Jacob Lurie - Bezout's theorem and nonabelian homological algebra (Derived algebraic geometry)

right well it's my great pleasure to introduce dick brewery from Harvard who he's going to tell us about kazoos theorem and non abelian homological ah good it's my great pleasure to be here this department has a lot of energy so my subject is the theory of what's called derived algebraic geometry which roughly speaking is what you get if you take the foundations of ordinary algebraic geometry and jazz them up a little bit by inserting some ideas from home Adobe theory before I get to that subject I'd like to begin with something very classical that will hopefully motivate why the foundations of ordinary abstraction on which we need to be jazzed up so I'd like to begin by thinking about who's there so what does bayes's theorem say well consider the complex projective plane and suppose I give you two smooth plane curves in the projective plane so these curves are defined by polynomial equations which have a particular degrees let's say C is a curve of degree N and C prime is a curve of degree so in its simplest form bayes's theorem tells you that if C and C prime meet transversely then the number of points of intersection of C and C prime is equal to the product and times M so let me just reformulate this statement a little bit using the language of Kohala G CNC primer smooth curves in the plane in particular they are smooth manifolds and they have fundamental classes which you can identify with elements in the Kohala g ring of c p2 so I'm gonna let C in brackets denote the fundamental class of C and C Prime in brackets denote the fundamental class of C prime then the product of their fundamental classes that's this number n times M and what are we saying what we're saying that that's equal to the fundamental class of the intersection of C and C Prime this is a zero dimensional manifold and taking the fundamental class in that case just amounts to counting the number of points that that manifold has so in some sense you should regard this equation of fundamental classes as obvious because what is the product on Co homology supposed to do it's supposed to intercept things so that might make you think that this is obvious but it can't be completely obvious because it's not necessarily true it's true of C meet C prime transversely but if C and C prime to not be transversely if they're tangent have one or more points that in fact this is always false the number of intersection points of C and C prime in that case is always smaller than the product end times out there why is that well it's because when C and C prime meet non transversely you don't just want to count the number of points of intersection you want to count them with the appropriate intersection multiplicities so in the non-trans first case you want another formula of this kind but where you do something less naive for the left-hand side here where you count points according to multiplicity so let me talk a little bit about how you get those intersection multiplicities so let's suppose that c and c prime are contained instead of in the projective plane that they're in the affine plane for simplicity so the affine plane on the fi plane we can choose coordinates x and y and then we can write the ring of functions on the affine plane as a polynomial range even the variables x and y and now c and c prime are defined by polynomial equations so c is defined by the equation f equals 0 where f is some polynomial in x and y and c prime will take to be defined by the equation G equals 0 so if we want to think about the ring of functions not on the entire plane but just on the curve see when we should take functions on the entire plane and then remember that on the curve see the function f is equal to zero so we divide out by the function f we get the ring of functions on the curve C similarly if we want the ring of functions on the curve C prime we should divide out by the element G so the ring of functions on C intersect C prime should be something like this polynomial ring but divided out by both the defining equation of C and the defining equation of C prime so this is the ring of functions on C intersect C prime where the intersection is interpreted schema theoretically not just set theoretically so if you don't know what that means well roughly speaking it just means we're not interested only in the points where C&Z crime intersects but also in the rings that come up when we impose the defining equation of cnc Prime and it turns out that the Rings which come up remember more than just the number of points where c and c prime intersect so let's consider some examples so first let's consider the example where cnc prime meat transversely at the origin so let's say c is the line where x equals 0 and c prime is the line where y equals 0 then function is on c intersect C prime would be like taking this polynomial ring in two variables and setting both of the variables equal to zero and in this case you just get the complex numbers which is what you expect that that's the ring of functions on a single point and the fact that it's just one point is reflected in the fact that this is one-dimensional as a vector space over the complex numbers so let's consider a slightly more complicated situation where cnc prime to not be transversely namely let's suppose that c is again given by the equation the line where x equals 0 but the c prime is given by a parabola find by the equation x equals y squared so here's the curve c here's