Why schemes

So John and I agreed and wrote the obituary below. Since the readership of Nature were more or less entirely made up of non-mathematicians, it seemed as though our challenge was to try to make some key parts of Grothendieck's work accessible to such an audience.

Obviously the very definition of a scheme is central to nearly all his work, and we also wanted to say something genuine about categories and cohomology. Here's what we came up with:. Although mathematics became more and more abstract and general throughout the 20th century, it was Alexander Grothendieck who was the greatest master of this trend. His unique skill was to eliminate all unnecessary hypotheses and burrow into an area so deeply that its inner patterns on the most abstract level revealed themselves -- and then, like a magician, show how the solution of old problems fell out in straightforward ways now that their real nature had been revealed.

His strength and intensity were legendary. He worked long hours, transforming totally the field of algebraic geometry and its connections with algebraic number theory. He was considered by many the greatest mathematician of the 20th century. Grothendieck was born in Berlin on March 28, to an anarchist, politically activist couple -- a Russian Jewish father, Alexander Shapiro, and a German Protestant mother Johanna Hanka Grothendieck, and had a turbulent childhood in Germany and France, evading the holocaust in the French village of Le Chambon, known for protecting refugees.

He received the Fields Medal in This is the field where one studies the locus of solutions of sets of polynomial equations by combining the algebraic properties of the rings of polynomials with the geometric properties of this locus, known as a variety.

Traditionally, this had meant complex solutions of polynomials with complex coefficients but just prior to Grothendieck's work, Andre Weil and Oscar Zariski had realized that much more scope and insight was gained by considering solutions and polynomials over arbitrary fields, e. The proper foundations of the enlarged view of algebraic geometry were, however, unclear and this is how Grothendieck made his first, hugely significant, innovation: he invented a class of geometric structures generalizing varieties that he called schemes.

In simplest terms, he proposed attaching to any commutative ring any set of things for which addition, subtraction and a commutative multiplication are defined, like the set of integers, or the set of polynomials in variables x,y,z with complex number coefficients a geometric object, called the Spec of the ring short for spectrum or an affine scheme, and patching or gluing together these objects to form the scheme.

The ring is to be thought of as the set of functions on its affine scheme. The geometric object to which this ring corresponds is an infinitesimal vector, a point which can move infinitesimally but to second order only. In effect, he is going back to Leibniz and making infinitesimals into actual objects that can be manipulated. A related idea has recently been used in physics, for superstrings.

To connect schemes to number theory, one takes the ring of integers. The corresponding Spec has one point for each prime, at which functions have values in the finite field of integers mod p and one classical point where functions have rational number values and that is 'fatter', having all the others in its closure.

Once the machinery became familiar, very few doubted that he had found the right framework for algebraic geometry and it is now universally accepted.

Going further in abstraction, Grothendieck used the web of associated maps -- called morphisms -- from a variable scheme to a fixed one to describe schemes as functors and noted that many functors that were not obviously schemes at all arose in algebraic geometry. This is similar in science to having many experiments measuring some object from which the unknown real thing is pieced together or even finding something unexpected from its influence on known things.

He applied this to construct new schemes, leading to new types of objects called stacks whose functors were precisely characterized later by Michael Artin. His best known work is his attack on the geometry of schemes and varieties by finding ways to compute their most important topological invariant, their cohomology.

A simple example is the topology of a plane minus its origin. Following the inspired suggestions of Grothendieck, Artin was able to show how with algebra alone that a suitably defined third cohomology group of this space has one generator, that is the sphere lives algebraically too.

Grothendieck went on to solve various deep conjectures of Weil, develop crystalline cohomology and a meta-theory of cohomologies called motives with a brilliant group of collaborators whom he drew in at this time. With a breathtakingly naive spririt that had served him well doing math he believed he could start a movement that would change the world. But when he saw this was not succeeding, he returned to math, teaching at the University of Montpellier.

There he formulated remarkable visions of yet deeper structures connecting algebra and geometry, e. Despite his writing thousand page treatises on this, still unpublished, his research program was only meagerly funded by the CNRS Centre Nationale de Recherche Scientifique and he accused the math world of being totally corrupt.

For the last two decades of his life he broke with the whole world and sought total solitude in the small village of Lasserre in the foothills of the Pyrenees. Here he lived alone in his own mental and spiritual world, writing remarkable self-analytic works.

He died nearby on Nov. As a friend, Grothendieck could be very warm, yet the nightmares of his childhood had left him a very complex person. He was unique in almost every way. His intensity and naivety enabled him to recast the foundations of large parts of 21st century math using unique insights that still amaze today. The power and beauty of Grothendieck's work on schemes, functors, cohomology, etc.

The dreams of his later work still stand as challenges to his successors. The sad thing is that this was rejected as much too technical for their readership. Their editor wrote me that 'higher degree polynomials', 'infinitesimal vectors' and 'complex space' even complex numbers were things at least half their readership had never come across. The gap between the world I have lived in and that even of scientists has never seemed larger. I am prepared for lawyers and business people to say they hated math and not to remember any math beyond arithmetic, but this!?

Very depressing. Well, Nature magazine really wanted to publish some obit on Grothendieck and wore us out until we agreed with a severely stripped down re-edit. The obit is coming out, I believe, in the Jan. The whole issue of trying to bridge the gap between the mathematician's world and that of other scientists or that of lay people is a serious one and I believe mathematicians could try harder to find bridges.

