The homology of GL_n(F) and SL_n(F) is studied, where F is an infinite field. Our main theorem states that the natural map H_4(GL_3(F), k) --> H_4(GL_4(F), k) is injective where k is a field with char(k) \neq 2, 3. For algebraically closed field F, we prove a better result, namely, H_4(GL_3(F), Z) --> H_4(GL_4(F), Z) is injective. We will prove a similar result replacing GL by SL. This is used to investigate the indecomposable part of the K-group K_4(F).
We improve the homology stability range for the 3rd integral homology of symplectic groups over commutative local rings with infinite residue field. As an application, we show that for local commutative rings containing an infinite field of characteristic not 2 the symbol map from Milnor-Witt K-theory to higher Grothendieck-Witt groups is an isomorphism in degrees 2 and 3.
For an infinite field $F$, we study the kernel of the map $H_{n}(\mathrm{GL}_{n-1}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big]) \to H_{n}(\mathrm{GL}_{n}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big])$ and the cokernel of $H_{n+1}\Big(\mathrm{GL}_{n-1}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big]\Big) \to H_{n+1}\Big(\mathrm{GL}_{n}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big]\Big)$. We give conjectural estimates of these kernels and cokernels and prove our conjectures for $n\leq 4$.
We prove without any assumption on the ground field that higher Hochschild homology groups do not vanish for two large classes of algebras whose global dimension in not finite.
For a spectrum $X$ represented by a special $Γ$-space $Y$ via the Segal machine, we give an elementary formula computing the homology groups of $X$ in terms of $Y$. Both the result and the method of proof are essentially due to T. Pirashvili, but the end result does not appear explicitly in his papers.
Let K(n) be the nth Morava K--theory at a prime p. This paper is a thorough study of questions like the following: to what extent does the K(n)--localization, or the K(n)--homology, of a spectrum X determine the K(n)--homology of its 0th space X_0? Our methods combine techniques from modern homotopical algebra with chromatic homotopy. In particular, we use the telescopic functors of Bousfield and the author (dependent on the Nilpotence Theorem of Devanitz, Hopkins, and Smith), as wel as Topological Andre--Quillen Homology and Goodwillie calculus in nonconnective settings.
We prove analogues of the fundamental theorem of algebraic K-theory for the second and third homology of SL_2 over an infinite field k. The statements involve Milnor-Witt K-theory and scissors congruence groups. We use these results to calculate the low-dimensional homology of SL_2 of Laurent polynomials over certain fields.
We prove that for any infinite field homology stability for the third integral homology of the special linear groups $SL(n,F)$ begins at $n=3$. When $n=2$ the cokernel of the map from the third homology of $SL(2,F)$ to the third homology of $SL(3,F)$ is naturally isomorphic to the square of Milnor $K_3$. We discuss applications to the indecomposable $K_3$ of the field and to Milnor-Witt K-theory.
This note explains how to prove that for any simply-connected reductive group G and any infinite field k, the inclusion of k in k[t] induces an isomorphism on homology. This generalizes results of Soule and Knudson.
