 very much. Let me start by thanking the organizers. It's a tremendous honor to be part of this. And I want to wish Ofer happy birthday. So I want to talk about log geometry, which, of course, is a subject that Gavir has had a tremendous influence on. In particular, I want to talk about how to think about Hockschild and cyclic homology for log schemes. That's my title. So let me give a little background. So let me start. So let me consider a ring homomorphism. I think in this area, one has to people consider non-commutative rings and things like this. Let me just stick with commutative rings with unit for now. Normally, you don't have to say that, but OK. All right, so what's the Hockschild complex? So it's just take the tensor products like this. And there's some formulas. Let me just write one of them here. So this is tensor product over R. And there's, well, you can write more complicated formulas if you want for the higher terms. But anyway, so you take the normalized complex and you get some complex. And it's a complex, because I'm in the commutative setting, complex of A modules. By the way, this is a simplification object, yes? So you get a chain complex that is the indexing of the differential laws of the grid by 1. Yeah, so I mean, there's a homological versus a homological issue that's sort of pervasive. Is that your question? Yeah, so. And it looks like a complex for simply commutative. Well, that also, yeah, I mean, if you put in the. It would be called to sort of change the order. I think it's OK, but. Well, you can define whatever you want. No, but the option, the option thing is much older. It goes back to the 50s, because that, I don't know. Yes. And it's for bimodules of a non-serialism, associated with non-serialic relative set. Yes. So this is a special case of this, where a is a bimodule of a set. Yes. And I, OK. Yeah. But anyway, this is a complex, well, so viewed comologically, it's a negative, yes? You can either view it as a comological negative degree. Sorry, what? You put the star up, so it must be comology. Well. Yeah, that's bad notation. Yeah. So you should, I mean, homological indexing, I think is the standard. OK, so let me, the Hochschild homology is, and now I will follow the, yeah, OK. I'm sorry, this. So you take the homology of this complex. And I, yeah. OK, and now you can also think of this as taking the tor over a tensor a of a. At least when the a is flat over r. Well, I think if a is not flat over r, yeah, then I have some, I can put it, I think people, some people want to put a derived there or another stone. So, yeah. Yeah, yeah, yeah. So if it's not flat, then I have to do more up there, yeah. OK, so this was generalized to schemes by Weibull and Geller. So let me set up the notation. So let me say s is spec r. And then you have x over s. It's an s scheme. And then, well, you can define, you just take for any affine open inside x. You can send that to this complex. Maybe I'll omit that. OK, so this is a complex of pre-sheves. This doesn't localize well. So you have to be a little careful. But then you can sheafify it to get some complex of sheaves. And then it will be different from writing structural sheaves. Oh, x for your hand up there. Why don't you take 0x and 0x over 0x and 0x. Just do it, well. In the flat case, then do we play that process by doing the opposite? Repeat the question? You're saying that that couldn't be 0x and 0x over 0x. So that's what you put here. Yeah, I mean that's. So it'll be important for me that I have this actual complex. So, well, it'll. It's good. You just lose the risk of such modification. You can lose the risk. You can also work with the etal topology. In other words. Yeah, yeah, I mean. Anyway, the homology localizes well anyway. The homology localizes well. So that's not a big issue. Yeah. All right, so then you define the Hock-Schild homology of x over s. Well, you have the sheaves. So they write it like this. Is that better? And then we get to the convergence issues like in Schultz's book about unbounded complexes. Yes, there's. Yeah? So unless, so of course, you can use plans and style definition. But anyway, as it was, there are some cases where people use things which are not Python-complete. I don't know if this is a right box as sophisticated as this, but the question is, which sense you take the etal homology, let me just plan this and try it out. So I think here. Maybe you don't have a problem here because. The homology sheaves are quasi-coherent. Yeah, yeah, yeah. OK, and then, well, there's also, I'll talk more about this towards the end, the cyclic homology. But let me write sort of the axioms up on this notation. OK, so there's a bunch of properties. And I'll write them once. And then, I'll say something about that later. But let me first go through some basic properties that I want to be true