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In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.

Some authors call a proper variety over a field k a complete variety. For example, every projective variety over a field k is proper over k. A scheme X of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space X(C) of complex points with the classical (Euclidean) topology is compact and Hausdorff.

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

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hello everyone and welcome to the newest correspond project polygraph introduction to higher mathematics math one fifty i'm very excited to be teaching this course and i look forward to sharing the beauty and arsonists of mathematics with all of you first of all let me introduce myself my name is bill short-term and i live in marietta georgia like some of you taking this course per watching these videos i consider myself an aspiring polygraph i've never been satisfied with dedicated my life to just one topic but rather have always a reveled in studying many things and seeing how they're all connected that being said math has always been a particular passion of mine because of how it so often seems to link everything together i graduated from georgia tech in two thousand eight with a bachelor of science and international affairs and modern language i concentrated in japanese and of travel to japan twice so far elsa speak spanish in fact i've always been interested in languages linguistics so i studied the workings of many other languages from french to chinese to ancient egyptian to even quite gna s for my current profession i've been teaching tutoring in math since two thousand nine and i'm about to pursue my master of arts in teaching so i can teach math at a high school or perhaps university level i've also unpublished a couple times in our local math teacher associations periodical reflections and i'm soon going to have an animated lessened featured on ted and showing a more intuitive way to do matrix most look asian one other particular interest of my name is closely related to math it's music i play both piano and viola and i think composing music since i was in elementary school most recently i've contributed music to a number of video games enough about me let's talk about us and what we're going to be doing in this course the purpose of this course is to help you build a strong foundation in mathematics you'll be developing your abstract thinking skills which will include learning how to write proofs one thing i'd like to point out is that this course requires absolutely no knowledge of calculus in fact some of the topics you encounter here will be quite beneficial to if you take calculus after this course because calculus rests upon the foundations that will build especially in the properties of real numbers in fact a goal for this course is that the topics will talk about and the skills to develop will prepare you for more advanced math courses you may take in your future studies as we progress for this course we're going to cover a number of topics that serve as some of the underpinnings of mathematics as a whole among these topics are set theory relations and functions number theory abstract algebra cardinality and real analysis note that as this is a ten-week course will only be hitting highlights of certain topics like number theory abstract algebra and real analysis these are very rich topics which we could devote an entire class however what we will be going to deeply into them what we will be discussing is why they're important and how they contribute to the cohesiveness of mathematics will also see what we might call the adhesive massive mathematics seen how math applies to topics such as physics computer science crypt hundred feet and music these connections will help cultivate a more complete understanding of the topics will discuss will delve a bit into the history of mathematics and meet the unique people who made all of what we're talking about possible anyway without further ado let's go to our first lecture problem solving one oh one so before we talk about how to solve problems we have to ask ourselves the question just what is a problem many students believe they know what the problem is math they claim they've been doing them all their lives i can't speak for the rest of the world but at least in the united states this is often not the case in my country what all the masquerade zoos problems would be better turned exercises these amount a little more than reading assessments usually called a work problem translating it into arithmetic doing one or two computations and arriving at the correct answer take a look at this so-called word problem johnny goes out with four of his friends the each by a slice of pizza for two dollars how much was the total bill this word problem is nothing more than a contrived way to ask the question what is five times too some they say there's a slight challenge because some students may see the word for i think the question dear supposed to do what is four times too but in reality that's rather lame attempt at making a question more challenging however before i turn this into a rant on the american educational system let's consider what a problem should be the national council of teachers of mathematics or n_c_t_a_ m sets of guidelines and what it considers worthwhile mathematical tasks the standard say that the math teachers should pose tasks that engage students intellect stimulate students to make connections and develop a coherent framework for mathematical reasoning and are based on knowledge of the range of ways a diverse students learn mathematics this begins to give us a better idea of what makes a good problem they should have answers that are not immediately obvious and there may be more than one valid way to approach them for example consider class learning about arian perimeter on one hand and exercise might look like this find the area and perimeter of each triangle this task just requires you to apply the formulas for air imprimatur a couple times now very interesting is that on the other hand true problem might instead look like this suppose he had sixty four meters offense with which you're going to build a pen for your large dog bones waterson different ends you can make a few years all the fencing what is the pen with the least play space what is the biggest pen