We prove homology stability for elementary and special linear groups over rings with many units improving known stability ranges. Our result implies stability for unstable Quillen K-groups and proves a conjecture of Bass. For commutative local rings with infinite residue fields, we show that the obstruction to further stability is given by Milnor-Witt K-theory. As an application we construct Euler classes of projective modules with values in the cohomology of the Milnor Witt K-theory sheaf. For d-dimensional commutative noetherian rings with infinite residue fields we show that the vanishing o…
We study the twisted K-theory and K-homology of some infinite dimensional spaces, like SU(\infty), in the bivariant setting. Using a general procedure due to Cuntz we construct a bivariant K-theory on the category of separable σ-C^*-algebras that generalizes both twisted K-theory and K-homology of (locally) compact spaces. We construct a bivariant Chern--Connes type character taking values in bivariant local cyclic homology. We analyse the structure of the dual Chern--Connes character from (analytic) K-homology to local cyclic cohomology under some reasonable hypotheses. We also investigate th…
The third homology group of GL_n(R) is studied, where R is a `ring with many units' with center Z(R). The main theorem states that if K_1(Z(R))_Q \simeq K_1(R)_Q, (e.g. R a commutative ring or a central simple algebra), then H_3(GL_2(R), Q) --> H_3(GL_3(R), Q) is injective. If R is commutative, Q can be replaced by a field k such that 1/2 is in k. For an infinite field R (resp. an infinite field R such that R*=R*^2), we get a better result that H_3(GL_2(R), Z[1/2] --> H_3(GL_3(R), Z[1/2]) (resp. H_3(GL_2(R), Z) --> H_3(GL_3(R), Z)) is injective. As an application we study the third homology gr…
We prove that for certain classes of graded algebras (Koszul, local, cellular), infinite global dimension implies that Hochschild homology does not vanish in high degrees, provided the characteristic of the ground field is zero. Our proof uses Igusa's formula relating the Euler characteristic of relative cyclic homology to the graded Cartan determinant.
Let E be an elliptic curve defined via a Weierstrass equation F(x,y)=0 over an infinite field k. Denote by A the coordinate ring of E. In this note we compute the integral homology of PGL_2(A). We obtain a rigidity result as a corollary.
We consider "Hopfological" techniques as in \cite{Ko} but for infinite dimensional Hopf algebras, under the assumption of being co-Frobenius. In particular, $H=k[{\mathbb Z}]\#k[x]/x^2$ is the first example, whose corepresentations category is d.g. vector spaces. Motivated by this example we define the "Homology functor" (we prove it is homological) for any co-Frobenius algebra, with coefficients in $H$-comodules, that recover usual homology of a complex when $H=k[{\mathbb Z}]\#k[x]/x^2$. Another easy example of co-Frobenius Hopf algebra gives the category of "mixed complexes" and we see (by c…
We show that for a smooth, projective variety X defined over a number field K, cyclic homology with coefficients in the ring of infinite adeles of K, provides the right theory to obtain, using the lambda-operations, Serre's archimedean local factors of the complex L-function of X as regularized determinants.
We prove that if R is a Hensel local ring with infinite residue field k, the natural map H_i(GL(n,R),Z/p) ---> H_i(GL(n,k),Z/p) is an isomorphism for i <=3, p distinct from char(k). This implies rigidity for H_i(GL_n), i <=3, which in turn implies the Friedlander-Milnor conjecture in positive characteristic in degrees <=3.
We prove a blow-up formula for cyclic homology which we use to show that infinitesimal $K$-theory satisfies $cdh$-descent. Combining that result with some computations of the $cdh$-cohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic $K$-theory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero.
We generalize a recent result by J.F. Carlson to finite tensor categories having finitely generated cohomology. Specifically, we show that if the Krull dimension of the cohomology ring is sufficiently large, then there exist infinitely many non-isomorphic and nontrivial bounded complexes of projective objects, and with small homology. We also prove a version for finite dimensional algebras with finitely generated cohomology.
We uncover several general phenomenas governing functor homology over additive categories. In particular, we generalize the strong comparison theorem of Franjou Friedlander Scorichenko and Suslin to the setting of Fp-linear additive categories. Our results have a strong impact in terms of explicit computations of functor homology, and they open the way to new applications to stable homology of groups or to K-theory. As an illustration, we prove comparison theorems between cohomologies of classical algebraic groups over infinite perfect fields, in the spirit of a celebrated result of Cline, Par…
We prove that Thompson's group $V$ is acyclic, answering a 1992 question of Brown in the positive. More generally, we identify the homology of the Higman-Thompson groups $V_{n,r}$ with the homology of the zeroth component of the infinite loop space of the mod $n-1$ Moore spectrum. As $V = V_{2,1}$, we can deduce that this group is acyclic. Our proof involves establishing homological stability with respect to $r$, as well as a computation of the algebraic K-theory of the category of finitely generated free Cantor algebras of any type $n$.