because then I want to talk about log schemes and some stacks. And so yeah, so we'll get there. OK, so the first point is that this is a commutative graded commutative algebra. Yeah, well, the sign convention is like for the Doraum conference. Yeah, but the element of it is an odd degree or an element of odd degree squared zero. I don't know. Is that yes? OK, yeah, thank you. Who said yes? Schultz. OK, so the second is that there carries a differential called the con operator. And with this, so this is a commutative differential graded algebra. Let me continue over here. So yeah, so then the next point is that if you look at B0, this is an R linear derivation. And so that by the universal property of the Doraum complex, you get a map here like that, which is an isomorphism. Morphism, if x over s is smooth. This here, this is usually referred to as the Hockschild Cost and Rosenberg isomorphism. All right, then there's some other things. So there's something called the SBI sequence. And I haven't told you what cyclic homology is, but no. And again here, so this is homological notation, so it's the m minus 1 up here. And finally, the last property, which HCM, HCM minus 2. And let's say x is apine, x over s is smooth. So then there's a spectral sequence, which I'll just draw in a picture. This is an E1 homological notation. And this converges to the cyclic homology. And if R is a Q algebra, then you have so-called lambda operations, which makes us degenerate. So this degenerates. So x is smooth here. And R is a Q algebra. And R is a Q algebra. Then this gives, degenerates and gives the cyclic homology. So you have the so-called what? Lamped operations. Lamped operations? Yeah. No, R is my base. S is spec R. So then this is, well, you read off from the picture there as you get differentials, mod boundaries. And then you have a sort of direct sum, i is at least 1. And then hm minus 2i d'Rom x over s. So it degenerates means add E1? Add E2, I think. You go to the next page. This is the one page. And that's why I draw the spectral sequence like this, because then you see. In the SBI sequence, why is it called the B SBI? Well, one of the B's and one of them is i of these maps here. And I can't remember. The B goes in your convention from hh to itself. And you don't have what you have here is hh going to hc, going to hc. There's no map points. So in the definition of the cyclic homology, there's a map, which is a B. Yeah. If there's time later, I will. But, well, I mean, it's a double complex. Yeah, it's a double complex with, yeah. Yeah, it's an explicit double complex. Well, OK, I think that's, well, I can do it. There seem to be a request, so let's see. So what is the cyclic homology? Give the flavor at least. I mean, I'd have to write out all the maps to really say what it is. But, yeah, yeah, so OK. So here's, you have some picture like this. So you just put the Hock-Schild complex in a double complex. So this is just copies of the Hock-Schild complex with shifts. And then you have maps here, which I think. I didn't label them here, but I believe they're usually called B. And they're built out of the cyclic. I mean, there was the B of the Carnot. Yes. It does, yeah. So, I mean, where did this thing, the B, wherever I drew it, comes out of this extra structure, the cyclic structure. And here you, I mean, this just comes from the permutation action on the self-tensor products. So this isn't necessarily lower on that pattern? Yes. And then you have to take the total complex and. The total complex in the sense of the finite in each total degree, or where we have to. Yes, so I guess the way I like to think of this is as an object in C of mod R, sort of indexed by the net. So think of this as a projective system, I think, in sort of this direction. Right. And then you have the sort of Huxbill terms sitting like this. We can draw them, that is to P2 times. I mean, where are the degrees? The degrees. I think, yeah, I'd have to, I'll get it wrong if I try to. I mean, there's a sign. Yeah, you mean when you take the total complex, is it? Let's see, so this way, is that what you're saying? This way, this way, yeah. OK, that was good. Thank you. It's fine, I think that was a great. Yeah, you mean in each term, yeah, yeah, yeah. OK, so there's no further way to take dykes out of that? No, you don't have to, but you still have to worry about the issue of taking comology. And now you've lost, this map is not a linear, so you have to, the horizontal maps. All right, yeah, so let me skip. So what I want to explain is how to do this for log schemes. Also, the other maps are only linear. All the maps are linear. So this is a double complex of our modules, the way, yeah. OK, so in general, I says to log schemes, which is actually, so I guess let me make a remark here, which is there's an alternate approach, which is