the conic c prime and their tangent at the origin so this is a non transverse intersection what happens on the level of rings well the intersection of C and C prime is encoded level that rings by taking this polynomial ring in X&Y and killing X and X minus y squared well of course once you've killed X you're then just killing in addition y squared so this is the ring C controlling Y modulo y squared and now this is not naively what you would think of as the ring of functions on a point it's a little bit bigger it's two-dimensional over the complex numbers and what that's reflecting is that this is a point of tangency and that in fact you really ought to be counting this point as being multiplicity 2 in the intersection of CMC prongs so what's true then well in general for plain curves the product end times M is equal to the number of intersection points counted with their multiplicities so we can write this as a sum over all points which lie in the intersection of C and C prime of an intersection multiplicity which according to the prescription which I've described here is given by the dimension of a certain ring you want to take the complex dimension of the ring of functions of the intersection at that point how do you get that ring well you take the ring of functions of the projective plane and you impose the defining equations of C and C prime simultaneously so in algebra we can write this what we're supposed to be doing is taking the ring of functions of C and ten Tsering it with the ring of functions of C Prime over the ring of functions of the projective plane and I'll write a little P here to denote that this should all be happening really locally at the point P so this dimension what computes the intersection multiplicities and if you add them up then you get the product end times M this is a more sophisticated version of base theorem where the curves don't have to intersect transversely so you might ask at this point why are we considering only curves in the plane let's consider something a little bit fancier suppose that C and C Prime our varieties of dimensions a and B and suppose that they're contained in a projective space of dimension a plus B so generically you'd expect that they intersect in finitely many points now two sub varieties of projective space of any dimension you can still associate a degree so let's suppose that C has degree n and C prime has degree M well once again if C intersects C Prime transversely then the number of intersection points of C and C prime is just given by the product n times n it's that's exactly what you would expect what if C and C prime do not meet transversely well once again there's still a formula like this but you have to consider points with the appropriate intersection multiplicities but in high dimensions extracting the correct intersection multiplicity is not as trivial a matter as it is in the plane it's not just given by the dimension of this ring so understand why that is I want to consider for some degenerate cases of this sort of situation so let me suppose that C is given by a single equation it's given by setting F equal to zero and let's suppose that C Prime as a ring of functions are so what is our prescription over here for finding the intersection multiplicity what we're supposed to do is take the ring of functions of C Prime and then intersect by imposing the defining equation of C so that means we're supposed to take R and divide out by the function f and then count the dimension of this ring we can identify our mod F with the co colonel of the multiplication by F map from R to R and let's consider a special case which is not really applicable to our situation where R as finite-dimensional so if R is finite dimensional multiplication by F we can think of that as just some linear transformation on a finite dimensional vector space it's given by a matrix and you might think that generically if F is chosen at random that matrix is probably invertible it probably has nonzero determinant and in that case the answer is just going to be zero but of course this is not necessarily the case for some degenerate s its determinant might be nonzero in which case this thing would be positive dimensional and in fact this F sort of wanders over the space of possible n by n matrices the rank of the matrix can jump and the size of the co kernel that complex dimension and vary and this isn't what we want because of important thing about intersection multiplicities in some sense the defining property of intersection multiplicities is that they don't really change if you perturb the situation a little bit another way of saying why why in this case where we have a per L parabola and a tangent line we said this is intersection multiplicity two and we computed that algebraically but just geometrically you can understand why it should be intersection multiplicity two because if you poked the line a little bit you can make it intersect the parabola transversely and if you do that you see that that one point of non transverse intersection perturbs to become two points of transverse intersection so really these intersection of multiplicities are determined by the fact that they don't really change when you return the situation and in the case of transverse intersections the multiplicity is 1 but if we just want to extract these multiplicities by taking some kind of Co kernel of a matrix and Counting the dimension that doesn't look like it's very invariant under perturbations as you perturb a matrix its rank can