you can make the one that allows bones the most play space which would be best for running the questions asked constitute through problems because they asked questions that require investigation conjecture in and critical thinking there are many ways to go about solving such a problem think about how you might eccles some of these questions good problems should also foster student's ability to reason and communicate mathematically mathematics is an ongoing human activity and as a social species is one that is often best tackle together this is why by the way hopefully encourage you to collaborate with each other for the homework or be assigning for this course discuss things reason out and don't be afraid to voice your concerns opinions and ideas now that we've established what a problem really is it's time to talk about how to go about solving them as we've seen mathematical problems often have many ways to tackle them but it's useful to study what most effective problem-solving strategies have in common so i'm going to introduced him problem solving framework adapted from a book i highly recommend called thinking mathematically by mason burton and stacy don't think of this is a step-by-step process that must be followed to the letter one's own problems but more of a suggested guideline for how to focus our efforts the problem solving process can generally be broken up into three parts and entreaties an attack face and a review face no matter how you tend to solve problems these three phases are also present some varying degree let's look at each phase intern the first phase is the entry face this is the phase entered when you first ponder problem and prepared to attack it some people just try to haphazardly dive right in after reading the question but it's usually ends in frustration it's important to recognize that the entry phase is a vital part of problem solving and should not be overlooked in the entry phase there are three questions but you must also consider what do i know what do i want and what can i introduce think about would do one no this is usually answered from two sources first of all what do you know from reading the question second of all what do you know from past experience often even countered exercises are problems or even real-world situations that involves similar principles to the ones on which your problems based the second question is what do i want this is where you isolate exactly what you're looking for in many cases it helps to write the essentials of the problem in your own words this helps to ensure that you understand your goal always make sure to watch out for possible ambiguities in the questions freezing them language can be tricky thing for example my question is talking about a circle do i care about just the outside edge of the circle or do i need to think about the interior of that circle which is usually called disk we often use circle to describe both concepts but in math it pays off to be precise finally you should ask what can i introduce this awful supplement your rephrasing of a question in one of two ways first of all in many cases it helps to make a diagram tell properly visualize the situation second of all it helps to represent the question in a mathematical notation to see what's going on under the hood and possibly eliminate extraneous details when i talk about reading the question i can't emphasize enough d very careful it's far too tempting to just try to put the numbers out of a question and slap some pluses and minuses between them without really understanding what the problems asking for instance consider the following classic question how much better is their in a whole three feet six inches wide four feet eight inches long and six feet three inches deep and of course mexican business can use one point two oh six meters by one point four two meters by two point but one meters as their dimensions feel free to pause the video if you'd like to give this question of shot okay now how many of you first had the font uh... i'd just need to multiply the numbers together and reach for your pencil are calculated to start to make the appropriate computations if you're careful you realize that before any of the numbers are even mentioned the important question is asked how much mdr tb is there in powerful well since it's a whole the answer is of course zero regardless of the dimensions of the whole if there were dirt in it it wouldn't be much of a whole now what it when you're in the entry things always beware hidden assumptions you may have oftentimes you'll be trying to solve the problem banging your head against the wall only to realize eventually got the solution is one you never thought of because an assumption you didn't even realize you were making for example consider this question given this three by three square arrangements but evenly spaced circles draw four straight lines them across to the centers of all four circles without picking up your pencil this problem called the nine dot puzzle originally comes from nineteen fourteen possible by american recreational mathematician sam lloyd all gun owners america's greatest puzzling again go ahead and positive you here if you want to try the problem out yourself so the key to solving this problem is to realize that the lines you draw can go outside the square arrangement so we can solve the problem with three four lines just like that this problem became particularly famous around the nineteen seventies when management consultants challenge their clients to solve it as part of their training in fact this is found to be the quite literal origin of the phrase thinking outside the box let's move on to the attack face this is where the meat of the actual problem-solving takes place and where you'll spend most of your time we'll talk about two contrasting ways to do this by brute force and by clever strategy one way to solve a problem is to try literally every single possibility creek toder first call this method brute force as is akin to trying to find some of these four-digit password by first-run zero zero zero zero then zero zero zero one then zero zero zero two and so on you might have seen this clip from family guy an important thing to do him one what what what do you this is in the briefing completed solve problems but for certain problems it does in fact work a famous example of a problem that was sold by brute force was the four color problem