due to Rognis, which is, I mean, this is somehow this story is closely related to the story of the cotangent complex. And for the log cotangent complex, there's kind of two things you can do. There's a Gavir cotangent complex, and then you can do this thing with stacks. So the Rognis approach is more like the Gavir cotangent complex, and I'll explain sort of the stack approach. So let's see. So let me just make a sort of one small remark just to get started. So let me just, so you can think about this for the log point. So this is spec k, k star, a copy of the natural numbers, just mapping to 0, spec k, k star. And if you write out, so the Dharam complex in this case is just k Dt over t. And so now you can think about it if you have a differential grade of k algebra, that B. Well, you really don't have enough information to talk about how to get a map there, because this is just a 0 map. So up here in 3, more work is needed to make sense of the universal property. So this may be a trivial remark, but I think one I found helpful in thinking about what should be the correct generalization. Let's see. So now actually, I'll say very little about log geometry. There's sort of a dictionary between log geometry and stack. So the approach is to define HH or the cyclic homology S for some morphism X to S, where now I want X to be an algebraic space. And this S will be an algebraic stack. Let me give an example of that. Again, sort of an easy example, but it has, well, anyway. So let's take a smooth group scheme over field k. And let's think about the trivial torsor, which corresponds to a map from spec k down to the classifying stack of the group. So then what you get here, the Diron complex. Well, it's k, but a 0 map to the dual of the Lie algebra. And then you have the second exterior power of the dual Lie algebra and so on. And this is what's, I think, classically called the Chivalier-Eilenberg complex. I think you can do this for representation, too. You can take the affine space defined by the representation quotient out by the group and go down. And now the main point I want to make about this is that this involves the Lie bracket. And so even, OK, you may appear for the log point. Well, you can go to degree 1, and you have the notion of a log derivation. But here you have to worry even about going further up in this complex. So spec k to bg is a random start. Yes. Which is like the reverse of G-forces. So it is maybe smooth. But why? How do you want the relation? So it is mostly you just have a, OK. And you claim that the random complex is identified with this. Yes? Because you can base it. Yeah, so maybe I should. OK, so let me at least say why the differentials. Yeah. OK, this is like the complex of invariant differential. Exactly, yeah. All right, so you have this Cartesian product. And then so you see that the pullback of the differentials here become the differentials of G. And then you have the Grubaction. So the differentials of this over that is exactly the invariant differentials. And from that, you can also read off what this complex is. Yeah. Yeah. So sort of a side question here is how to think about is the universal property complex. All right, so in the log case, we have a good answer. It has involved log derivations and so on. All right, so maybe I'll just say a word about the dictionary. So if I have S, I have a fine log scheme. Well, then you can think about the stack. What does it do? An object over some x to f. Some fx to s here is a fine log structure on x. And this map f upper star is sort of an old story. But let me write it out anyway. So that's a fiber, which means that, so let me sort of have a better board work. But anyway, so that's a continuation of the two columns. So that means that if you have a map like this of log schemes, that's the same as a map to the stack. And then the Huck Shield homology will be just defined to be the Huck Shield homology of x over the stack, and similarly for the cyclic homology. If I don't see in the stack context, you have to. Yeah, so now that I've looked at the rest of the talk, yeah. OK, so maybe it's better explained by example. So let's say v is a DVR, pi is a uniformizer, and let's say we have f from x to spec v with this regular closed fiber divisor with symbol of normal crossings. And let me write it as d1. So then the basic thing to study is then the stack here, which is you take the local model, this pi, and then you push it out by the torus. And here the torus acts i times xi for i 1 up to pi minus 1. And then you take the diagonal inverse on the xr. OK, so you take this little sort of torque stack, and you, so this is not quite the log s, but it's a tall over the other stack. So it turns out you can just work with this. And so if you take, for example, the ROM homology of this over that, you get the log to ROM complexes. But