change but the observation that I want to make in this case the generic were that the transverse case is where F has nonzero determinant and so this Co kernel is actually 0 so the answer we expect in this case is 0 and if we just think about the co kernel we don't necessarily get 0 but whenever an N by n matrix has coke kernel it also has kernel so in fact if we look at the the dimension of the toke kernel of F thinking of that is sort of the first approximation to what we want and then we subtract off the dimension of the kernel of F well it's just linear algebra that will get zero now of course you might say this isn't very interesting why am I going through all of this just to define something which is always zero the answer is well this is very special and very degenerate because I'm saying that R is finite dimensional but in more realistic situations when we're actually intersecting these curves it will often happen happen or it does happen when we're intersecting curves but R is infinite dimensional but the Kurylenko kernel multiplication by F are both finite dimensional so in that case this number is still defined and it turns out to be more interesting than considering either the dimension of this kernel or the dimension of the kernel individually it's somehow more robust or more invariant under perturbations but of course this discussion has all been very special to considering the case when C is given by a single equation but the idea that somehow there's this naive approximation to the answer and then there are section terms that you should add if you want things to be invariant under perturbations is actually very general so let me remind you review for you a little bit of algebra namely the existence of what are called torsion product functors so let a be a commutative ring and let M&N be a modules then you can define some new a modules which are called tour i over a of m with n and these have the property that tour 0 over a of m with n is just the tensor product of M&M and these higher tour eyes although i won't review the definition there's some kind of correction terms that's going to help you out when this naive tensor product is behaving badly on its own so in many situations considering the tensor product alone turns out to be a bad idea but considering all the tour eyes simultaneously will somehow fix that up so with this in mind let me state a a fancier version of bayes's theorem that works in high dimensions so have i erased so let's see in c prime be contained in projective a plus b space remember the C is dimension a and degree n C prime is to mention B degree M then n times M is equal to well it's the number of intersection points I'm going to assume I should mention that C and C prime meet and only finitely many points but I'm not going to assume that they meet transversely that's the whole point I can compute the right intersection number by adding up the intersection multiplicities that all the points of intersection but now the intersection multiplicities themselves so should credits therefore this probably the intersection multiple seasons the point themselves are given by Sara's intersection formula that they're given by a sum of contributions from these different tour groups so it's an alternating sum minus 1 to the I times the dimension of the group tour I of the ring of functions of C with the ring of functions of C product so what is this say the tour I over the ring o of projective space I'm sorry this is probably invisible to the camera and that probably didn't help but so what is this saying well if you look only at the term where I equals zero the tour group here specializes to just a tensor product and what we're doing is saying impo take the ring of functions on projective space impose the defining equations of C and C prime simultaneously and then count dimensions so in other words we recover this formula if we ignore the terms where I is greater than zero but of course we're not supposed to ignore the terms where I is greater than zero or we'll get something which is false so in general you should include the dimensions of all these tour groups so this is good on the one hand because we get a formula which we can use and we can compute with we want to know an intersection of multiplicity we can write down our rings and try and compute these groups and in some situations we might be able to do it but it's a little bit bad from the point of view of this very conceptual looking formula over here this very conceptual looking formula we want the product of the fundamental class of C in this fundamental class of C prime to be the fundamental class of the intersection so if C and Z prime meet transversely that's true in the most naive sense if C and C prime are plane curves and they need non transverse thing then it's true if we interpret the right hand side in a less naive sense that is we don't just want to take the set theoretic intersection of C and C prime but we have to take the scheme erratic intersection we need to remember the rings of functions on the intersection and of course we have to take that scheme structure into account when we compute fundamental classes we have to interpret this as something which measures the sizes of those rings of functions but even if we think about the scheme theoretic intersection of C and C Prime we can't interpret the entire right hand side of Sarah's intersection formula the scheme theoretic intersection will tell us about the tour 0 terms which appear here but these higher tour