the question was given a map like for example geography map is always possible to color the entire map using only four colors so they know to neighboring regions or countries share the same color the problem was solved in nineteen seventy six my kenneth apple and full-time hochiminh at the university of illinois dissolution involved imagining each a region as a single point anisha border between countries as a line connecting those points they use these more easy to manipulate graphs to classify a total of one thousand nine hundred thirty six graphs such that any maps could be represented in terms of the smaller graphs they then proceeded to check every single one of those graphs using a computer one-by-one which took over a thousand dollars and this was the first time in history the theorem was proven by computer we'll talk more about theorems in the next lecture when we delve into pricks often brute force is a problem solving method is either inefficient or completely inadequate so we need to try to find a more sophisticated approach nature is full of patterns and as humans were wired to recognize them sapan recognition is often a go-to for math problems for instance let's look at staircases into the world i can build a three-step staircase with six blocks as you can see here using the same sort of configuration i met wonder how many blocks would take to make a staircase with four steps what about five steps or ten steps pause the video to think about how to solve this problem by the way i've been telling you to pause whenever you should try problem on your own analytics was at least a distant future lectures but i encourage you to do so whatever questions posed that you should think about back to the staircases this is a great place to look at a pattern a one step staircase of course takes only one block how can turn this into a two-step staircase i can add two blocks for a total of three to turn this into a three-step staircase i had three blocks for a total of six four steps ad four blocks for a total of ten and so on and so forth each time to add another step to a staircase the number of blocks i have to add goes up by one i can follow this pattern to find out how many blocks will be in my five steps staircase fifteen of them and my ten steps staircase fifty five blocks once we feel we've got a viable solution to the problem it's time to move on to the review face there are three important parts to the review face check the solution reflect on the key ideas and key moments that led to the solution extend the solution to a wider context let's look at these parts in the context of the staircase problem check the solution that is make sure it in fact does solve the problem we can easily drawn out a staircase with ten steps or even take some blocks of actually start building one and then count the number of steps that has just to make sure it actually has fifty five reflect on the key ideas moments what were the big realizations that led to your solution the important idea when we saw the staircase problem was that every time you add another step to your staircase you have to add one more blocked and you had to add the previous time finally and this is essential extend the solution to a wider context this is often called generalizing your solution could we come up with a formula that would give us the number of steps needed to construct a staircase with any number of steps or as we might see in math n number of steps looking again at her pattern we can see that for one step we needed one block two steps one plus two blocks three steps one plus two plus three blocks four steps one plus two plus three plus four blocks and so on so for n steps we need to add up all the whole numbers from one to end to find a total number of blocks if you remember arithmetic series from high school you may know that there's a single formula for the some and times and plus one all divided by two see if you can figure out where this formula comes from and why it works one final word of warning or patterns are a great thing to look for they can also be misleading consider the following situation say we have a circle and replace some number of points say employment on the road now we draw all possible chords between these points questionnaires what is the maximum number of reasons that can be formed pause the video and see if you can come up with a pattern before you move on let's take a look at some examples and see if we can find a pattern suppose there's only one point on the circle well then we can't draw any chords so the entire circle is still intact and we have only one region next suppose there's two points on circle we can draw one court which will split the circle into two regions now three points we control three quarts so four regions starting to see a pattern here four points six chords so it regions five points ten chords some sixteen regions by now he probably see an apparent pattern the number of reasons seems to double everytime so we can conclude that for endpoints the circle can be divided into a maximum of two to the end minus one power regions or can it let's take a look at what happens with six points this is a drawing which there are six points on the edge circle note that to get the maximum number of reasons we should make sure that no more than two cords ever meet a single point in the interior of the circle let's cover regions six seven nine and thirteen fourteen fifteen sixteen seventeen eighteen nineteen twenty when twenty-two twenty-three twenty-four twenty five twenty-six point seven twenty-eight twenty nine thirty-one you were probably expecting thirty to one of you since our formula was to the end minus one as it turns out the actual formula for the number of regions the circle be split into is much more complicated and that's why it's important not to forget to review your problem more important than simply noticing a pattern is understanding why it works in this case our proposed solution failed when we checked it because it didn't extend all numbers of points so it might have been beneficial to reflect on why we copy answer we did how to find the actual solution to this problem might be explored in a class called competent or x that deals with the mathematics of counting that's it for this lecture next time we'll talk about how to communicate the solutions and mathematically sound way using something called proofs see you next time

