:
884:
879:
865:
849:
825:
816:
804:
788:
783:
782:
781:
749:
744:
706:
694:
673:Artinian module
656:
651:
639:
594:
576:
561:
522:
514:
497:An affine ring
494:
478:
471:excellent rings
463:
462:acceptable ring
401:
389:
384:
364:
341:
322:
311:polynomial ring
303:
267:3. (
244:2. (
227:field extension
210:
206:
201:
200:
199:
198:
59:
56:
12:
11:
5:
8454:
8452:
8444:
8443:
8438:
8428:
8427:
8424:
8423:
8419:978-0470628652
8418:
8402:
8360:
8318:
8276:
8234:
8192:
8150:
8108:
8066:
8053:
8023:
8010:
7991:
7986:
7960:
7957:
7954:
7953:
7946:
7928:
7905:
7904:
7902:
7899:
7898:
7897:
7890:
7887:
7886:
7885:
7878:
7875:
7873:
7854:
7837:5.
7835:
7830:4.
7828:
7817:
7810:
7804:
7801:
7797:
7794:
7793:
7792:
7785:
7782:
7780:
7768:
7748:
7743:
7740:
7737:
7733:
7729:
7726:
7723:
7718:
7714:
7710:
7707:
7704:
7699:
7695:
7691:
7688:
7683:
7680:
7677:
7673:
7669:
7666:
7663:
7658:
7654:
7650:
7647:
7644:
7641:
7617:
7597:
7592:
7588:
7584:
7581:
7578:
7573:
7569:
7565:
7554:
7551:
7548:
7544:
7541:
7540:
7539:
7532:
7513:valuation ring
7509:
7498:
7495:
7491:
7488:
7487:
7486:
7483:
7475:
7472:
7470:
7451:
7448:
7446:
7443:
7440:
7438:
7431:
7428:
7426:
7419:
7416:
7414:
7400:
7396:
7392:
7389:
7386:
7381:
7377:
7365:
7363:unimodular row
7362:
7360:
7353:
7350:
7348:
7341:
7338:
7334:
7331:
7330:
7329:
7313:
7287:
7284:
7282:
7279:
7276:
7274:
7263:
7260:
7258:
7251:
7244:
7241:
7234:
7231:
7224:
7221:
7219:
7212:
7209:
7207:
7162:* of an ideal
7153:
7150:
7148:
7141:
7138:
7134:
7131:
7130:
7129:
7126:
7123:
7121:
7094:
7091:
7089:
7051:symbolic power
7047:
7045:symbolic power
7044:
7042:
7020:
7017:
7015:
7012:
7009:
7007:
6968:
6965:
6963:
6960:
6958:strictly local
6957:
6955:
6948:
6941:
6935:
6932:
6930:
6903:
6900:
6898:
6888:
6878:
6868:
6865:
6863:
6856:
6853:prime spectrum
6849:
6846:
6844:
6837:
6834:
6832:
6824:
6821:
6819:
6812:
6809:
6806:
6803:
6801:
6794:
6791:
6789:
6782:
6779:
6777:
6774:
6771:
6769:
6757:
6754:
6749:
6745:
6724:
6721:
6716:
6712:
6685:
6681:
6677:
6672:
6668:
6640:
6637:
6635:
6628:
6625:semilocal ring
6621:
6618:
6616:
6613:
6611:
6608:
6605:
6603:
6568:
6565:
6563:
6545:
6541:The condition
6539:
6533:
6529:
6525:
6522:
6521:
6520:
6513:
6510:
6508:
6505:
6502:
6500:
6494:
6491:
6489:
6462:
6452:
6445:
6438:
6424:
6411:
6404:
6397:
6383:
6364:
6361:
6354:
6347:
6344:
6342:
6330:
6327:
6324:
6321:
6318:
6315:
6312:
6309:
6304:
6301:
6297:
6293:
6290:
6271:
6268:
6266:
6259:
6248:
6245:
6242:
6239:
6236:
6233:
6230:
6227:
6224:
6221:
6218:
6215:
6210:
6206:
6200:
6196:
6190:
6185:
6182:
6179:
6175:
6155:
6149:
6146:
6144:
6111:
6108:
6106:
6099:
6096:
6094:
6083:
6076:
6073:
6071:
6044:
6041:
6037:
6034:
6031:
6028:
6026:
5995:
5992:
5990:
5983:
5956:
5949:
5938:
5931:
5924:
5921:
5919:
5897:
5893:The condition
5891:
5885:
5881:
5877:
5874:
5873:
5872:
5869:quotient field
5865:
5862:
5859:
5857:
5850:4.
5848:
5843:3.
5841:
5834:
5827:
5824:
5820:
5817:
5816:
5815:
5812:
5792:
5788:
5783:
5755:
5752:
5750:
5719:
5702:
5675:
5645:pure submodule
5641:
5638:
5636:
5629:
5622:
5615:
5603:
5600:
5578:
5556:
5553:
5547:
5542:
5522:
5519:
5517:
5510:
5507:
5505:
5498:
5491:
5484:
5481:
5479:
5472:
5465:
5458:
5455:
5453:
5444:
5367:
5364:
5362:
5359:
5352:
5345:
5338:
5335:
5333:
5330:
5327:
5325:
5316:is a map from
5303:
5300:
5298:
5291:
5288:
5286:
5271:
5268:
5266:
5263:
5261:
5258:
5239:
5236:
5229:
5226:
5224:
5217:
5214:
5212:
5203:) â
5181:
5178:
5174:
5171:
5170:
5169:
5158:
5155:
5151:
5148:
5147:
5146:
5135:
5132:
5129:
5127:
5090:
5087:
5085:
5078:
5071:
5068:
5066:
5061:3.
5059:
5052:
5045:
5039:
5036:
5034:
5031:
5029:
5022:
5019:
5017:
5010:
5007:
5005:
5002:
4999:
4997:
4975:
4972:
4970:
4963:
4960:
4958:
4951:
4948:
4946:
4939:
4936:
4932:
4929:
4928:
4927:
4921:
4912:
4882:
4879:
4877:
4874:
4869:
4867:
4860:
4857:
4855:
4852:
4849:
4847:
4844:
4842:modular system
4841:
4839:
4832:
4829:
4827:
4824:Cohen-Macaulay
4816:
4813:
4811:
4808:
4805:
4802:
4795:
4792:
4790:
4779:
4772:
4769:
4767:
4760:
4757:Matlis duality
4755:1.
4753:
4747:
4744:
4742:
4737:3.