here, are you looking at the number of 1 points and the r components of the component meeting this point? Yeah, so OK. Well, I didn't say that the closed point has to go to the vertex of this torque. But yeah, this is a local picture in any case. And so acting kind of smooth over this kind of local model. Right. Well, oh no. I mean, no, you don't need to do that. I mean, if you think about a1 mod gm, to give a map to this stack is a line bundle with a map to the structure sheet. And so if you have a divisor on a scheme that's equivalent to giving a map to this a1 mod gm. So even though I don't have to choose coordinates, but I do need it to be simple normal crossing, so I can choose a line bundle for each divisor. But you just add from x, which I believe you didn't say that add is generally smooth or flat. You just add. Yeah, OK. I do mean it should be locally like this. Yes, yeah, yeah. It should be locally. That should be local coordinates. It should look like this. It should be something smooth or a local model which is like this. That should be the local picture. And if I put simple normal crossings, then I can look at each divisor and look at the line bundle given by that, and then that gives me a map to this stack. To this quotient stack. No, but what I worried about could be divisors in many places. So the number of divisors could be directly in the dimensions. Oh, yeah, OK. Just looking at the ones that occur locally. So are you just? But I think that's OK, because I'm killing it off with the, you know, let's say I'm at a point where one of the x is not zero, then it goes away because I'm acting by the torus. It's pi, yeah. And x over v is the generic fiber smooth? So here, yeah, so the generic fiber is smooth. So it really is in local coordinates, like this, yeah. I should flat, you know, there's missing adjectives. All right, OK, so let me now go to sort of the general part here. So I guess to define, so let me start with, so s, I'll still work over in ring r. There will be an algebraic stack with finite diagonal, with quasi-compact diagonal, sorry. Yes, I think I want, well, I don't necessarily want the diagonal to be separated, but I want the diagonal to be of finite type. Is that what you, I forget the, so the diagonal, I think, for this thing here is not separated. I believe it's, the diagonal is quasi-separated. Of course, it's finite type because they are working. Yeah, so you're using the very general definition of stacks out depending on the reference, but like in the stack, but with plus-dose condition. Yes. OK, and then I have a map like this, where x should be a quasi-compact, at least, algebraic space. OK, so, well. So let's see, so options for defining s, and again I'll sort of carry this, even though I didn't really define it, let me carry that forward. OK, so I have three options. I'll get to that when I need it. Option one, well, it's just to do descent. What do I mean? Let's see, I have, do I have 23 minutes? Is that the, yeah, OK, so, well, the point is simply that this complex is funtorial. So if I have x over spec r, spec r prime, f, x prime, then you get a map, g upper star, c, x over r, to c, x prime over r prime. And so you can choose a hypercover. You're trying to define the kind of name without derived terms of what you can do. Yes, let me, let me, yeah. Just this one, because for the other one, we have extra choices in which kind of machine you use. Yes, yes, yes. Simplification resolution of some out here. Yeah, yeah. OK, so let's take a hypercover with the sn affine scheme, say. And then you have your x, and you take your x dot. And you just take level by level, c, well, all right, x dot over s dot. This is a complex on this simplitial scheme with restriction to xn, the complex c, xn over sn. And then you have a comological descent problem. But it turns out to be OK in this case, because you have quasi-query or chromology sheaves. And so this descends to some complex h, h, x over s on the tau side of x. So OK, so that's, then you have to check independence of covers and so on. This is like a plot of smooth. Yeah, let me do a smooth hypercover. Probably flat is OK for that one. And descends to, and so this seems not, I think this descends thing without the smear quasi-compact and quasi-separate, the smear proof thing works because, of course, you work with such a black space you are supposed to develop a quasi, but then you just need the comological descent for the sub-qc and this should work, because as we said, it's kind of not complete. It's certainly, yeah, I mean, it would not surprise me if it works in general. With quasi-coherent chromology? Yes, yes. So this, let's say, lies in d, quasi-coherent. No, is it the complex of all x-materials? Only the comologies are quasi-coherent. i's plus is, because the complex itself, the option complex itself, didn't