Definition

A morphism f: X → Y of schemes is called universally closed if for every scheme Z with a morphism Z → Y, the projection from the fiber product

X\times _{Y}Z\to Z

is a closed map of the underlying topological spaces. 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. In particular, a variety X over a field k is said to be proper over k if the morphism X → Spec(k) is proper.

Examples

For any natural number n, projective space Pⁿ over a commutative ring R is proper over R. Projective morphisms are proper, but not all proper morphisms are projective. For example, there is a smooth proper complex variety of dimension 3 which is not projective over C.^[1] Affine varieties of positive dimension over a field k are never proper over k. More generally, a proper affine morphism of schemes must be finite.^[2] For example, it is not hard to see that the affine line A¹ over a field k is not proper over k, because the morphism A¹ → Spec(k) is not universally closed. Indeed, the pulled-back morphism

\mathbb {A} ^{1}\times _{k}\mathbb {A} ^{1}\to \mathbb {A} ^{1}

(given by (x,y) ↦ y) is not closed, because the image of the closed subset xy = 1 in A¹ × A¹ = A² is A¹ − 0, which is not closed in A¹.

Properties and characterizations of proper morphisms

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

The composition of two proper morphisms is proper.
Any base change of a proper morphism f: X → Y is proper. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X ×_Y Z → Z is proper.
Properness is a local property on the base (in the Zariski topology). That is, 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.
More strongly, properness is local on the base in the fpqc topology. For example, if X is a scheme over a field k and E is a field extension of k, then X is proper over k if and only if the base change X_E is proper over E.^[3]
Closed immersions are proper.
More generally, finite morphisms are proper. This is a consequence of the going up theorem.
By Deligne, a morphism of schemes is finite if and only if it is proper and quasi-finite.^[4] This had been shown by Grothendieck if the morphism f: X → Y is locally of finite presentation, which follows from the other assumptions if Y is noetherian.^[5]
For X proper over a scheme S, and Y separated over S, the image of any morphism X → Y over S is a closed subset of Y.^[6] This is analogous to the theorem in topology that the image of a continuous map from a compact space to a Hausdorff space is a closed subset.
The Stein factorization theorem states that any proper morphism to a locally noetherian scheme can be factored as X → Z → Y, where X → Z is proper, surjective, and has geometrically connected fibers, and Z → Y is finite.^[7]
Chow's lemma says that proper morphisms are closely related to projective morphisms. One version is: if X is proper over a quasi-compact scheme Y and X has only finitely many irreducible components (which is automatic for Y noetherian), then there is a projective surjective morphism g: W → X such that W is projective over Y. Moreover, one can arrange that g is an isomorphism over a dense open subset U of X, and that g⁻¹(U) is dense in W. One can also arrange that W is integral if X is integral.^[8]
Nagata's compactification theorem, as generalized by Deligne, says that a separated morphism of finite type between quasi-compact and quasi-separated schemes factors as an open immersion followed by a proper morphism.^[9]
Proper morphisms between locally noetherian schemes 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). (Analogously, for a proper map between complex analytic spaces, Grauert and Remmert showed that the higher direct images preserve coherent analytic sheaves.) As a very special case: the ring of regular functions on a proper scheme X over a field k has finite dimension as a k-vector space. By contrast, the ring of regular functions on the affine line over k is the polynomial ring k[x], which does not have finite dimension as a k-vector space.
There is also a slightly stronger statement of this:(EGA III, 3.2.4) let $f\colon X\to S$ 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.
For a scheme X of finite type over the complex numbers, the set X(C) of complex points is a complex analytic space, using the classical (Euclidean) topology. For X and Y separated and of finite type over C, a morphism f: X → Y over C is proper if and only if the continuous map f: X(C) → Y(C) is proper in the sense that the inverse image of every compact set is compact.^[10]
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

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 fraction field 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). More generally, a quasi-separated morphism f: X → Y of finite type (note: finite type includes quasi-compact) of 'any' schemes X, Y is proper if and only if for all valuation rings R with fraction field 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 ${\overline {x}}\in X(R)$ . (Stacks project Tags 01KF and 01KY). 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 every such diagram, there is at most one lift ${\overline {x}}\in X(R)$ .

