arXiv subjects
Publications and source records attributed to Igor Baskov.
Each commutative algebra $A$ gives rise to a representation $\mathcal{L}_A$, which we call the Loday functor of $A$, of the category $\Omega$ of finite sets and surjective maps. In this paper we present two (infinite-dimensional) non-isomorphic algebras over $\mathbb{C}$ with isomorphic Loday functors -- the algebras of continuous functions on the M\"obius strip and on the cylinder.
Let $A$ be a real soft function algebra. In arXiv:2208.11431 we have obtained a canonical splitting $\mathrm{H}^* (\Omega ^\bullet _{A|\mathrm{R}}) \cong \mathrm{H} ^* (X,\mathrm{R})\oplus \text{(something)}$ via the canonical maps $\Lambda_A:\mathrm{H} ^* (X,\mathrm{R})\to\mathrm{H} ^* (\Omega ^\bullet _{A|\mathrm{R}})$ and $\Psi_A:\mathrm{H} ^* (\Omega ^\bullet _{A|\mathrm{R}})\to\mathrm{H} ^* (X,\mathrm{R})$. In this paper we prove that these maps are multiplicative.
We consider the algebra $A^0 (X)$ of polynomial functions on a simplicial complex $X$. The algebra $A^0 (X)$ is the $0$th component of Sullivan's dg-algebra $A^\bullet (X)$ of polynomial forms on $X$. Our main interest lies in computing the de Rham cohomology of the algebra $A^0(X)$, that is, the cohomology of the universal dg-algebra $\Omega ^\bullet _{A^0(X)}$. There is a canonical morphism of dg-algebras $P:\Omega ^\bullet _{A^0(X)} \to A^\bullet (X)$. We prove that $P$ is a quasi-isomorphism. Therefore, the de Rham cohomology of the algebra $A^0 (X)$ is canonically isomorphic to the cohomology of the simplicial complex $X$ with coefficients in $k$. Moreover, for $k=\mathbb{Q}$ the dg-algebra $\Omega ^\bullet _{A^0 (X)}$ is a model of the simplicial complex $X$ in the sense of rational homotopy theory.
We study the dg-algebra $\Omega ^\bullet_{A|\mathbb{R}}$ of algebraic de Rham forms of a real soft function algebra $A$, i.e., the algebra of global sections of a soft subsheaf of $C_X$, the sheaf of continuous functions on a space $X$. We obtain a canonical splitting $\mathrm H ^n (\Omega ^\bullet_{A|\mathbb{R}}) \cong \mathrm H ^n (X,\mathbb{R})\oplus V$, where $V$ is some vector space. In particular, we consider the cases $A=C(X)$ for $X$ a compact Hausdorff space and $A = C^\infty (X)$ for $X$ a compact smooth manifold. For the algebra $\mathrm{PPol}_K (|K|)$ of piecewise polynomial functions on a polyhedron $K$ the above splitting reduces to a canonical isomorphism $\mathrm H ^* (\Omega ^\bullet_{\mathrm{PPol}_K (|K|)|\mathbb{R}}) \cong \mathrm H ^* (|K|,\mathbb{R})$. We also prove that the algebraic de Rham cohomology $\mathrm H ^n (\Omega ^\bullet_{C(X)|\mathbb{R}})$ is nontrivial for each $n\geq 1$ if $X$ is an infinite compact Hausdorff space.