it? Oh yes, so you have to work, I mean, yeah. But if you think about the unbounded comological descent, if you put some conditions on the h i's, yeah. So it is not in d of all x, it is d of all x. I guess you want to put r like that? And the one-coherent is all over all x. It's comodations, that's what it is. OK, so that's option one, and one could do that. Tell me what is g in this? Expand u to x, d dot. Thank you. Thank you very much. All right, there's another approach in the case when, now the shadow is back, 2. If the diagonal of s over r is affine, then you can sort of copy the case of schemes. So let me write it this way. In the diagonal of x over s? No, of s over r. Yeah, the stack over the base of my s is spec r. And then x is arbitrary. Yeah, so let me write s, I'm right. I keep writing this because if I want s to be a log scheme, I don't know. r is not so good notation for a log scheme, but s could be anything. Yeah. OK, so then you can think about these fiber products. So this is like taking the tensor part, right? So if x is affine, and you have various maps with the i's. You can write formulas if you copy down the formulas for what you do with rings. And then if you look at an affine x, then you have c a over s. You just define this to be the sections of xn o xn. So if x is affine, if x is that, and then you get a complex using these maps like this, and then you sheafify. So OK, so now you have this. So now for x arbitrary, you can start looking at an affine over x. Do this construction, get the complex, and it's functorial. And so then you can sheafify it to get a complex cx over s for the etal topology. So this? Cx is a classical cycle complex. So it'll be, yeah. In the case of things. Yeah, I mean, OK, so let me do an example. So this is closely related to the log diagonal. So related to the, so let's just take a1 to a1 mod gm. So the quotient map. And then xn, in this case, is a1 with a bunch of copies of the torus. And so you get some complex in this case, kt u1 plus or minus ur plus or minus, and so on. So it has this nice form, and it goes on. So this is sort of the affine. In this local kind of situation, it's given in a way similar. And so in fact, this gives the equivalent answer, gives the equivalent complex to the first approach, yeah. And now I'm going to go over here. So I should say, I mean, the end conclusion is that there is a theory where we have the axioms that are listed above in either the log setting or sort of the setting of an algebraic space to a stack. But let me go back to this question of the universal property of, so let me go back to universal property omega dot x over s, which is then also the third approach. So the point here is to think about, so let me get my notation, right? So we have a fibered etal topos. So I'll write this as x over s, so I'll write it this way. So we have the category of smooth schemes, smooth, maybe affine, there are some choices there, s schemes. And for each s in here, so that's a morphin, maybe I should write it this way, s to s, I can think about the fiber product, which I'll write as x sub s. And you can think about the etal topos on that. And that's what's called a fiber topos in the sense of sj4. Anyway, so I mean, it's nothing fancy really. You just take, for each scheme, your base change, and you have the etal topos. So a sheaf on this thing, so a sheaf, so well, OK. And then let me write x over s etal for the total topos. So what is it? It just means it's a category of systems where for every s to s, you have a sheaf of fs on the xs etal. And if you have a morphism, s prime, maybe g to s over s. And then you have a map from g, well, base change up to the xs, d inverse fs to fs prime, x prime. Plus, they satisfy some co-cycle conditions. And the target identity goes identity, which is omitted in some places in the book, right? OK, yeah. OK, I guess I should write that. Plus, identity, OK, yeah. All right, so that's that. Yeah? x etal, we need to factor over where? Well, there's n plus 1. So maybe it, I mean, maybe let me clarify. Let me do 3, just as an example. So n is 2. And so now you have sort of 1. So I mean, you think of this categorically. So you have one object, and then you have an sort of arrow, which I'll write as u1 to sort of u1 x1. And then you have a second arrow, which maybe is u2, which is given by unit 2, u1, u2x, or maybe x, right? And so you have sort of the object plus 2 units. In topology, is it a passive processing pool? Yeah. Or not the space in that action of the pool. Right, right, yeah. That's exactly the thing. OK, so well, so now we have this thing, x over s. And it has a sheet of rings, which is just for each xs. You have O xs. It maps down to the etal topos, x etal. And it maps down to the etal topos of the stack. And I'll just write it like that, OK? And so, right. And so the