An example is Gower's work on bases in Banach spaces: when he received the Fields Medal, no one to my knowledge used the example of musical notes to explain Fourier series and thus bases of function spaces to the general public. No: excised by Nature. I had hoped that the "web of maps" was an excellent metaphor for the functor represented by an object in a category and gave one the gist. I had hoped that the "symmetry group of the set of all algebraic numbers" might pass muster to define this Galois group.

To be fair, they did need to cut down the length and they didn't want to omit the personal details. The essential minimum I thought for a Grothendieck obit was to make some attempt to explain schemes and say something about cohomology. To be honest, the central stumbling block for explaining schemes was the word "ring". If you haven't taken an intro to abstract algebra, where to begin? The final draft settled on mentioning in passing three examples -- polynomials leaving out the frightening phrase "higher degree" , the dual numbers and finite fields.

We batted about Spec of the dual numbers until something approaching an honest description came out, using "very small" and "infinitesimal distance". As for finite fields, in spite of John's discomfort, I thought the numbers on a clock made a decent first exposure. Then, clearly, if is another polynomial in the ideal , then we can use the axioms of commutative algebra which are basically the axioms of high school algebra to obtain the syntactic deduction.

In particular, we have the semantic deduction. If we restrict to lie in only, then even if is an algebraically closed field , the converse of the above statement is false; there can exist polynomials outside of for which 1 holds for all assignments in. For instance, we have. Of course, the nullstellensatz again explains what is going on here; 1 holds whenever lies in the radical of , which can be larger than itself.

For instance, since. On the other hand, we still have. Comments feed for this article. Qiaochu Yuan. In general this is an application of the Yoneda lemma, but in the particular case of schemes it really does tie in quite closely to the most naive perspective on algebraic geometry as being about zero sets of polynomials, and arguably does this better than the standard description of schemes as locally ringed spaces.

Terence Tao. This post is absolutely awesome! It clarified in 20 minutes one graduate course of algebraic geometry. Thank you very much! Can I translate into Spanish? My mathematical intuition has always been strongest in analysis; algebra and geometry have always […]. Thank you for a very nice explanation! It is so refreshing to read about the actual underlying ideas of some widely used notion, instead of a mere definition and examples that explains nothing.

A trivial remark about schemes « Guzman's Mathematics Weblog. Trying to understand the Galois correspondence What's new. There is a technical set-theoretic issue here because the class of integral domains is a proper […]. You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account.

Notify me of new comments via email. Notify me of new posts via email. Blog at WordPress. Ben Eastaugh and Chris Sternal-Johnson. Subscribe to feed. What's new Updates on my research and expository papers, discussion of open problems, and other maths-related topics. A trivial remark about schemes 5 September, in expository , math. Such varieties can be viewed in at least four different ways: Algebraic geometry One can view a variety through the set of points over in that variety.

Commutative algebra One can view a variety through the field of rational functions on that variety, or the subring of polynomial functions in that field. Dual algebraic geometry One can view a variety through a collection of polynomials that cut out that variety.

Dual commutative algebra One can view a variety through the ideal of polynomials that vanish on that variety. For instance, the unit circle over the reals can be thought of in each of these four different ways: Algebraic geometry The set of points. Commutative algebra The quotient of the polynomial ring by the ideal generated by or equivalently, the algebra generated by subject to the constraint , or the fraction field of that quotient.

Dual algebraic geometry The polynomial. Dual commutative algebra The ideal generated by. Any set of polynomials in indeterminate variables with coefficients in determines, on the one hand, an ideal in , and also cuts out a zero locus since each of the polynomials clearly make sense as maps from to. Of course, one can also write in terms of : Thus the ideal uniquely determines the zero locus , and we will emphasise this by writing as.

Indeed, suppose we have two ideals of that cut out the same non-empty zero locus for all extensions of , thus for all extensions of. Over , this becomes Note that the polynomial vanishes to order on this locus, but fails to lie in the ideal. Then, clearly, if is another polynomial in the ideal , then we can use the axioms of commutative algebra which are basically the axioms of high school algebra to obtain the syntactic deduction since is just a sum of multiples of.

In particular, we have the semantic deduction for any assignment of indeterminates in or in any extension of. For instance, we have for all in an algebraically closed field, despite not lying in the ideal. For instance, since we no longer have a counterexample to the converse coming from and once we work in instead of.

On the other hand, we still have so the extension is not powerful enough to detect that does not actually lie in ; a larger ring which is less easy to assign an analytic interpretation to is needed to achieve this.

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Categories expository tricks 11 guest blog 10 Mathematics math. AC 8 math. AG 42 math. IMO you should try to get hold of it and read through at least its first few chapters. I personally think this is not a good question for this site, for it more or less requires a chapter-length essay to be answered sensibly This tool is extremely important for the study of elliptic curves and for arithmetic geometry.

Add a comment. Active Oldest Votes. There are three distinct aspects of schemes that each have their own purpose: 1 Affine schemes generalizing affine varieties by allowing nilpotent elements in the coordinate ring. There are a couple of really good sources to turn to learn more about this: 1 Dieudonne's classical book, History of Algebraic Geometry 2 Ravi Vakil's book in progress, Foundations of Algebraic Geometry.

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