4735:
4728:
4721:
4715:
4712:
4708:
4705:
4704:
4703:
4700:
4697:
4695:
4688:
4685:
4682:
4675:
4657:
4651:
4640:
4633:
4630:
4628:
4625:
4623:
4620:
4618:
4615:
4612:
4610:
4607:
4605:
4582:
4579:
4577:
4566:
4563:
4561:
4558:Laskerian ring
4554:
4552:Laskerian ring
4551:
4547:
4544:
4543:
4542:
4535:
4532:
4530:
4523:
4520:
4518:
4515:Koszul complex
4511:
4509:Koszul complex
4508:
4506:
4499:
4496:
4494:
4489:The module of
4487:
4484:
4480:
4477:
4476:
4475:
4438:
4435:
4433:
4426:
4419:
4413:
4410:
4408:
4397:
4386:Jacobian ideal
4382:
4375:
4372:
4370:
4363:
4360:
4358:
4351:
4348:
4346:
4339:
4336:
4332:
4329:
4328:
4327:
4324:isolated prime
4320:
4317:
4315:
4308:
4305:
4303:
4296:
4288:
4281:
4278:
4276:
4273:
4270:
4268:
4261:
4258:
4255:
4248:
4229:
4226:
4219:
4216:
4213:
4211:
4208:
4201:
4198:injective hull
4190:
4183:
4180:
4178:
4143:
4116:
4113:
4111:
4104:
4101:
4099:
4092:
4089:
4087:
4084:indecomposable
4080:
4078:indecomposable
4077:
4075:
4026:The extension
4024:
4021:
4019:
4004:
4001:
3999:
3967:
3964:
3961:
3958:
3955:
3950:
3945:
3940:
3936:
3930:
3927:
3924:
3920:
3897:
3871:
3860:
3857:
3855:
3817:
3814:
3811:
3807:
3804:
3803:
3802:
3799:injective hull
3795:
3792:
3790:
3783:
3781:W. V. D. Hodge
3777:
3774:
3771:
3766:3.
3764:
3757:
3751:
3748:
3746:
3739:
3732:
3725:
3720:4.
3718:
3713:3.
3711:
3704:
3699:1.
3697:
3691:
3688:
3686:
3679:
3676:Henselian ring
3672:
3637:Hensel's lemma
3635:1.
3633:
3627:
3624:
3622:
3619:
3617:
3614:
3612:
3609:
3602:
3599:
3597:
3591:
3588:
3584:
3581:
3580:
3579:
3574:2.
3572:
3565:
3559:
3556:
3554:
3547:
3544:
3542:
3539:graded algebra
3535:
3532:
3530:
3499:
3476:
3475:) is non-zero.
3460:
3436:) of a module
3430:
3406:) of an ideal
3396:
3393:
3390:
3388:
3381:
3374:
3370:1.
3368:
3365:
3363:
3348:
3344:
3333:
3326:
3319:
3308:
3301:
3290:
3279:
3268:
3261:
3243:
3240:
3238:
3215:
3211:
3200:
3193:
3186:
3175:
3168:
3157:
3146:
3135:
3128:
3110:
3107:
3105:
3088:
3069:
3066:
3064:
3057:
3051:
3048:
3046:
3028:
3025:
3023:
3016:
3013:
3009:
3006:
3005:
3004:
2998:
2995:
2993:
2954:
2951:
2949:
2942:
2935:
2913:
2895:
2880:formally etale
2856:
2853:
2851:
2836:
2795:
2788:
2781:
2774:
2771:
2769:
2752:) of a module
2743:
2734:
2731:
2729:
2726:
2719:
2712:
2709:
2707:
2704:
2701:
2699:
2692:
2689:
2687:
2684:
2681:
2674:
2671:
2668:
2666:
2655:
2644:
2641:
2639:
2632:
2629:
2627:
2624:Factorial ring
2621:
2618:
2616:
2609:
2606:
2602:
2599:
2598:
2597:
2578:
2575:
2568:
2565:
2563:
2556:
2553:
2551:
2548:excellent ring
2544:
2541:
2539:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2471:
2467:
2446:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2390:
2386:
2365:
2362:
2342:
2339:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2272:
2268:
2243:
2232:
2229:
2227:
2216:
2213:
2211:
2204:
2197:
2194:
2192:
2189:
2186:
2184:
2161:
2142:
2139:
2137:
2134:
2131:
2129:
2122:
2119:
2117:
2110:
2107:
2105:
2102:
2099:
2097:
2094:
2087:
2080:
2073:
2067:
2064:
2062:
2050:
2030:
2010:
2007:
2004:
1989:
1986:
1982:
1979:
1978:
1977:
1970:
1967:
1965:
1958:
1920:
1914:vector space
1888:
1883:3.
1881:
1878:Matlis duality
1876:2.
1874:
1869:1.
1867:
1864:
1862:
1859:
1857:
1854:
1852:
1829:
1826:
1824:
1817:
1814:
1812:
1805:
1802:
1800:
1796:3.
1794:
1787:
1784:
1781:
1779:
1772:
1769:
1767:
1760:
1757:
1755:
1752:
1737:
1730:
1723:
1716:
1709:
1702:
1695:
1692:flat dimension
1688:weak dimension
1684:
1677:
1674:
1672:
1665:
1662:
1660:
1653:
1650:
1648:
1641:
1638:
1636:
1629:
1626:
1624:
1593:
1590:
1588:
1548:
1528:) of a module
1518:
1515:
1513:
1459:
1456:
1454:
1451:
1449:
1442:
1439:
1437:
1430:
1427:
1425:
1418:
1415:
1411:
1408:
1407:
1406:
1399:
1396:
1394:
1375:
1372:
1370:
1367:
1364:
1361:
1359:
1352:
1349:
1347:
1340:
1337:
1335:
1332:
1329:
1327:
1320:
1317:
1315:
1306:is the module
1300:
1297:
1295:
1284:
1281:
1279:
1252:
1249:
1247:
1216:
1213:
1211:
1192:
1189:
1186:
1184:
1158:
1155:
1153:
1126:
1123:
1121:
1114:
1107:
1104:
1102:
1095:
1088:
1085:
1083:
1068:
1065:CohenâMacaulay
1061:
1059:CohenâMacaulay
1058:
1056:
1049:
1045:1.
1043:
1040:
1038:
1035:
1032:
1029:
1026:
1024:
1017:
1014:
1012:
1009:
1006:
1004:
997:
994:
992:
989:CohenâMacaulay
985:
982:
980:
976:
941:
938:
936:
925:
923:characteristic
922:
920:
917:
914:
912:
909:
906:
904:
897:
894:
892:
885:
882:
878:
875:
874:
873:
870:Buchsbaum ring
866:
864:Buchsbaum ring
863:
861:
850:
848:Bourbaki ideal
847:
845:
826:
823:
821:
817:
814:
812:
805:
802:
800:
784:
750:
747:
743:
740:
739:
738:
707:
704:
702:
695:
692:
690:
683:
676:
669:
665:2.