eyes are not reflected in the scheme threading intersection so what I'd like to talk about in this lecture is how to fix that problem namely it's not good enough to think about this intersection set theoretically or even scheme theoreticaly but maybe if we jazz up our notions enough then we can put ourselves in a world where this intersection really does have all the information that we need to extract the right intersection multiplicity so this is where bright algebraic geometry comes in so to get an idea of what that world might look like I'd like to begin with a very degenerate situation so let's go back to basics and think about the Xu's gira curves in the plane again and let's even think about a simplest possible case of facing new spirit namely two lines of the plane so if L and L prime are lines in the plane then it's the easiest thing in the world two lines in the projective plane always intersect at exactly one point except when they're the same and if LM l prime happen to be the same line then of course this formula over here will fail dramatically the intersection of the fundamental classes of al-adel prime that would be one point but that's nothing like the fundamental class of the intersection of L and L prime that's just the line elegant and that has the wrong dimension so what's going on at the level of ranked so once again let's put ourselves in the affine plane where the ring of functions is this polynomial ring in two variables over the complex numbers now if L and L prime are two lines in general position then we might as well take them to be given by the equations where L is given by the equation x equals 0 and L prime is given by the equation y equals 0 then what is the intersection well the ring of functions on the intersection where it wants to divide out by the defining equation of both lines and we just get the complex numbers and that's exactly what we want it's one-dimensional everything works very nicely but now let's consider what happens when L equals L prime so that both of these lines are given by the equation x equals 0 so in that case we're supposed to take a polynomial ring in two variables x and y and we're supposed to mod out by x and x and what do we get but we get a polynomial ring in one variable and it looks like the ring of functions on a line it doesn't look like it's the right dimension in particular we can't date its dimension it's infinite dimensional over the complex numbers and what went wrong well we've taken a polynomial ring in two variables and impose two equations but the two equations were not independent they're the same equation and in classical commutative algebra it doesn't matter whether you impose an equation once or impose it twice it's the same ring that you obtain so whatever world we need to be in it needs to be a world in which dividing out by X twice is different from dividing out by X once so how are we going to find such a world well of course we're doing algebraic geometry so we want everything to be commutative rings in the end but let's just think about a baby example of the same problem in the world of Seth suppose I give you a set which I'm going to draw on the board is a bunch of points and suppose in my set I have two elements a and B and I want to identify them well of course everybody knows that I can identify them by collapsing them to a single point I can make a smaller set in which a and B have been identified with one another but now if I'm a homotopy theorist I might do something a little bit different instead of making a smaller set where am ii have identified i might think about this set as a discrete topological space and i might instead make a larger space by adjoining a path from A to B now this is in some sense an equivalent construction because to a woman to be theorists this interval from A to B that might as well be a point it's contractible so the space that we make is actually home to the equivalence the space that we made with more naive construction what then is the point well the point is that this is a construction that it's interesting to get irate if you want to set a equal to B twice you instead of join two paths from A to B and this is now interesting because we have two paths from A to B we have a non trivial path from a to itself this space is not how much the people on two discrete space it's got a loop so this might seem a little strange to you I want to if it does I want to convince you that in fact this is maybe a good way to think about what homophobic theory is so you're all familiar with the idea that if you have various kinds of algebraic structure like groups and rings then it makes sense to give examples of this kind of structure by presentation so you give generators and you give relations so you can also do this when there's no algebraic structure when you just have sets that is you can take a set you can make a set by writing down elements those are your generators and then writing down relations collapsing some of them into the same point but now in home atomic theory there's this notion of a CW complex which just needs a space which is built by attaching cells so what is what does that mean well you start by it writing down some zero cells that is plunking down some points and then you attach one cells that is you join the points some of the points to other points by paths then you would join two cells which means you pick some loose saying this loop here and enjoin a disk that's bounded by that loop and then you're allowed to proceed and