For example, given the valuative criterion, it becomes easy to check that projective space Pⁿ is proper over a field (or even over Z). One simply observes that for a discrete valuation ring R with fraction field K, every K-point [x₀,...,x_n] of projective space comes from an R-point, by scaling the coordinates so that all lie in R and at least one is a unit in R.

Geometric interpretation with disks

One of the motivating examples for the valuative criterion of properness is the interpretation of ${\text{Spec}}(\mathbb {C} [[t]])$ as an infinitesimal disk, or complex-analytically, as the disk $\Delta =\{x\in \mathbb {C} :|x|<1\}$ . This comes from the fact that every power series

$f(t)=\sum _{n=0}^{\infty }a_{n}t^{n}$

converges in some disk of radius $r$ around the origin. Then, using a change of coordinates, this can be expressed as a power series on the unit disk. Then, if we invert $t$ , this is the ring $\mathbb {C} [[t]][t^{-1}]=\mathbb {C} ((t))$ which are the power series which may have a pole at the origin. This is represented topologically as the open disk $\Delta ^{*}=\{x\in \mathbb {C} :0<|x|<1\}$ with the origin removed. For a morphism of schemes over ${\text{Spec}}(\mathbb {C} )$ , this is given by the commutative diagram

${\begin{matrix}\Delta ^{*}&\to &X\\\downarrow &&\downarrow \\\Delta &\to &Y\end{matrix}}$

Then, the valuative criterion for properness would be a filling in of the point $0\in \Delta$ in the image of $\Delta ^{*}$ .

Example

It's instructive to look at a counter-example to see why the valuative criterion of properness should hold on spaces analogous to closed compact manifolds. If we take $X=\mathbb {P} ^{1}-\{x\}$ and $Y={\text{Spec}}(\mathbb {C} )$ , then a morphism ${\text{Spec}}(\mathbb {C} ((t)))\to X$ factors through an affine chart of $X$ , reducing the diagram to

${\begin{matrix}{\text{Spec}}(\mathbb {C} ((t)))&\to &{\text{Spec}}(\mathbb {C} [t,t^{-1}])\\\downarrow &&\downarrow \\{\text{Spec}}(\mathbb {C} [[t]])&\to &{\text{Spec}}(\mathbb {C} )\end{matrix}}$

where ${\text{Spec}}(\mathbb {C} [t,t^{-1}])=\mathbb {A} ^{1}-\{0\}$ is the chart centered around $\{x\}$ on $X$ . This gives the commutative diagram of commutative algebras

${\begin{matrix}\mathbb {C} ((t))&\leftarrow &\mathbb {C} [t,t^{-1}]\\\uparrow &&\uparrow \\\mathbb {C} [[t]]&\leftarrow &\mathbb {C} \end{matrix}}$

Then, a lifting of the diagram of schemes, ${\text{Spec}}(\mathbb {C} [[t]])\to {\text{Spec}}(\mathbb {C} [t,t^{-1}])$ , would imply there is a morphism $\mathbb {C} [t,t^{-1}]\to \mathbb {C} [[t]]$ sending $t\mapsto t$ from the commutative diagram of algebras. This, of course, cannot happen. Therefore $X$ is not proper over $Y$ .

Geometric interpretation with curves

There is another similar example of the valuative criterion of properness which captures some of the intuition for why this theorem should hold. Consider a curve $C$ and the complement of a point $C-\{p\}$ . Then the valuative criterion for properness would read as a diagram

${\begin{matrix}C-\{p\}&\rightarrow &X\\\downarrow &&\downarrow \\C&\rightarrow &Y\end{matrix}}$

with a lifting of $C\to X$ . Geometrically this means every curve in the scheme $X$ can be completed to a compact curve. This bit of intuition aligns with what the scheme-theoretic interpretation of a morphism of topological spaces with compact fibers, that a sequence in one of the fibers must converge. Because this geometric situation is a problem locally, the diagram is replaced by looking at the local ring ${\mathcal {O}}_{C,{\mathfrak {p}}}$ , which is a DVR, and its fraction field ${\text{Frac}}({\mathcal {O}}_{C,{\mathfrak {p}}})$ . Then, the lifting problem then gives the commutative diagram

