Commalg - User contributions [en]

Generalized local ring

2009-01-03T19:14:11Z

Masnevets: /* Stronger properties */ multivariate link

==Definition==

===Symbol-free definition===

A [[graded ring|positively graded]] [[commutative unital ring]] ''R'' is termed a '''generalized local ring''' if its degree 0 part is [[local ring|local]] and [[Noetherian ring|Noetherian]], and ''R'' is a finitely generated as an algebra over its degree 0 part.

==Metaproperties==

===Uniqueness of maximal ideal===

A generalized local ring has a unique maximal homogeneous ideal.

==Relation with other properties==

===Stronger properties===

* [[Multivariate polynomial ring over a field]]

Minimal resolution

2009-01-03T19:13:16Z

Masnevets: /* Metaproperties */ formatting

==Definition==

===Symbol-free definition===

A '''minimal resolution''' of a [[module]] over a [[generalized local ring]] is a graded [[free resolution]] (possibly infinite in length) terminating at 0, with the second last member being the given module, such that the differentials of the resolution become 0 after tensoring with the ring modulo its unique homogeneous maximal ideal.

==Metaproperties==

===Uniqueness===

Given a fixed module <math>M</math>, minimal resolutions of <math>M</math> are unique up to isomorphism.

===Bounds on length===

If the generalized local ring is a [[multivariate polynomial ring over a field]] in <math>n</math> variables, then the [[Hilbert syzygy theorem]] says that the minimal resolution of a finitely generated module has length less than or equal to <math>n</math>.

Hilbert syzygy theorem

2009-01-03T19:12:10Z

Masnevets: multivariate polynomial ring link

{{curing metaproperty satisfaction}}

==Name==

This result is termed the '''Hilbert syzygy theorem'''.

==Statement==

===Verbal statement===
Minimal resolutions over a [[multivariate polynomial ring over a field]] of finitely generated [[graded module]]s have length bounded by the number of variables.

===Symbolic statement===

Let <math>A = k[x_1, \dots, x_n]</math> be a polynomial ring over a field <math>k</math> and <math>M</math> a finitely generated graded <math>A</math>-module. Then there exists an exact sequence with degree 0 maps

::<math> 0 \to F_n \to F_{n-1} \to \cdots \to F_1 \to F_0 \to M \to 0 </math>

where the <math>F_i</math> are free modules.

==Proof==

===Proof idea===

The key tools of the proof are the symmetry of [[Tor]], and the [[Koszul complex]].

The idea is that the Koszul complex of <math>k</math> has length <math>n</math> and hence <math>\mathrm{Tor}^A_i(k,M) = \mathrm{Tor}^A_i(M,k) = 0</math> for <math>i>n</math>. Using this symmetry, one can also compute the Tor groups by tensoring a free resolution of <math>M</math> by <math>k</math>. In particular, taking a minimal free resolution (it is easy to see that minimal free resolutions exist) <math>F_\bullet</math> of <math>M</math>, the differentials become 0 upon tensoring with <math>k</math> by definition of minimal resolution. Hence it follows that <math>F_i = \mathrm{Tor}^A_i(k,M) = 0</math> for <math>i>0</math>.

==Remarks==

We are assuming that <math>A</math> has the canonical grading, i.e., that the degree <math>i</math> part is the vector space generated by the monomials of degree <math>i</math>.

Hilbert syzygy theorem

2009-01-03T19:11:14Z

Masnevets: New page: {{curing metaproperty satisfaction}} ==Name== This result is termed the '''Hilbert syzygy theorem'''. ==Statement== ===Verbal statement=== Minimal resolutions over a [[polynomial ring ...

{{curing metaproperty satisfaction}}

==Name==

This result is termed the '''Hilbert syzygy theorem'''.

==Statement==

===Verbal statement===
Minimal resolutions over a [[polynomial ring over a field]] of finitely generated [[graded module]]s have length bounded by the number of variables.

===Symbolic statement===

Let <math>A = k[x_1, \dots, x_n]</math> be a polynomial ring over a field <math>k</math> and <math>M</math> a finitely generated graded <math>A</math>-module. Then there exists an exact sequence with degree 0 maps

::<math> 0 \to F_n \to F_{n-1} \to \cdots \to F_1 \to F_0 \to M \to 0 </math>

where the <math>F_i</math> are free modules.

==Proof==

===Proof idea===

The key tools of the proof are the symmetry of [[Tor]], and the [[Koszul complex]].