attach cells of higher and higher dimensions but the way I want you to think about this is you're really presenting a set by generators and relations so the generators are the zero cells the points the relations are the one cells the arcs that you add to join different points together but you're not just thinking about generators and relations but you're thinking about incredibly generalization of that where you also think about relations between relations likeness to sell here and relations between the relations between the relations and so on so that's how I want you to think about the world of home NOP theory it's a generalization of the world of sets where you have generators and relations and relations between relations and so on and what I'd like to get at is that this is an interesting thing to do not just with sex but also with other kinds of algebraic structures like commutative rings so drive does grant geometry is based on the idea of doing classical algebraic geometry but replacing commutative rings by something that simultaneously has the flavor of communitive ring theory and Homa tobe theory the basic objects that you work with are going to be something that sort of a space and a commutative ring simultaneously so there are a number of ways that you can try and make that precise but for this lecture we're only going to use the most naive of them we're going to consider topological commutative rings so a topological commutative ring is just what you might think it is it's a topological space equipped with the structure of a commutative ring for which the ring operations addition and multiplication are continuous so what are some examples well here's a silly example let R be a commutative ring then we can take our and endow it with the discreet topology and it becomes a topological commutative ring it is somewhat trivial way so one thing to keep in mind about this business of doing topological community rings is that we're really interested in these objects from the eyes of a homotopy theorist meaning we're going to regard to topological commutative rings as equivalent if they have roughly speaking the same moments will be tough that is if we have a map from R to s of topological commutative rings which induces isomorphisms on all homotopy groups then we think of aren't assets equivalents and this map although it might not be a nice more for sum of topological commutative rings it's somehow just as good as an isomorphism now a consequence of this is that it's really hard for me to give you any interesting examples I'd really like to give you a non discrete example here to get your mind around but it's difficult because although topological computer Tabriz occur all over the place in other areas of mathematics the ones that appear are not interesting from our point of view so for example if you're an analyst then you might like topological commutative rings like the real numbers or the complex numbers or various function spaces with Fermanagh elements but all of those are completely uninteresting for us because as topological spaces they're contractable so in other words the real numbers of the complex numbers those are equivalent to 0 in this world on the other hand if you're coming from algebra then you might see a lot of topological commutative rings because you get topologies whenever you take a completion so for example when you make poppy on an integers this has a p-adic topology but as a topological space the p-adic integers are totally disconnected there are no non-trivial paths and write down in the P adding integers so from our point of view that ring might as well be discrete so if these aren't good examples where are the good examples what sorts of topological community friends do we actually want to work with to do algebraic geometers well the answer is that they arise from doing certain constructions so let's take for example this polynomial ring functions on the plane see you join x and y and I'm just going to give it the discrete topology and now suppose that so here's a picture of it just a bunch of points one of which is Y and one which is 0 and now suppose I want to intersect the line with itself in other words I want to impose the equation y equals 0 twice so if this was just the world of sets rather than the world of rings I just attached two paths from 0 to Y but the problem is that the space that I make by doing that certainly no longer a ring however I can always take the sort of three commutative ring that's generated now by those two paths so you get a lot more junk in there because we can take sums and products of any points that lie along these paths so the thing that you get is a big mess and it's not something that you can really visualize geometrically at all however it's something that has good formal properties it's good for doing algebra as we're going to see in a minute so let's return to this formula over here this naive formula for the intersection multiplicities which only counted the dimension of the toward zero term so what I'd like to say is the difference between this naive formula and Sarah's embellishment of this formula Sarah's inclusion of these higher tor terms the difference is really that in this naive formula you're taking the tensor product in the wrong world you really ought to be working not in the world of ordinary commutative rings but in the world of topological commutative rings so so let's consider this situation so this universal topological ring see you join your price around I wonder both