663:
659:1.
657:
654:
652:
649:
647:
640:
637:
635:
628:
619:
610:
601:
595:
592:
590:
583:
580:
577:
574:
572:
569:
562:
559:
557:
542:
523:
520:
518:
515:
512:
510:
495:
492:
490:
479:
476:
474:
464:
461:
459:
454:is one with a
448:
439:5.
437:
432:4.
430:
423:
416:
405:
402:
399:
397:
390:
387:
383:
380:
379:
365:
362:
360:
342:
339:
337:
323:
320:
318:
304:
302:
300:
277:ideal quotient
265:
242:
213:1.
211:
208:
205:
202:
197:
196:
191:
186:
181:
176:
171:
166:
161:
156:
151:
146:
141:
136:
131:
126:
121:
116:
111:
106:
101:
96:
91:
86:
81:
76:
71:
66:
60:
55:
54:
52:
13:
10:
9:
6:
4:
3:
2:
8453:
8442:
8439:
8437:
8434:
8433:
8431:
8421:
8415:
8411:
8407:
8403:
8399:
8395:
8391:
8387:
8383:
8379:
8378:
8373:
8369:
8365:
8361:
8357:
8353:
8349:
8345:
8341:
8337:
8336:
8331:
8327:
8323:
8319:
8315:
8311:
8307:
8303:
8299:
8295:
8294:
8289:
8285:
8281:
8277:
8273:
8269:
8265:
8261:
8257:
8253:
8252:
8247:
8243:
8239:
8235:
8231:
8227:
8223:
8219:
8215:
8211:
8210:
8205:
8201:
8197:
8193:
8189:
8185:
8181:
8177:
8173:
8169:
8168:
8163:
8159:
8155:
8151:
8147:
8143:
8139:
8135:
8131:
8127:
8126:
8121:
8117:
8113:
8109:
8105:
8101:
8097:
8093:
8089:
8085:
8084:
8079:
8075:
8071:
8067:
8064:
8060:
8056:
8050:
8046:
8042:
8038:
8034:
8033:
8028:
8024:
8021:
8017:
8013:
8007:
8003:
7999:
7998:
7992:
7989:
7983:
7979:
7975:
7971:
7967:
7963:
7962:
7958:
7949:
7943:
7939:
7932:
7929:
7925:
7921:
7917:
7910:
7907:
7900:
7896:
7893:
7892:
7888:
7883:
7879:
7874:
7871:
7867:
7863:
7859:
7855:
7852:
7848:
7844:
7840:
7836:
7833:
7829:
7826:
7822:
7818:
7815:
7811:
7809:
7808:Oscar Zariski
7805:
7800:
7795:
7790:
7786:
7781:
7766:
7741:
7738:
7735:
7731:
7727:
7724:
7721:
7716:
7712:
7705:
7697:
7693:
7689:
7681:
7678:
7675:
7671:
7667:
7664:
7661:
7656:
7652:
7642:
7639:
7631:
7630:weak sequence
7615:
7590:
7586:
7582:
7579:
7576:
7571:
7567:
7555:
7552:
7547:
7542:
7537:
7533:
7530:
7526:
7522:
7519:such that if
7518:
7514:
7510:
7507:
7503:
7499:
7494:
7489:
7484:
7480:
7476:
7471:
7468:
7464:
7460:
7456:
7452:
7447:
7444:
7439:
7436:
7432:
7427:
7424:
7420:
7415:
7398:
7394:
7390:
7387:
7384:
7379:
7375:
7366:
7361:
7358:
7354:
7349:
7346:
7342:
7337:
7332:
7327:
7323:
7316:
7308:
7304:
7300:
7296:
7292:
7288:
7283:
7280:
7275:
7272:
7268:
7264:
7259:
7256:
7252:
7249:
7245:
7242:
7239:
7235:
7232:
7229:
7225:
7220:
7217:
7213:
7208:
7205:
7201:
7197:
7193:
7189:
7185:
7181:
7177:
7173:
7169:
7165:
7161:
7158:
7157:tight closure
7154:
7151:tight closure
7149:
7146:
7142:
7137:
7132:
7127:
7122:
7119:
7115:
7111:
7107:
7103:
7099:
7096:A set of dim
7095:
7090:
7087:
7083:
7079:
7075:
7071:
7067:
7063:
7059:
7055:
7052:
7048:
7043:
7040:
7036:
7032:
7028:
7025:
7021:
7016:
7013:
7008:
7005:
7001:
6997:
6993:
6989:
6985:
6981:
6977:
6973:
6969:
6964:
6961:
6956:
6953:
6949:
6946:
6942:
6940:
6936:
6931:
6928:
6924:
6920:
6916:
6912:
6908:
6904:
6899:
6896:
6891:
6887:
6881:
6877:
6873:
6869:
6864:
6861:
6857:
6854:
6850:
6845:
6842:
6838:
6833:
6829:
6825:
6820:
6817:
6813:
6810:
6807:
6802:
6799:
6795:
6790:
6787:
6783:
6778:
6775:
6770:
6755:
6752:
6747:
6743:
6722:
6719:
6714:
6710:
6701:
6683:
6679:
6675:
6670:
6666:
6657:
6653:
6649:
6645:
6641:
6636:
6633:
6629:
6626:
6622:
6617:
6612:
6609:
6604:
6601:
6597:
6594:implies that
6593:
6589:
6585:
6581:
6577:
6573:
6569:
6564:
6561:
6557:
6553:
6548:
6544:
6540:
6536:
6532:
6528:
6523:
6518:
6514:
6509:
6506:
6503:residue field
6501:
6498:
6495:
6490:
6487:
6483:
6479:
6475:
6471:
6467:
6463:
6460:
6455:
6451:
6444:
6437:
6433:
6427:
6423:
6419:
6414:
6410:
6403:
6396:
6392:
6388:
6384:
6381:
6377:
6373:
6369:
6365:
6362:
6359:
6355:
6352:
6348:
6343:
6325:
6322:
6319:
6310:
6307:
6302:
6299:
6295:
6288:
6280:
6276:
6272:
6267:
6264:
6260:
6246:
6240:
6234:
6231:
6225:
6222:
6216:
6213:
6208:
6204:
6198:
6194:
6183:
6180:
6177:
6173:
6164:
6160:
6156:
6154:
6150:
6145:
6142:
6138:
6135:
6131:
6128:
6124:
6120:
6116:
6112:
6107:
6104:
6100:
6095:
6092:
6088:
6084:
6081:
6077:
6072:
6069:
6065:
6061:
6057:
6053:
6049:
6045:
6042:
6038:
6035:
6032:
6027:
6024:
6021:>1. (When
6020:
6016:
6012:
6008:
6004:
6000:
5996:
5991:
5988:
5984:
5981:
5977:
5973:
5969:
5965:
5961:
5957:
5954:
5950:
5947:
5943:
5939:
5936:
5932:
5929:
5925:
5920:
5917:
5913:
5909:
5905:
5900:
5896:
5892:
5888:
5884:
5880:
5875:
5870:
5866:
5863:
5858:
5855:
5854:
5853:quasi-unmixed