point is, so let's call this r. So there's a little lemma, is that our upper star, our lower star, define an equivalence between quasi-coherent sheaves on x and quasi-coherent sheaves on this topos, which just means it's a sheet of O modules, such that each fs is a quasi-coherent sheave. And the transition maps are isomorphisms. The transitions are in the tensile language there. With tensoring, with tensoring, yeah, of course, yeah. And that's, again, sort of an exercise in flat descent. But the point is that, so in the Dirom complex, we have these quasi-coherent sheaves. But the maps are not, of course, linear. And so where should the maps live? And so one answer that at least works is to think of them in here. So we can talk about, well, let's see. Let me write it over here. So we have a category of quasi-coherent commutative differential graded algebras in sort of x over s. So that means, maybe I should give this a name, pi. So I mean, one reason to work with a small etal site there is that you can then talk about pi inverse. This is sort of well-behaved inverse of the structure sheet here. So I'm only considering the smaller etal site of the x of s's. And so on each of those, I have the pullback of the structure sheet on s to x of s. So I don't have, I mean, I shouldn't have, I shouldn't really have, yeah, maybe I shouldn't even have discussed the Lisa-Tal topology here. But anyway, I mean, the point is that I'm just thinking that here, I'm just using it as a category. It's a fibered category. And so in this kind of language, for each s, I have the pullback of os to x of s. And that maps to o x. Yeah, so I can talk about quasi-coherent commutative differential graded algebras in this category, which I mean that they're, with respect to this map of rings in the topos. And the individual terms are quasi-coherent. And then this has an initial object. Just take the wrong complex. So the differences are a lot, but it's not enough to go. Right, so the point is that even though the notion quasi-coherent is the same, but there's more structure to the differentials than just looking at them as maps on. So think of the case of the point mapping to bg. So the d is 0. And the second map is this map you get from the lee bracket. So where should that, where is that sort of the universal object? Well, as was pointed out, what you're really doing is sort of taking invariant differentials from something up in some higher level. So this is some formal way to formalize that. But you're speaking about differentials like the algebra consisting of quasi-coherent shifts in your sense. And all the structures are quite differentially respecting their own. No, no, differentials not. So the differentials are pi inverse os linear. The terms are o x over s. Yeah, like the Dirac complex, right? That's what it is. It's just a Dirac complex in a, yeah, yeah, yeah. And so I'm maybe, I think I'm essentially out of time. So maybe let me just say at least this is a part philosophy, but the punchline is really, I think, one approach to this is just to upgrade the con differential to get a structure, which is b, in this commutative dga. And I think I don't have time to say much about the cyclic homology, but you kind of do this. Everything is defined on very explicit complexes. So once you're in the right topos, you kind of just follow along. Eventually, you will find some more. So I'm comparing this to this one. Yeah, so all of my properties, one through five, hold it. Log smooth is fine, yeah, yeah. So I didn't want to rewrite them, because I think that would be redundant. This gives another definition of the. So this is the third definition of the. Equivalent to the first. Which is equivalent to the first two, yeah. And then you have the sequence, and you can calculate some examples. Yeah, so, well anyway, so I think I'm out of time. But I'm happy to show people examples. Are there any further questions? Look, so I'm using the non-jump tool to get the omega. Yes, so I think this is a bug of, yeah, yeah, yeah. Yeah, yeah, yeah, yeah, no, I mean. Yeah, this is sort of. Yeah, in theory we are worrying in terms of always linear. But there is a finite answer like the uncompact. Yes, and they are only. Yes? But in characteristic T everything becomes tricer. The non-contact is almost weaker. So what is the store in characteristic T? Well, I haven't thought about it. I mean. You still have the second context, you still have the uncompact, and then you have Cartier. Yes, yes, I mean, there was this very nice paper of, I think it's Kaledin on Dillini-Luzi in characteristic P for cyclic homology, I think. So I mean, one motivation here in the large context is that you want to compactify and then reduce mod P and do things like this. But I don't have anything to say, but no. So there are no further questions. Let's thank Martin again.