look like Co join X so you told us that so I can see why you'd want to join those two arcs yes lieutenant you could build a ring yes there's that larger yes oh I'm sorry so the question is what is the HOH motoki type of the object that I get when I do this massive construction of adjoining these two paths and then taking the free commutative algebra that they generate and the answer is that the ho motoki type is not as bad as you might think so let me let's let's give this thing a name so this maps to our by sort of sending Y to 0 twice so R is this topological commutative ring that's obtained by joining those two paths what do you get well hi zero bar is just the ordinary commutative ring that you would get by dividing out by Y twice we're dividing out by Y once equivalently so this ring just looks like C you join X by one of our well there's a canonical element in PI one of our which is given we have now two paths from Y to 0 so we have a non trivial loop from zero to itself but PI 1 of R is also a module over PI 0 of R in a natural way so if I let epsilon denote that natural path that we can write down it turns out that PI 1 of R is a free C a joint X module on the element Epsilon and there are no higher home it'll be groups above 1 so even though it looks like we did a completely horrendous construction and even though we can't get our hands on this thing geometrically at all for a harmon to be theorist this is not bad at all this is a very simple space which has a bunch of connected components and each connected component is has only pi 1 no higher home to be groups so let me return now to this formula over here and let's try thinking about what would happen if instead of taking the tensor products in the world of ordinary community brings we took the tensor product in the world of topological community links so so consider the ring of functions on the curve see the ring of functions on the curve C Prime and the ring of functions on the or they're not curves actually they're varieties of higher dimensions a and B sorry and take their tensor product over the ring of functions on projective space of dimension a plus B now I'm going to think of these things as ordinary commutative rings regarded this having the discreet topology what I'm going to use the right notion of tensor product construction in the world of topological commutative rings so this is something that a lot of algebra is would denote by putting a little L above the tensor product side the reason is this is the Left derived functor of the tensor product operation now usually those left-right functors are done in the context of home illogical algebra so you imagine that these guys are modules over over this ring and then you perform some kind of construction which involves resolving these modules by free modules but it turns out that this construction is equivalent to doing some kind of construction the world of topological modules and you can just as well work in the world of topological modules that happen to have ring structures and that's a little different from the usual homological algebra because the category of algebra over this ring is not an additive category you can't add morphisms together so this is what some people call non abelian homological algebra or what some people call home a topical algebra something that very much has a flavor of homotopy theory in it but anyway what I want to say is that if you take the tensor product in this world of topological commutative rings you get a topological commutative ring which I'm going to call R and now what does our look like well once again so let me erase this to avoid computing so now let's contemplate this remar well first of all if we just look at the level of connected components if we forget that we're working in the world of topological commutative rings then we just get what we would expect from ordinary algebra we just get the naive tensor product of OC with OC prime over o a projective space and now you might guess what happens the IPO motoki group of the Ring R is exactly what appears as the eye correction in stares intersection formula it can be identified with tour I of OC with itself with R's with OC prime over O of projective space so working in this world of topological commutative rings really solves the problem that I mentioned earlier that the higher terms in series intersection formula cannot be interpreted as invariants of the intersection of C with C prime to solve this problem what we have to do is intersect C with C Prime in a more sophisticated version of algebraic geometry where commutative rings have been replaced by topological communities so let me give a few definitions to make you a little bit more comfortable with this world and then state a generalization of Bayes theorem which will make it look very nice so suppose that R is a topological commutative ring so what does that mean well first of all we could make an ordinary commutative ring by collapsing all of the path components of our to a point high zero of our is just a commutative ring an ordinary algebra and what I want you to think is that PI zero bar is the underlying commutative ring of our the difference between R and PI zero bar as a discrete brand is that our might also have some higher homotopic groups and those higher or multiple groups might carry some interesting information that's sort of the whole point of this business but somehow PI zero contains the most interesting or the most fundamental information so how are we going to do