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A theorem of O. Forster says that if R is a noetherian ring of Krull dimension d, then every projective R-module of rank n can be generated by d+n elements. S. Chase and R. Swan subsequently showed that this bound is sharp: there exist examples that cannot be generated by fewer than d+n elements. We view rank-n projective R-modules as R-forms of the non-unital R-algebra R^n where the product of any two elements is 0. The first two authors generalized Forster's theorem to forms of other algebras (not necessarily commutative, associative or unital); A. Shukla and the third author then showed that this generalized Forster bound is optimal for finite étale algebras. In this paper, we prove new upper and lower bounds on the number of generators of an R-form of a k-algebra, where k is an infinite field and R is a finitely generated k-ring of Krull dimension d. In particular, we show that, contrary to expectations, for most types of algebras, the generalized Forster bound is far from optimal. Our results are particularly detailed in the case of Azumaya algebras. Our proofs are based on reinterpreting the problem as a question about approximating the classifying stack BG, where G is the automorphism group of the algebra in question, by algebraic spaces of a certain type. [ABSTRACT FROM AUTHOR]

Let G be a finite group. A natural place to study the Chow ring of the classifying space BG is Voevodsky's triangulated category of motives, inside which Morel-Voevodsky and Totaro have defined motives M.BG/ and Mc.BG/, respectively. We show that, for any group G of order p³ over a field of characteristic not equal to p which contains a primitive p³-th root of unity, the motive M.BG/ is a mixed Tate motive. We also show that, for a finite group G over a field of characteristic zero, M.BG/ is a mixed Tate motive if and only if Mc.BG/ is a mixed Tate motive. [ABSTRACT FROM AUTHOR]

We prove that the D'ıaz-Park sharpness conjecture holds for saturated fusion systems defined on a Sylow p-subgroup of the group G2(p), for p 5. [ABSTRACT FROM AUTHOR]

For a finite group G, we define the G-cobordism category in dimension two. We show there is a one-to-one correspondence between the connected components of its classifying space and the abelianization of G. Also, we find an isomorphism of its fundamental group onto the direct sum Z OSO 2 (BG), whereOSO2 (BG) is the free oriented G-bordism group in dimension two, and we study the classifying space of some important subcategories. We obtain the classifying space has the homotopy type of the product G/[G,G] × S1 × XG, where p1(XG) = OSO 2 (BG). Finally, we present some results about the classification of Gtopological quantum field theories in dimension two. [ABSTRACT FROM AUTHOR]

For each n ≥ 3 we give examples of infinitesimally rigid projective manifolds of general type of dimension n with non-contractible universal cover. We provide examples with projective and examples with non-projective universal cover. [ABSTRACT FROM AUTHOR]

For every one-relator monoid M = ⟨ A ∣ u = v ⟩ with u , v ∈ A ∗ we construct a contractible M-CW complex and use it to build a projective resolution of the trivial module which is finitely generated in all dimensions. This proves that all one-relator monoids are of type FP ∞ , answering positively a problem posed by Kobayashi in 2000. We also apply our results to classify the one-relator monoids of cohomological dimension at most 2, and to describe the relation module, in the sense of Ivanov, of a torsion-free one-relator monoid presentation as an explicitly given principal left ideal of the monoid ring. In addition, we prove the topological analogues of these results by showing that all one-relator monoids satisfy the topological finiteness property F ∞ , and classifying the one-relator monoids with geometric dimension at most 2. These results give a natural monoid analogue of Lyndon’s Identity Theorem for one-relator groups. [ABSTRACT FROM AUTHOR]