5849:
5846:
5842:
5839:
5835:
5832:
5828:
5823:
5818:
5813:
5810:
5790:
5786:
5781:
5772:
5768:
5764:
5760:
5756:
5751:
5748:
5744:
5740:
5736:
5732:
5728:
5724:
5720:
5717:
5713:
5709:
5705:
5700:
5696:
5692:
5688:
5684:
5680:
5676:
5673:
5669:
5665:
5661:
5657:
5653:
5649:
5646:
5642:
5637:
5634:
5630:
5627:
5623:
5620:
5616:
5601:
5598:
5554:
5551:
5540:
5531:
5527:
5523:
5518:
5515:
5514:PrĂŒfer domain
5511:
5508:PrĂŒfer domain
5506:
5503:
5499:
5496:
5492:
5489:
5485:
5480:
5477:
5473:
5470:
5466:
5463:
5459:
5454:
5451:
5450:
5445:
5442:
5438:
5434:
5430:
5426:
5422:
5419:such that if
5418:
5414:
5410:
5406:
5402:
5398:
5394:
5390:
5386:
5383:such that if
5382:
5378:
5374:
5373:
5372:primary ideal
5368:
5363:
5360:
5357:
5353:
5350:
5349:prime element
5346:
5343:
5339:
5334:
5331:
5326:
5323:
5319:
5315:
5311:
5308:
5304:
5299:
5296:
5292:
5287:
5284:
5280:
5276:
5272:
5267:
5262:
5259:
5256:
5252:
5248:
5244:
5240:
5237:
5234:
5230:
5225:
5222:
5218:
5213:
5210:
5206:
5202:
5198:
5194:
5190:
5189:parafactorial
5186:
5182:
5179:parafactorial
5177:
5172:
5167:
5163:
5159:
5154:
5149:
5144:
5140:
5136:
5133:
5128:
5125:
5122:
5118:
5115:
5111:
5107:
5103:
5099:
5095:
5091:
5088:normally flat
5086:
5083:
5079:
5076:
5075:normal domain
5072:
5067:
5064:
5060:
5057:
5053:
5050:
5046:
5044:
5040:
5035:
5030:
5027:
5023:
5018:
5015:
5011:
5006:
5003:
4998:
4995:
4991:
4987:
4983:
4979:
4976:
4971:
4968:
4964:
4959:
4956:
4952:
4947:
4944:
4940:
4935:
4930:
4924:
4920:
4915:
4911:
4907:
4903:
4899:
4895:
4891:
4887:
4883:
4878:
4875:
4872:
4868:
4865:
4861:
4856:
4853:
4848:
4845:
4840:
4837:
4833:
4828:
4825:
4821:
4817:
4812:
4809:
4806:
4803:
4800:
4799:minimal prime
4796:
4791:
4788:
4784:
4780:
4777:
4776:maximal ideal
4773:
4768:
4765:
4764:Matlis module
4761:
4758:
4754:
4752:
4748:
4743:
4740:
4736:
4733:
4729:
4726:
4725:Macaulay ring
4722:
4720:
4716:
4711:
4706:
4701:
4696:
4693:
4689:
4686:
4683:
4680:
4676:
4673:
4669:
4665:
4660:
4654:
4649:
4645:
4641:
4638:
4634:
4629:
4624:
4619:
4616:
4611:
4606:
4603:
4599:
4595:
4591:
4588:over a field
4587:
4583:
4578:
4575:
4571:
4567:
4562:
4559:
4555:
4550:
4545:
4540:
4536:
4531:
4528:
4524:
4519:
4516:
4512:
4507:
4504:
4500:
4495:
4492:
4488:
4483:
4478:
4473:
4469:
4465:
4461:
4457:
4454:
4450:
4447:
4443:
4442:Japanese ring
4439:
4436:Japanese ring
4434:
4431:
4430:Jacobson ring
4427:
4424:
4420:
4418:
4414:
4409:
4406:
4402:
4398:
4395:
4391:
4387:
4383:
4380:
4376:
4371:
4368:
4364:
4359:
4356:
4352:
4347:
4344:
4340:
4335:
4330:
4325:
4321:
4316:
4313:
4309:
4304:
4301:
4297:
4293:
4289:
4286:
4282:
4277:
4274:
4269:
4266:
4262:
4259:
4256:
4253:
4249:
4246:
4242:
4238:
4234:
4230:
4227:
4224:
4220:
4217:
4212:
4209:
4206:
4202:
4199:
4195:
4191:
4188:
4184:
4179:
4176:
4172:
4164:
4160:
4152:
4151:initial ideal
4148:
4147:Gröbner bases
4144:
4141:
4137:
4125:
4124:initial ideal
4121:
4117:
4114:initial ideal
4112:
4109:
4105:
4100:
4097:
4096:inertia group
4093:
4090:inertia group
4088:
4085:
4081:
4076:
4073:
4069:
4065:
4061:
4057:
4053:
4049:
4045:
4041:
4037:
4033:
4029:
4025:
4020:
4017:
4013:
4009:
4005:
4000:
3997:
3993:
3992:maximal ideal
3989:
3985:
3981:
3965:
3962:
3956:
3953:
3938:
3934:
3928:
3925:
3922:
3918:
3885:
3869:
3861:
3856:
3853:
3849:
3845:
3842:
3838:
3834:
3830:
3826:
3822:
3818:
3815:
3810:
3805:
3800:
3796:
3791:
3788:
3787:Hodge algebra
3784:
3782:
3778:
3773:
3769:
3765:
3762:
3758:
3756:
3752:
3747:
3744:
3740:
3737:
3733:
3730:
3726:
3723:
3719:
3716:
3712:
3709:
3705:
3702:
3698:
3696:
3695:David Hilbert
3692:
3687:
3684:
3683:Henselization
3680:
3677:
3673:
3670:
3666:
3662:
3658:
3654:
3650:
3646:
3642:
3638:
3634:
3632:
3628:
3625:Henselization
3623:
3618:
3613:
3610:
3607:
3603:
3598:
3596:
3592:
3587:
3582:
3577:
3573:
3570:
3566:
3564:
3560:
3555:
3552:
3551:Gröbner basis
3548:
3545:Gröbner basis
3543:
3540:
3536:
3531:
3528:
3524:
3520:
3516:
3512:
3508:
3504:
3500:
3497:
3493:
3489:
3485:
3481:
3477:
3474:
3470:
3463:
3456:such that Ext
3455:
3451:
3447:
3444:is grade(Ann
3443:
3439:
3435:
3431:
3429:
3425:
3421:
3417:
3413:
3409:
3405:
3401:
3397:
3394:
3389:
3386:
3382:
3379:
3375:
3373:
3369:
3364:
3361:
3357:
3353:
3349:
3343:
3339:
3332:
3325:
3318:
3314:
3307:
3300:
3296:
3289:
3285:
3278:
3274:
3267:
3260:
3256:
3252:
3248:
3244:
3239:
3236:
3232:
3228:
3224:
3220:
3216:
3210:
3206:
3199:
3192:
3185:
3181:
3174:
3167:
3163:
3156:
3152:
3145:
3141:
3134:
3127:
3123:
3119:
3115:
3111:
3106:
3103:
3099:
3095:
3091:
3086:
3082:
3078:
3075:over a field
3074:
3070:
3067:geometrically
3065:
3062:
3058:
3056:
3052:
3047:
3044:
3040:
3037:
3033:
3032:Gaussian ring
3029:
3024:
3021:
3017:
3012:
3007:
3003:
2999:
2994:
2991:
2987:
2983:
2980:contained in
2979:
2975:
2971:
2967:
2963:
2959:
2955:
2950:
2947:
2943:
2940:
2936:
2933:
2929:
2925:
2921:
2916:
2911:
2907:
2903:
2898:
2893:
2889:
2885:
2882:if for every
2881:
2877:
2873:
2869:
2865:
2861:
2857:
2852:
2849:
2845:
2841:
2837:
2834:
2831:
2827:
2824:
2820:
2816:
2812:
2808:
2807:normally flat
2804:
2800:
2796:
2793:
2789:
2786:
2782:
2779:
2775:
2770:
2767:
2763:
2759:
2756:generated by
2755:
2751:
2746:
2742:
2739:
2738:Fitting ideal
2735:
2732:Fitting ideal
2730:
2727:
2724:
2720:
2717:
2713:
2708:
2705:
2700:
2697:
2693:
2688:
2685:
2682:
2679:
2675:
2672:
2667:
2664:
2660:
2656:
2653:
2649:
2645:
2640:
2637:
2633:
2628:
2625:
2622:
2617:
2614:
2610:
2605:
2600:
2595:
2591:
2587:
2583:
2579:
2576:
2573:
2569:
2564:
2561:
2557:
2552:
2549:
2545:
2540:
2525:
2522:
2519:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2486:
2480:
2474:
2469:
2465:
2444:
2441:
2438:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2405:
2399:
2393:
2388:
2384:
2363:
2360:
2340:
2337:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2287:
2281:
2275:
2270:
2266:
2257:
2241:
2233:
2228:
2225:
2221:
2217:
2212:
2209:
2208:Ă©tale algebra
2205:
2202:
2198:
2193:
2190:
2185:
2182:
2178:
2174:
2170:
2166:
2162:
2159:
2155:
2152:is called an
2151:
2147:
2143:
2138:
2135:
2130:
2127:
2123:
2118:
2115:
2111:
2106:
2103:
2098:
2095:
2092:
2088:
2085:
2081:
2078:
2074:
2072:
2068:
2063:
2048:
2028:
2008:
2005:
2002:
1994:
1990:
1985:
1980:
1975:
1971:
1966:
1963:
1959:
1956:
1952:
1948:
1944:
1938:
1934:
1930:
1923:
1913:
1909:
1905:
1901:
1897:
1893:
1889:
1886:
1882:
1879:
1875:
1872:
1868:
1863:
1858:
1853:
1850:
1846:
1842:
1838:
1834:
1831:A local ring
1830:
1825:
1822:
1818:
1813:
1810:
1806:
1801:
1799:
1795:
1792:
1788:
1785:
1780:
1777:
1773:
1768:
1765:
1761:
1756:
1753:
1750:
1746:
1742:
1738:
1735:
1731:
1728:
1724:
1721:
1717:
1714:
1710:
1707:
1703:
1700:
1696:
1693:
1689:
1685:
1682:
1678:
1673:
1670:
1666:
1661:
1658:
1654:
1651:determinantal
1649:
1646:
1642:
1637:
1634:
1630:
1625:
1622:
1618:
1614:
1610:
1606:
1602:
1598:
1594:
1589:
1586:
1582:
1578:
1574:
1570:
1566:
1562:
1558:
1551:
1544:such that Ext
1543:
1539:
1535:
1531:
1527:
1524:(also called
1523:
1519:
1514:
1511:
1507:
1503:
1499:
1496:. The number
1495:
1491:
1487:
1483:
1479:
1476:is given by =
1475:
1471:
1468:
1464:
1460:
1455:
1450:
1447:
1443:
1438:
1435:
1431:
1426:
1423:
1419:
1414:
1409:
1404:
1400:
1395:
1392:
1388:
1384:
1380:
1376:
1371:
1368:
1365:
1360:
1357:
1353:
1348:
1345:
1341:
1336:
1333:
1328:
1325:
1321:
1318:constructible
1316:
1313:
1309:
1305:
1301:
1296:
1293:
1289:
1285:
1280:
1277:
1273:
1269:
1265:
1261:
1257:
1253:
1248:
1245:
1241:
1237:
1233:
1229:
1225:
1221:
1217:
1212:
1209:
1205:
1201:
1197:
1193:
1190:
1185:
1182:
1179:
1175:
1171:
1167:
1163:
1159:
1154:
1151:
1147:
1143:
1139:
1135:
1131:
1127:
1122:
1119:
1115:
1112:
1108:
1103:
1100:
1099:coherent ring
1096:
1093:
1089:
1084:
1081:
1077:
1073:
1069:
1066:
1062:
1057:
1054:
1050:
1048:
1044:
1039:
1036:
1033:
1030:
1025:
1022:
1018:
1013:
1010:
1005:
1002:
998:
993:
990:
986:
981:
977:
974:
970:
966:
962:
958:
954:
950:
946:
942:
937:
934:
930:
926:
921:
918:
913:
910:
905:
902:
898:
893:
890:
886:
881:
876:
871:
867:
862:
859:
855:
851:
846:
843:
839:
835:
831:
827:
824:Boolean ring
822:
818:
813:
810:
809:BĂ©zout domain
806:
803:BĂ©zout domain
801:
798:
794:
787:
779:
775:
771:
767:
763:
759:
755:
751:
746:
741:
736:
732:
728:
724:
720:
716:
712:
708:
703:
700:
696:
691:
688:
684:
681:
680:Artinian ring
677:
674:
670:
668:
667:Michael Artin
664:
662:
658:
653:
648:
645:
641:
636:
633:
629:
626:
625:
620:
617:
616:
611:
608:
607:
602:
600:
596:
591:
588:
587:analytic ring
584:
581:
578:
573:
570:
567:
563:
558:
555:
551:
547:
543:
540:
536:
532:
528:
524:
519:
516:
511:
508:
504:
500:
496:
491:
488:
484:
480:
475:
472:
468:
465:
460:
457:
453:
449:
446:
442:
438:
435:
431:
428:
424:
421:
417:
414:
410:
406:
403:
398:
395:
391:
386:
381:
377:
373:
369:
366:
361:
358:
354:
350:
346:
343:
338:
335:
331:
327:
324:
319:
316:
312:
308:
305:
301:
298:
294:
290:
286:
282:
278:
274:
270:
266:
263:
259:
256:generated by
255:
252:,...) is the
251:
247:
243:
240:
236:
233:generated by
232:
228:
224:
220:
216:
212:
207:
203:
195:
192:
190:
187:
185:
182:
180:
177:
175:
172:
170:
167:
165:
162:
160:
157:
155:
152:
150:
147:
145:
142:
140:
137:
135:
132:
130:
127:
125:
122:
120:
117:
115:
112:
110:
107:
105:
102:
100:
97:
95:
92:
90:
87:
85:
82:
80:
77:
75:
72:
70:
67:
65:
62:
61:
51:
49:
44:
42:
38:
34:
30:
26:
21:
19:
8409:
8381:
8375:
8339:
8333:
8297:
8291:
8255:
8249:
8213:
8207:
8171:
8165:
8129:
8123:
8087:
8081:
8031:
7996:
7969:
7937:
7931:
7915:
7909:
7882:zero divisor
7876:zero divisor
7869:
7865:
7861:
7850:
7842:
7814:Zariski ring
7628:is called a
7535:
7528:
7524:
7520:
7516:
7505:
7466:
7462:
7458:
7454:
7325:
7321:
7314:
7306:
7302:
7298:
7294:
7290:
7203:
7199:
7195:
7191:
7187:
7183:
7179:
7175:
7171:
7167:
7163:
7159:
7117:
7113:
7109:
7105:
7101:
7097:
7085:
7081:
7077:
7073:
7069:
7065:
7061:
7057:
7053:
7041:is non-zero.
7038:
7034:
7030:
7026:
7003:
6999:
6995:
6991:
6987:
6983:
6975:
6971:
6970:A submodule
6926:
6922:
6918:
6910:
6909:over a ring
6906:
6894:
6889:
6885:
6879:
6875:
6871:
6798:simple field
6797:
6699:
6655:
6651:
6648:reduced ring
6632:Zariski ring
6599:
6595:
6591:
6587:
6583:
6579:
6575:
6571:
6555:
6551:
6546:
6542:
6534:
6530:
6485:
6481:
6477:
6473:
6469:
6458:
6453:
6449:
6442:
6435:
6431:
6425:
6421:
6417:
6412:
6408:
6401:
6394:
6390:
6379:
6375:
6371:
6367:
6358:regular ring
6274:
6162:
6161:of an ideal
6159:Rees algebra
6140:
6136:
6133:
6129:
6126:
6122:
6121:is an ideal
6118:
6114:
6090:
6086:
6080:reduced ring
6067:
6063:
6059:
6055:
6051:
6047:
6022:
6018:
6014:
6010:
6006:
6002:
5979:
5975:
5971:
5967:
5963:
5962:of a module
5959:
5945:
5941:
5911:
5907:
5903:
5898:
5894:
5886:
5882:
5851:
5844:
5808:
5770:
5766:
5758:
5746:
5742:
5738:
5734:
5730:
5726:
5725:over a ring
5722:
5715:
5711:
5707:
5703:
5698:
5694:
5690:
5686:
5682:
5678:
5671:
5667:
5663:
5659:
5655:
5651:
5650:of a module
5647:
5625:
5618:
5529:
5525:
5447:
5440:
5439:annihilates
5436:
5432:
5428:
5427:then either
5424:
5420:
5416:
5412:
5408:
5404:
5400:
5396:
5392:
5391:then either
5388:
5384:
5380:
5376:
5370:
5355:
5321:
5317:
5313:
5309:
5282:
5281:) of a ring
5278:
5275:Picard group
5269:Picard group
5254:
5250:
5246:
5242:
5233:perfect ring
5208:
5204:
5200:
5197:Picard group
5184:
5165:
5161:
5123:
5120:
5116:
5113:
5109:
5105:
5101:
5097:
5096:over a ring
5093:
5043:Emmy Noether
5041:Named after
4993:
4989:
4985:
4984:is equal to
4981:
4922:
4918:
4913:
4909:
4905:
4901:
4897:
4893:
4889:
4885:
4880:multiplicity
4786:
4782:
4749:Named after
4717:Named after
4691:
4671:
4667:
4663:
4658:
4652:
4647:
4646:of a module
4626:localization
4601:
4597:
4589:
4585:
4527:Krull domain
4471:
4467:
4463:
4459:
4455:
4448:
4415:Named after
4393:
4389:
4244:
4240:
4236:
4232:
4169:(this is an
4157:for a given
4153:of an ideal
4150:
4134:(this is an
4126:of an ideal
4123:
4071:
4067:
4063:
4059:
4055:
4051:
4047:
4043:
4039:
4038:if whenever
4031:
4027:
4015:
4011:
4007:
3995:
3987:
3979:
3883:
3851:
3847:
3843:
3840:
3836:
3832:
3828:
3824:
3823:of a module
3820:
3701:Hilbert ring
3693:Named after
3668:
3664:
3660:
3656:
3652:
3648:
3644:
3640:
3561:Named after
3557:Grothendieck
3526:
3522:
3518:
3514:
3510:
3506:
3502:
3495:
3491:
3487:
3483:
3479:
3472:
3468:
3461:
3453:
3449:
3445:
3441:
3440:over a ring
3437:
3433:
3427:
3423:
3419:
3415:
3411:
3407:
3403:
3399:
3359:
3355:
3341:
3337:
3330:
3323:
3316:
3312:
3305:
3298:
3294:
3287:
3283:
3276:
3272:
3265:
3258:
3257:if whenever
3250:
3246:
3234:
3230:
3226:
3222:
3208:
3204:
3197:
3190:
3183:
3179:
3172:
3165:
3161:
3154:
3150:
3143:
3139:
3132:
3125:
3124:if whenever
3117:
3113:
3101:
3097:
3093:
3089:
3084:
3076:
3072:
3042:
3038:
2989:
2985:
2981:
2977:
2973:
2969:
2961:
2957:
2931:
2927:
2923:
2919:
2914:
2909:
2905:
2901:
2896:
2891:
2887:
2883:
2867:
2863:
2859:
2847:
2843:
2842:of a module
2832:
2829:
2825:
2822:
2818:
2814:
2810:
2802:
2801:over a ring
2798:
2765:
2761:
2757:
2753:
2749:
2744:
2740:
2696:finite field
2662:
2661:over a ring
2651:
2650:over a ring
2593:
2589:
2585:
2584:of a module
2560:Ext functors
2255:
2180:
2176:
2172:
2171:is a module
2168:
2167:of a module
2157:
2149:
2145:
2069:Named after
1954:
1950:
1946:
1942:
1941:vanishes if
1936:
1932:
1928:
1921:
1911:
1907:
1903:
1899:
1895:
1848:
1844:
1840:
1836:
1832:
1819:A domain or
1791:Weil divisor
1632:
1620:
1616:
1612:
1608:
1604:
1600:
1596:
1584:
1580:
1576:
1572:
1568:
1564:
1560:
1556:
1549:
1541:
1537:
1533:
1532:over a ring
1529:
1525:
1509:
1505:
1501:
1497:
1493:
1489:
1485:
1481:
1477:
1473:
1469:
1466:
1462:
1422:decomposable
1416:decomposable
1390:
1386:
1382:
1311:
1307:
1303:
1291:
1287:
1275:
1271:
1267:
1263:
1259:
1243:
1239:
1235:
1231:
1227:
1223:
1207:
1203:
1199:
1195:
1180:
1177:
1173:
1169:
1168:at an ideal
1165:
1149:
1145:
1141:
1137:
1129:
1079:
1075:
972:
968:
964:
960:
956:
952:
948:
944:
932:
857:
853:
841:
837:
833:
830:Boolean ring
796:
792:
785:
777:
773:
765:
761:
757:
753:
734:
730:
726:
722:
718:
717:over a ring
714:
713:of a module
697:Acronym for
631:
622:
613:
604:
593:analytically
586:
565:
553:
549:
545:
538:
534:
530:
526:
506:
502:
498:
486:
482:
451:
440:
433:
426:
419:
375:
367:
356:
352:
348:
344:
333:
325:
314:
306:
296:
292:
288:
284:
280:
272:
268:
261:
257:
249:
245:
238:
234:
230:
222:
218:
214:
45:
22:
17:
15:
8410:Local rings
7482:unramified.
7429:universally
7010:superheight
6980:superfluous
6966:superfluous
6915:stably free
6901:stably free
6698:, there is
6103:irreducible
5633:Nagata ring
5530:pseudo-zero
5342:prime ideal
5328:presentable
5156:orientation
5082:normal ring
4967:Nagata ring
4961:Nagata ring
4864:Mori domain
4858:Mori domain
4751:Eben Matlis
4592:are called
4300:irreducible
4295:submodules.
4285:irreducible
4279:irreducible
4120:graded ring
4006:An element
3631:Kurt Hensel
2778:flat module
2702:finite type
2175:containing
1500:is a power
1338:contraction
1047:Irvin Cohen
1015:codimension
770:Bass number
764:, then the
748:Bass number
644:annihilator
638:annihilator
493:affine ring
225:,...) is a
48:commutative
8430:Categories
7924:0768.12001
7901:References
7473:unramified
7457:of a ring
7064:such that
6978:is called
6913:is called
6638:seminormal
6619:semi-local
6606:saturation
6511:resolution
6492:regularity
6153:David Rees
5937:of a ring.
5935:nilradical
5930:of a ring.
5681:of a ring
5591:of height
5528:is called
5482:projective
5379:of a ring
5245:of a ring
5191:if it has
5187:is called
5037:Noetherian
5026:nilradical
5020:nilradical
4904:occurs in
4892:or a ring
4637:local ring
4521:Krull ring
4306:irrelevant
4271:invertible
4002:idempotent
3984:local ring
3839:such that
3629:Named for
3366:Gorenstein
3229:such that
3108:going down
3061:GCD domain
2840:flat cover
2805:is called
2642:faithfully
2065:Eisenstein
1745:local ring
1591:derivation
1457:deficiency
1156:completion
1132:is called
1053:Cohen ring
725:such that
705:associated
661:Emil Artin
400:absolutely
372:completion
291:such that
194:References
16:This is a
7739:−
7725:⋯
7706:⊂
7690::
7679:−
7665:⋯
7643:⋅
7580:⋯
7502:valuation
7496:valuation
7453:An ideal
7441:universal
7388:…
7357:unibranch
7351:unibranch
7293:of depth
7072:for some
6905:A module
6786:Hausdorff
6780:separated
6772:separable
6614:semilocal
6570:A subset
6566:saturated
6329:⟩
6320:⋅
6317:⟨
6314:↦
6303:∗
6300:∗
6292:→
6279:reflexive
6273:A module
6269:reflexive
6232:⊂
6189:∞
6174:⊕
6109:reduction
6097:reducible
5714:-modules
5599:≤
5456:principal
5215:parameter
5199:Pic(Spec(
5092:A module
5014:nilpotent
5008:nilpotent
4539:dimension
4181:injective
3954:⊗
3926:≥
3919:⋂
3862:A module
3850:for some
3620:Henselian
2984:for some
2964:, then a
2934:-algebra.
2886:-algebra
2792:dimension
2619:factorial
2607:face ring
2566:extension
2542:excellent
2502:∣
2496:∈
2475:
2421:∣
2415:∈
2394:
2303:∣
2297:∈
2276:
2140:essential
2114:dimension
2006:⊂
1945:â height(
1865:dualizing
1770:divisible
1713:dimension
1675:dimension
1663:deviation
1373:cotangent
1350:coprimary
1282:connected
1220:conductor
1214:conductor
1187:composite
1128:A domain
883:canonical
599:#formally
275:) is the
23:See also
8408:(1962),
8370:(1967).
8328:(1966).
8286:(1965).
8244:(1964).
8202:(1963).
8160:(1961).
8118:(1961).
8076:(1960).
8029:(1995),
7968:(1998),
7889:See also
7759:for all
7309:) of Ext
7194:, where
6994:implies
6847:spectrum
6804:singular
6658:satisfy
6040:numbers.
5860:quotient
5733:) = dim(
5710:for all
5126:is flat.
4996:is zero.
4955:N-2 ring
4943:N-1 ring
4850:monomial
4713:Macaulay
4474:module.