The idea is that the Koszul complex of <math>k</math> has length <math>n</math> and hence <math>\mathrm{Tor}^A_i(k,M) = \mathrm{Tor}^A_i(M,k) = 0</math> for <math>i>n</math>. Using this symmetry, one can also compute the Tor groups by tensoring a free resolution of <math>M</math> by <math>k</math>. In particular, taking a minimal free resolution (it is easy to see that minimal free resolutions exist) <math>F_\bullet</math> of <math>M</math>, the differentials become 0 upon tensoring with <math>k</math> by definition of minimal resolution. Hence it follows that <math>F_i = \mathrm{Tor}^A_i(k,M) = 0</math> for <math>i>0</math>.

==Remarks==

We are assuming that <math>A</math> has the canonical grading, i.e., that the degree <math>i</math> part is the vector space generated by the monomials of degree <math>i</math>.

Minimal resolution

2009-01-03T18:54:30Z

Masnevets: New page: ==Definition== ===Symbol-free definition=== A '''minimal resolution''' of a module over a generalized local ring is a graded free resolution (possibly infinite in length) ter...

==Definition==

===Symbol-free definition===

A '''minimal resolution''' of a [[module]] over a [[generalized local ring]] is a graded [[free resolution]] (possibly infinite in length) terminating at 0, with the second last member being the given module, such that the differentials of the resolution become 0 after tensoring with the ring modulo its unique homogeneous maximal ideal.

==Metaproperties==

Given a fixed module <math>M</math>, minimal resolutions of <math>M</math> are unique up to isomorphism.

If the generalized local ring is a [[polynomial ring over a field]] in <math>n</math> variables, then the [[Hilbert syzygy theorem]] says that the minimal resolution of a finitely generated module has length less than or equal to <math>n</math>.

Free resolution

2009-01-03T18:46:20Z

Masnevets: added uniqueness up to chain homotopy

==Definition==

===Symbol-free definition===

A '''free resolution''' of a [[module]] over a [[commutative unital ring]] is an [[exact sequence of modules]] (possibly infinite in length) terminating at 0, with the second last member being the given module, and where all preceding members are [[free module]]s.

==Metaproperties==

Given a fixed module <math>M</math>, free resolutions of <math>M</math> are unique up to [[chain homotopy]].

==Related notions==

* [[Injective resolution]]
* [[Koszul complex of a module]]
* [[Minimal resolution]]
* [[Projective resolution]]

Generalized local ring

2009-01-03T18:44:04Z

Masnevets: formatting

Generalized local ring

2009-01-03T18:41:56Z

Masnevets: New page: ==Definition== ===Symbol-free definition=== A positively graded commutative unital ring ''R'' is termed a '''generalized local ring''' if its degree 0 part is [[local...

==Definition==

===Symbol-free definition===

A [[graded ring|positively graded]] [[commutative unital ring]] ''R'' is termed a '''generalized local ring''' if its degree 0 part is [[local ring|local]] and [[Noetherian ring|Noetherian]], and ''R'' is a finitely generated as an algebra over its degree 0 part.

===Metaproperties===

==Uniqueness of maximal ideal==

A generalized local ring has a unique maximal homogeneous ideal.

==Relation with other properties==

===Stronger properties===

* [[Polynomial ring over a field]]

Graded ring

2009-01-03T18:41:05Z

Masnevets: /* Definition */ added notion of positively graded

{{ring with addl structure}}

==Definition==

A '''graded ring''' is a [[commutative unital ring]] <math>A</math> equipped with a direct sum decomposition as a sum of Abelian subgroups:

<math>A = \oplus_{i=-\infty}^\infty A_i = \cdots A_{-2} \oplus A_{-1} \oplus A_0 \oplus A_1 \oplus A_2 \oplus \cdots</math>

such that the following hold:

* Each <math>A_i</math> is a subgroup under addition
* <math>1 \in A_0</math>
* <math>A_mA_n \subset A_{m+n}</math>. In other words, if <math>a \in A_m</math> and <math>b \in A_n</math> then <math>ab \in A_{m+n}</math>

A structure of the above sort on a ring is termed a '''gradation''', also a <math>\mathbb{Z}</math>-gradation. The ring <math>A</math> is '''positively graded''' if <math>A_i = 0</math> for all <math>i<0</math>.

There are related notions for noncommutative rings.

==Related notions==

===Weaker notions===

* [[Filtered ring]]

Local

2009-01-03T18:33:59Z

Masnevets: redirect

#REDIRECT [[Local ring]]

Injective resolution

2009-01-03T18:25:56Z

Masnevets: New page: ==Definition== ===Symbol-free definition=== An '''injective resolution''' of a module over a commutative unital ring is an exact sequence of modules (possibly infinite in len...

==Definition==

===Symbol-free definition===

An '''injective resolution''' of a [[module]] over a [[commutative unital ring]] is an [[exact sequence of modules]] (possibly infinite in length) beginning at 0, with the second member being the given module, and such that all successive members of the exact sequence are [[injective module]]s.

==Related notions==

* [[Free resolution]]
* [[Projective resolution]]