algebraic geometry with these topological community rings well let me review for you the definition of a scheme the most basic definition in algebraic geometry and then explaining how to generalize it by inserting topological so definition a scheme it's a topological space X with a sheaf of rings o X such that the pair x o x locally looks like speck a Oh Becca we're here a is a commutative ring so what's the derived algebraic geometry version of this definition we insert the word derived before scheme and now everywhere that we see the notion of commutative rings appearing we replace them by topological commutative bridges so [Applause] there are a lot of things to remark on here one of them is you have to ask what I exactly mean by a sheaf of topological commutative rings because it's not what you might think interpreting that most literally but let me just comment on what's most important which is how do I take speck in a toast met a when a is a topological commutative ring and not an ordinary computer great and the answer is it's not nearly as bad as you might think so let a be a topological commutative ring then speck a as a topological space is just defined to be speck of pi naught obey the spec of the underlying ring so let me remind you that's the set of prime ideals in the ordinary ring of Pi not of a and world or we're going to give speck a its usual topology does risky's apology and you recall that the zariski topology has a basis of open sets u sub F namely the set of prime ideals but you cannot contain the element F and here F is allowed to be anything that lives in PI now today so as couple of those places these drive schemes are just like schemes they look like Zoo risky spectra of rings what's different well what's different is the structure sheet it's no longer a sheaf of commutative rings it's now a sheaf of topological Commuter friends but it's not very different if you consider Oh Becca on one of these open sets use of that well this was ordinary algebraic geometry I would just take the ring hay and invert the element F and now let me simply remark that this is a construction which makes perfectly good sense in the world of topological commutative rings there's a good construction point for inverting an element so this is the definition of a derive scheme what does the derive scheme look like well locally it just looks like an ordinary scheme the only difference is that its structure sheaf has some higher homotopy groups which are scurrying some kind of mysterious extra information and the extra information that they're carrying is precisely the information that you need to extract the higher terms in Sears intersection formula so what happened is now also what should I call this improved Bayes theorem you can extract a version of a Zeus theorem which has essentially no hypotheses suppose that we have two smooth varieties in a projective space any dimensions that you like well their intersection doesn't have to be smooth and in the case when the intersection is is bad we really have to intersect them in this world of derive schemes not in the world of ordinary schemes and you also have to jazz up what you mean by the fundamental class of such an intersection so there's a generalization of the theory of fundamental classes to this world of derive schemes and the things which you obtain are usually called virtual fundamental classes but so you have to interpret this in the sense of virtual fundamental classes but provided that you do that this formula is valid with essentially no hypotheses on the way that C and C prime intersect they don't have to intersect transversely they don't have to even intersect in the right dimension they can C and C prime can coincide for all you care if C equals C prime then this thing interpreted classically is certainly not going to give you the right answer but as we saw earlier this derived algebra geometry working with topological commutative rings knows the difference between imposing the defining equations of C ones and imposing them multiple times so this idea of derived algebraic geometry has several applications this improved basis theorem is only one of them the applications in both directions so here's an application of Roma topi theory to algebraic geometry it is we can extract out better versions of some classical theorems in algebraic geometry and in my electric tomorrow I'll be talking about some applications which run in the other direction namely applying ideas from algebraic geometry enriched in yes [Music] so the question is to what extent do the virtual fundamental classes that I mentioned agree with the notion of virtual fundamental class as it exists in for example gromov-witten theory and the answer is that they're the same notion that these spaces that appear and grow up would in theory that do not have the expected dimension actually are the underlying schemes of certain natural derived schemes that you get by thinking about the problem in the world of derived algebraic geometry and that the virtual fundamental classes that appear in groom of wouldn't theory are precisely the same virtual fundamental classes that appear when you go through the process that I have a not described like check [Applause]

Channel: Márton Vaitkus

Views: 6,912

Rating: 4.970149 out of 5

Keywords: lurie, algebraic geometry, math, homotopy, derived geometry, homological algebra, intersection theory, serre, bezout, tor, derived functor

Id: htTL0VvfsvM

Length: 53min 33sec (3213 seconds)

Published: Thu Nov 16 2017