4411:Jacobson
4373:Jacobian
4367:J-2 ring
4361:J-2 ring
4355:J-1 ring
4349:J-1 ring
4343:J-0 ring
4337:J-0 ring
4318:isolated
4214:integral
3749:Hironaka
3241:going up
3026:Gaussian
2976:-module
2912:) to Hom
2854:formally
2835:is flat.
2630:faithful
2120:envelope
2100:embedded
1953:=height(
1827:dominate
1536:, where
1403:Cox ring
1397:Cox ring
1298:conormal
1238:, where
1230:-module
1164:or ring
1144:module,
1105:complete
1092:coherent
1086:coherent
901:catenary
895:catenary
840:for all
820:regular.
655:Artinian
575:analytic
566:altitude
560:altitude
189:See also
8398:0238860
8356:0217086
8314:0199181
8272:0173675
8230:0163911
8188:0217085
8146:0217084
8104:0217083
8063:1322960
8020:1251956
7802:Zariski
7449:unmixed
7277:trivial
7222:torsion
7139:tangent
7076:not in
7018:support
6933:Stanley
6921:⊕
6831:smooth.
6558:. (cf.
6345:regular
6074:reduced
6058:⊗
5922:radical
5701:⊗
5693:⊗
5666:⊗
5658:⊗
5365:primary
5227:perfect
5104:if the
5032:Noether
4814:miracle
4793:minimal
4770:maximal
4631:locally
4613:linkage
4058:, then
3689:Hilbert
3087:⊗
2813:if the
1860:duality
1782:divisor
1627:derived
1522:I-depth
1362:coprime
1330:content
1021:#height
1007:codepth
776:is the
370:is the
328:] is a
8416:
8396:
8354:
8312:
8270:
8228:
8186:
8144:
8102:
8061:
8051:
8018:
8008:
7984:
7944:
7922:
7182:is in
7124:syzygy
7068:is in
6866:stable
6822:smooth
6792:simple
6598:is in
5753:purely
5520:pseudo
5431:is in
5423:is in
5403:is in
5395:is in
5387:is in
5320:âȘâ to
5069:normal
4988:where
4745:Matlis
4608:linked
4564:length
4175:monoid
4149:, the
4140:monoid
4122:, the
3615:Hensel
3606:height
3600:height
3533:graded
3014:G-ring
2690:finite
1906:, the
1815:domain
1480:where
1452:defect
1148:is in
907:center
521:almost
7261:total
6874:) if
6835:socle
6702:with
6476:with
6448:,...,
6407:,...,
6125:with
5978:into
5974:maps
5825:quasi
5336:prime
5301:place
5193:depth
4621:local
4171:ideal
4136:ideal
4010:with
3812:ideal
3775:Hodge
3391:grade
3315:with
3286:with
3182:with
3153:with
2878:, or
2669:field
2201:Ă©tale
2195:Ă©tale
1987:Eakin
1743:of a
1526:grade
1516:depth
1041:Cohen
1001:CoCoA
995:CoCoA
939:clean
915:chain
650:Artin
332:over
313:over
309:is a
254:ideal
8414:ISBN
8049:ISBN
8006:ISBN
7982:ISBN
7942:ISBN
7845:-th
7549:weak
7285:type
7265:The
7214:The
7155:The
7143:The
7049:The
7022:The
6839:The
6735:and
6586:and
6147:Rees
6101:See
6029:rank
5773:and
5639:pure
5277:Pic(
5273:The
5219:See
5024:The
5000:neat
4834:The
4568:The
4537:See
4513:The
4501:The
4310:The
4106:Not
4066:and
4042:and
3994:and
3990:its
3793:hull
3659:in (
3647:and
3329:and
3275:and
3196:and
3142:and
3030:The
2772:flat
2736:The
2558:The
2457:and
2112:See
1991:The
1855:dual
1643:The
1631:The
1520:The
1478:defg
1461:The
1377:The
1342:The
1218:The
1160:The
999:The
959:for
927:The
642:The
481:The
477:adic
392:The
264:,...
241:,...
229:of
204:!$ @
64:!$ @
39:and
8386:doi
8344:doi
8302:doi
8260:doi
8218:doi
8176:doi
8134:doi
8092:doi
8041:doi
7920:Zbl
7796:XYZ
7632:if
7339:UFD
7269:or
7210:Tor
7190:of
7037:at
6982:if
6974:of
6917:if
6590:in
6582:in
6578:if
6478:xyx
6416:of
6277:is
6165:is
6066:of
5761:is
5745:of
5532:if
5415:of
5289:PID
5264:Pic
4953:An
4949:N-2
4941:An
4937:N-1
4674:).
4656:Hom
4466:in
4322:An
4196:or
4094:An
4054:in
4044:Q'
3797:An
3772:.
3589:HCF
3490:on
3426:on
3311:of
3178:of
3100:of
3049:GCD
2988:in
2968:of
2956:If
2554:Ext
2546:An
2466:Ann
2385:Ann
2267:Ann
2148:of
2124:An
1968:DVR
1916:Ext
1690:or
1465:or
1206:in
815:big
807:A
772:of
768:th
752:If
709:An
693:ASL
374:of
340:{}
283:by
279:of
184:XYZ
8432::
8394:MR
8392:.
8384:.
8382:32
8380:.
8374:.
8366:;
8352:MR
8350:.
8342:.
8340:28
8338:.
8332:.
8324:;
8310:MR
8308:.
8300:.
8298:24
8296:.
8290:.
8282:;
8268:MR
8266:.
8258:.
8256:20
8254:.
8248:.
8240:;
8226:MR
8224:.
8216:.
8214:17
8212:.
8206:.
8198:;
8184:MR
8182:.
8174:.
8172:11
8170:.
8164:.
8156:;
8142:MR
8140:.
8132:.
8128:.
8122:.
8114:;
8100:MR
8098:.
8090:.
8086:.
8080:.
8072:;
8059:MR
8057:,
8047:,
8039:,
8016:MR
8014:,
8004:,
7980:,
7972:,
7880:A
7787:A
7328:).
7206:.
7180:cz
7066:xy
7006:).
6886:IM
6882:+1
6826:A
6796:A
6654:,
6642:A
6580:xs
6515:A
6434:/(
6428:+1
6127:JI
5997:A
5512:A
5421:rm
5385:rm
5305:A
5080:A
5073:A
4986:IM
4965:A
4862:A
4556:A
4440:A
4365:A
4353:A
4341:A
4241:ax
4068:Q'
4064:Q'
3986:,
3910:,
3848:IM
3549:A
3537:A
3043:ni
2982:kR
2922:,
2908:,
2874:,
2376::
2218:A
2021:,
1957:).
1807:A
1774:A
1762:A
1667:A
1623:).
1617:bd
1615:)+
1609:ad
1607:)=
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