${\begin{matrix}{\text{Spec}}({\text{Frac}}({\mathcal {O}}_{C,{\mathfrak {p}}}))&\rightarrow &X\\\downarrow &&\downarrow \\{\text{Spec}}({\mathcal {O}}_{C,{\mathfrak {p}}})&\rightarrow &Y\end{matrix}}$

where the scheme ${\text{Spec}}({\text{Frac}}({\mathcal {O}}_{C,{\mathfrak {p}}}))$ represents a local disk around ${\mathfrak {p}}$ with the closed point ${\mathfrak {p}}$ removed.

Proper morphism of formal schemes

Let $f\colon {\mathfrak {X}}\to {\mathfrak {S}}$ 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 $f_{0}\colon X_{0}\to S_{0}$ is proper, where $X_{0}=({\mathfrak {X}},{\mathcal {O}}_{\mathfrak {X}}/I),S_{0}=({\mathfrak {S}},{\mathcal {O}}_{\mathfrak {S}}/K),I=f^{*}(K){\mathcal {O}}_{\mathfrak {X}}$ and K is the ideal of definition of ${\mathfrak {S}}$ .(EGA III, 3.4.1) The definition is independent of the choice of K.

For example, if g: Y → Z is a proper morphism of locally noetherian schemes, Z₀ is a closed subset of Z, and Y₀ is a closed subset of Y such that g(Y₀) ⊂ Z₀, then the morphism ${\widehat {g}}\colon Y_{/Y_{0}}\to Z_{/Z_{0}}$ on formal completions is a proper morphism of formal schemes.

Grothendieck proved the coherence theorem in this setting. Namely, let $f\colon {\mathfrak {X}}\to {\mathfrak {S}}$ be a proper morphism of locally noetherian formal schemes. If F is a coherent sheaf on ${\mathfrak {X}}$ , then the higher direct images $R^{i}f_{*}F$ are coherent.^[11]

References

^ Hartshorne (1977), Appendix B, Example 3.4.1.
^ Liu (2002), Lemma 3.3.17.
^ Stacks Project, Tag 02YJ.
^ Grothendieck, EGA IV, Part 4, Corollaire 18.12.4; Stacks Project, Tag 02LQ.
^ Grothendieck, EGA IV, Part 3, Théorème 8.11.1.
^ Stacks Project, Tag 01W0.
^ Stacks Project, Tag 03GX.
^ Grothendieck, EGA II, Corollaire 5.6.2.
^ Conrad (2007), Theorem 4.1.
^ SGA 1, XII Proposition 3.2.
^ Grothendieck, EGA III, Part 1, Théorème 3.4.2.

SGA1 Revêtements étales et groupe fondamental, 1960–1961 (Étale coverings and the fundamental group), Lecture Notes in Mathematics 224, 1971
Conrad, Brian (2007), "Deligne's notes on Nagata compactifications" (PDF), Journal of the Ramanujan Mathematical Society, 22: 205–257, MR 2356346
Grothendieck, Alexandre; Dieudonné, Jean (1961). "Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes". Publications Mathématiques de l'IHÉS. 8: 5–222. doi:10.1007/bf02699291. MR 0217084., section 5.3. (definition of properness), section 7.3. (valuative criterion of properness)
Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11: 5–167. doi:10.1007/bf02684274. MR 0217085.
Grothendieck, Alexandre; Dieudonné, Jean (1966). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie". Publications Mathématiques de l'IHÉS. 28: 5–255. doi:10.1007/bf02684343. MR 0217086., section 15.7. (generalizations of valuative criteria to not necessarily noetherian schemes)
Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32: 5–361. doi:10.1007/bf02732123. MR 0238860.
Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
Liu, Qing (2002), Algebraic geometry and arithmetic curves, Oxford: Oxford University Press, ISBN 9780191547805, MR 1917232

External links

V.I. Danilov (2001) [1994], "Proper morphism", Encyclopedia of Mathematics, EMS Press
The Stacks Project Authors, The Stacks Project

This page was last edited on 6 May 2024, at 01:06

Proper morphism

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