Magnification

In algebraic geometry, a proper morphism between schemes is a scheme-theoretic analogue of a proper map between complex-analytic varieties.

A basic example is a complete variety (e.g., projective variety) in the following sense: a k-variety X is complete in the classical definition if it is universally closed. A proper morphism is a generalization of this to schemes.

A closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite.

Definition

A morphism f : X → Y of algebraic varieties or more generally of schemes, is called universally closed if for all morphisms Z → Y, the projections for the fiber product

$X\times _{Y}Z\to Z$

are closed maps of the underlying topological spaces. A morphism f : X → Y of algebraic varieties is called proper if it is separated and universally closed. A morphism of schemes is called proper if it is separated, of finite type and universally closed ([EGA] II, 5.4.1 [1]). One also says that X is proper over Y. A variety X over a field k is complete when the structural morphism from X to the spectrum of k is proper.

Examples

The projective space P^d over a field K is proper over a point (that is, Spec(K)). In the more classical language, this is the same as saying that projective space is a complete variety. Projective morphisms are proper, but not all proper morphisms are projective. For example, it can be shown that the scheme obtained by contracting two disjoint projective lines in some P³ to one is a proper, but non-projective variety. Affine varieties of non-zero dimension are never complete. More generally, it can be shown that affine proper morphisms are necessarily finite. For example, it is not hard to see that the affine line A¹ is not complete. In fact the map taking A¹ to a point x is not universally closed. For example, the morphism

$f\times {\textrm {id}}:\mathbb {A} ^{1}\times \mathbb {A} ^{1}\to \{x\}\times \mathbb {A} ^{1}$

is not closed since the image of the hyperbola uv = 1, which is closed in A¹ × A¹, is the affine line minus the origin and thus not closed.

Properties and characterizations of proper morphisms

In the following, let f : X → Y be a morphism of schemes.

• Properness is a local property on the base, i.e. if Y is covered by some open subschemes Y_i and the restriction of f to all f^-1(Y_i) is proper, then so is f.
• Proper morphisms are stable under base change and composition.
• Closed immersions are proper.
• More generally, finite morphisms are proper. This is a consequence of the going up theorem.
• Conversely, every quasi-finite, locally of finite presentation and proper morphism is finite. (EGA III, 4.4.2 in the noetherian case and EGA IV, 8.11.1 for the general case)
• Stein factorization theorem states that any proper morphism to a locally noetherian scheme can be factorized into $X\to Z\to Y$, where the first morphism has geometrically connected fibers and the second on is finite.
• Proper morphisms are closely related to projective morphisms: If f is proper over a noetherian base Y, then there is a morphism: g: X' →X which is an isomorphism when restricted to a suitable open dense subset: g^-1(U) ≅ U, such that f'  := fg is projective. This statement is called Chow's lemma.
• Nagata's compactification theorem says that a separated morphism of finite type between quasi-compact and quasi-separated schemes (e.g., noetherian schemes) factors as an open immersion followed by a proper morphism.
• Proper morphisms between locally noetherian schemes or complex analytic spaces preserve coherent sheaves, in the sense that the higher direct images Rⁱf_∗(F) (in particular the direct image f_∗(F)) of a coherent sheaf F are coherent (EGA III, 3.2.1). This boils down to the fact that the cohomology groups of projective space over some field k with respect to coherent sheaves are finitely generated over k, a statement which fails for non-projective varieties: consider C^∗, the punctured disc and its sheaf of holomorphic functions ${\mathcal {O}}$. Its sections ${\mathcal {O}}(\mathbb {C} ^{*})$ is the ring of Laurent polynomials, which is infinitely generated over C.
• There is also a slightly stronger statement of this:Template:Harv let $f:X\to S$ be a morphism of finite type, S locally noetherian and $F$ a ${\mathcal {O}}_{X}$-module. If the support of F is proper over S, then for each $i\geq 0$ the higher direct image $R^{i}f_{*}F$ is coherent.:
• Template:Harv If X, Y are schemes of locally of finite type over the field of complex numbers $\mathbb {C}$, f induces a morphism of complex analytic spaces

  $f(\mathbb {C} ):X(\mathbb {C} )\to Y(\mathbb {C} )$

between their sets of complex points with their complex topology. (This is an instance of GAGA.) Then f is a proper morphism defined above if and only if $f(\mathbb {C} )$ is a proper map in the sense of Bourbaki and is separated.

• If f: X→Y and g:Y→Z are such that gf is proper and g is separated, then f is proper. This can for example be easily proven using the following criterion

Valuative criterion of properness

Valuative criterion of properness

There is a very intuitive criterion for properness which goes back to Chevalley. It is commonly called the valuative criterion of properness. Let f: X → Y be a morphism of finite type of noetherian schemes. Then f is proper if and only if for all discrete valuation rings R with fields of fractions K and for any K-valued point x ∈ X(K) that maps to a point f(x) that is defined over R, there is a unique lift of x to ${\overline {x}}\in X(R)$. (EGA II, 7.3.8). Noting that Spec K is the generic point of Spec R and discrete valuation rings are precisely the regular local one-dimensional rings, one may rephrase the criterion: given a regular curve on Y (corresponding to the morphism s : Spec R → Y) and given a lift of the generic point of this curve to X, f is proper if and only if there is exactly one way to complete the curve.

Similarly, f is separated if and only if in all such diagrams, there is at most one lift ${\overline {x}}\in X(R)$.

For example, the projective line is proper over a field (or even over Z) since one can always scale homogeneous co-ordinates by their least common denominator.

Proper morphism of formal schemes

Let $f:{\mathfrak {X}}\to {\mathfrak {S}}$ be a morphism between locally noetherian formal schemes. We say f is proper or ${\mathfrak {X}}$ is proper over ${\mathfrak {S}}$ if (i) f is an adic morphism (i.e., maps the ideal of definition to the ideal of definition) and (ii) the induced map $f_{0}:X_{0}\to Y_{0}$ is proper, where $X_{0}=({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}}/I),S_{0}=({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}}/K),I=f^{*}(K){\mathcal {O}}_{\mathfrak {X}}$ and K is the ideal of definition of ${\mathfrak {S}}$.Template:Harv The definition is independent of the choice of K. If one lets $X_{n}=({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}}/I^{n+1}),S_{n}=({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}}/K^{n+1})$, then$f_{n}:X_{n}\to S_{n}$ is proper.

For example, if $g:Y\to Z$ is a proper morphism, then its extension ${\widehat {g}}:{\widehat {Y}}\to {\widehat {Z}}$ between formal completions is proper in the above sense.

As before, we have the coherence theorem: let $f:{\mathfrak {X}}\to {\mathfrak {S}}$ be a proper morphism between locally noetherian formal schemes. If F is a coherent ${\mathcal {O}}_{\mathfrak {X}}$-module, then the higher direct images $R^{i}f_{*}F$ are coherent.

See also

• Proper base change theorem
• Stein factorization

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, section 5.3. (definition of properness), section 7.3. (valuative criterion of properness)
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, section 15.7. (generalisations of valuative criteria to not necessarily noetherian schemes)
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

• Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.
  
  my web-site http://himerka.com/